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Number bases

• Cuneiform numerals
• Greek numerals
• Roman numerals
• Multiplicative grouping systems
• Ciphered numeral systems
• Positional numeral systems
• The Hindu-Arabic system
• The binary system

• How do Roman numerals work?
• Where do Roman numerals come from?
• Is it still important to learn Roman numerals?

numerals and numeral systems

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• Numbers and Number Systems - Children's Encyclopedia (Ages 8-11)
• numeration systems and numbers - Student Encyclopedia (Ages 11 and up)
• Table Of Contents

Numerals are the symbols used to represent small numbers , and numeral systems are collections of these symbols together with systems of rules for representing larger numbers.

Just as the first attempts at writing came long after the development of speech, so the first efforts at the graphical representation of numbers came long after people had learned how to count . Probably the earliest way of keeping record of a count was by some tally system involving physical objects such as pebbles or sticks. Judging by the habits of indigenous peoples today as well as by the oldest remaining traces of written or sculptured records, the earliest numerals were simple notches in a stick, scratches on a stone, marks on a piece of pottery , and the like. Having no fixed units of measure, no coins, no commerce beyond the rudest barter, no system of taxation, and no needs beyond those to sustain life, people had no necessity for written numerals until the beginning of what are called historical times. Vocal sounds were probably used to designate the number of objects in a small group long before there were separate symbols for the small numbers, and it seems likely that the sounds differed according to the kind of object being counted. The abstract notion of two, signified orally by a sound independent of any particular objects, probably appeared very late.

When it became necessary to count frequently to numbers larger than 10 or so, the numeration had to be systematized and simplified; this was commonly done through use of a group unit or base , just as might be done today counting 43 eggs as three dozen and seven. In fact, the earliest numerals of which there is a definite record were simple straight marks for the small numbers with some special form for 10. These symbols appeared in Egypt as early as 3400 bce and in Mesopotamia as early as 3000 bce , long preceding the first known inscriptions containing numerals in China ( c. 1600 bce ), Crete ( c. 1200 bce ), and India ( c. 300 bce ). Some ancient symbols for 1 and 10 are shown here .

The special position occupied by 10 stems from the number of human fingers, of course, and it is still evident in modern usage not only in the logical structure of the decimal number system but in the English names for the numbers. Thus, eleven comes from Old English endleofan , literally meaning “[ten and] one left [over],” and twelve from twelf , meaning “two left”; the endings -teen and -ty both refer to ten, and hundred comes originally from a pre-Greek term meaning “ten times [ten].”

It should not be inferred, however, that 10 is either the only possible base or the only one actually used. The pair system, in which the counting goes “one, two, two and one, two twos, two and two and one,” and so on, is found among the ethnologically oldest tribes of Australia , in many Papuan languages of the Torres Strait and the adjacent coast of New Guinea , among some African Pygmies , and in various South American tribes. The indigenous peoples of Tierra del Fuego and the South American continent use number systems with bases three and four. The quinary scale, or number system with base five, is very old, but in pure form it seems to be used at present only by speakers of Saraveca, a South American Arawakan language; elsewhere it is combined with the decimal or the vigesimal system , where the base is 20. Similarly, the pure base six scale seems to occur only sparsely in northwest Africa and is otherwise combined with the duodecimal, or base 12, system.

In the course of history, the decimal system finally overshadowed all others. Nevertheless, there are still many vestiges of other systems, chiefly in commercial and domestic units, where change always meets the resistance of tradition. Thus, 12 occurs as the number of inches in a foot, months in a year, ounces in a pound ( troy weight or apothecaries’ weight ), and twice 12 hours in a day, and both the dozen and the gross measure by twelves. In English the base 20 occurs chiefly in the score (“Four score and seven years ago…”); in French it survives in the word quatre-vingts (“four twenties”), for 80; other traces are found in ancient Celtic, Gaelic, Danish, and Welsh. The base 60 still occurs in measurement of time and angles.

Numerical Expression – Definition with Examples

What is numerical expression, examples of numerical expressions, writing numerical expression.

The term numerical expression is made up of two words, numerical meaning  numbers , and  expression  meaning phrase. Thus, it is a phrase involving numbers. 

A numerical expression in mathematics can be a combination of numbers, and integers combined using mathematical operators such as addition, subtraction, multiplication, or division.

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We can form a numerical expression by combining numbers with various mathematical operators. There is no limit to the number of operators a numerical expression may contain. Some numerical expressions use only one operator between two numbers, and some may contain more. 

Some examples of numerical expression are given below:  

250 – 75

60 × 5 + 10 

72 ÷ 8 × 5 – 4 + 1 

82 + 4 – 10 

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Any mathematical word problem is solved by first converting into a numerical expression. Below are some examples. 

Candice has 10 chocolate bars. She gives 3 to her sister, 1 to her friend and eats 2 of them. Later she visits her grandmother, and she (grandmother) offers Candice 12 more chocolate bars. How many chocolate bars does Candice have now? 

Let’s look at the numbers involved in the above problem. Candice starts with 10 bars, gives away 4 (3+1), eats 2 and then again gets 12 more from her grandmother. So, the numerical expression is

    10 – 3 – 1 – 2 + 12     

    = 7 – 1 – 2 + 12     

    = 6 – 2 + 12     

    = 4 + 12     

    = 16

This gives us 16. 

For example:  

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Number Systems

Number systems are systems in mathematics that are used to express numbers in various forms and are understood by computers. A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. Numbers have various categories like natural numbers, whole numbers, rational and irrational numbers, and so on. Similarly, there are various types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

In this article, we will explore different types of number systems that we use such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. We will learn the conversions between these number systems and solve examples for a better understanding of the concept.

What are Number Systems?

A number system is a system representing numbers. It is also called the system of numeration and it defines a set of values to represent a quantity. These numbers are used as digits and the most common ones are 0 and 1, that are used to represent binary numbers. Digits from 0 to 9 are used to represent other types of number systems.

Number Systems Definition

A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The numbers are represented in a unique manner and allow us to operate arithmetic operations like addition, subtraction, and division.

Types of Number Systems

There are different types of number systems in which the four main types are as follows.

• Binary number system (Base - 2)
• Octal number system (Base - 8)
• Decimal number system (Base - 10)
• Hexadecimal number system (Base - 16)

We will study each of these systems one by one in detail after going through the following number system chart.

Number System Chart

Given below is a chart of the main four types of number system that we use to represent numbers.

Binary Number System

The binary number system uses only two digits: 0 and 1. The numbers in this system have a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. The binary number system does not deal with other numbers such as 2,3,4,5 and so on. For example: 10001 2 , 111101 2 , 1010101 2 are some examples of numbers in the binary number system.

Octal Number System

The octal number system uses eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system. Just like the binary, the octal number system is used in minicomputers but with digits from 0 to 7. For example, 35 8 , 23 8 , and 141 8 are some examples of numbers in the octal number system.

Decimal Number System

The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example, 723 10 , 32 10 , and 4257 10 are some examples of numbers in the decimal number system.

Hexadecimal Number System

The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example, 7B3 16 , 6F 16 , and 4B2A 16 are some examples of numbers in the hexadecimal number system.

Conversion of Number Systems

A number can be converted from one number system to another number system using number system formulas. Like binary numbers can be converted to octal numbers and vice versa, octal numbers can be converted to decimal numbers and vice versa, and so on. Let us see the steps required in converting number systems.

Steps for Conversion of Binary to Decimal Number System

To convert a number from the binary to the decimal system, we use the following steps.

• Step 1: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• Step 2: The exponents should start with 0 and increase by 1 every time we move from right to left.
• Step 3: Simplify each of the above products and add them.

Let us understand the steps with the help of the following example in which we need to convert a number from binary to decimal number system.

Example: Convert 100111 2 into the decimal system.

Step 1: Identify the base of the given number. Here, the base of 100111 2 is 2.

Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left. Since the base is 2 here, we multiply the digits of the given number by 2 0 , 2 1 , 2 2 , and so on from right to left.

Step 3: We just simplify each of the above products and add them.

Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified.

100111 = (1 × 2 5 ) + (0 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 )

= (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)

= 32 + 0 + 0 + 4 + 2 + 1

Thus, 100111 2 = 39 10 .

Conversion of Decimal Number System to Binary / Octal / Hexadecimal Number System

To convert a number from the decimal number system to a binary/octal/hexadecimal number system, we use the following steps. The steps are shown on how to convert a number from the decimal system to the octal system.

Example: Convert 4320 10 into the octal system.

Step 1: Identify the base of the required number. Since we have to convert the given number into the octal system, the base of the required number is 8.

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.

Step 3: The given number in the octal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Therefore, 4320 10 = 10340 8

Conversion from One Number System to Another Number System

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

Example: Convert 1010111100 2 to the hexadecimal system.

Step 1: Convert this number to the decimal number system as explained in the above process.

Thus, 1010111100 2 = 700 10 → (1)

Step 2: Convert the above number (which is in the decimal system), into the required number system (hexadecimal).

Here, we have to convert 700 10 into the hexadecimal system using the above-mentioned process. It should be noted that in the hexadecimal system, the numbers 11 and 12 are written as B and C respectively.

Thus, 700 10 = 2BC 16 → (2)

From the equations (1) and (2), 1010111100 2 = 2BC 16

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Number Systems Examples

Example 1: Convert 300 10 into the binary number system with base 2.

Solution: 300 10 is in the decimal system. We divide 300 by 2 and note down the quotient and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2.

The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above.

Thus, 300 10 = 100101100 2

Example 2: Convert 5BC 16 into the decimal system.

Solution: 5BC 16 is in the hexadecimal system. We know that B = 11 and C = 12 in the hexadecimal system. So we get the equivalent number in the decimal system using the following process:

Thus, 5BC 16 = 1468 10

Example 3: Convert 144 8 into the hexadecimal system.

Solution: The base of 144 8 is 8. First, we will convert this number into the decimal system as follows:

Thus, 144 8 = 100 10 → (1). Now we will convert this into the hexadecimal system as follows:

Thus, 100 10 = 64 16 → (2)

From the equations (1) and (2), we can conclude that: 144 8 = 64 16

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Practice Questions on Number Systems

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FAQs on Number Systems

What are number systems with examples.

A number system is a system of writing or expressing numbers. In mathematics, numbers are represented in a given set by using digits or symbols in a certain manner. Every number has a unique representation of its own and numbers can be represented in the arithmetic and algebraic structure as well. There are different types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. Some examples of numbers in different number systems are 10010 2 , 234 8 , 428 10 , and 4BA 16 .

What are the Different Types of Number Systems?

There are four main types of number systems:

What are the Conversion Rules of Number Systems?

To convert a number from binary/octal/hexadecimal system to a decimal number system, we use the following steps:

• Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• The exponents should start with 0 and increase by 1 every time we move from right to left.
• Simplify each of the above products and add them.

To convert a number from decimal system to binary/octal/hexadecimal system, we use the following steps:

• Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.
• Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.
• The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems:

• We first convert it into the decimal system.
• Then we convert it to the required system.

What are the Uses of Each Number System?

There are different purposes of each number system, such as:

• The binary number system is used to store the data in computers.
• The advantage of the octal number system is that it has fewer digits when compared to several other systems, hence, there would be fewer computational errors.
• The decimal number system is the system that we use in daily life.
• The hexadecimal number system is used in computers to reduce the large-sized strings of the binary system.

What is the Importance of Number Systems?

Number systems help in representing the numbers in a small symbol set. Binary numbers are mostly used in computers that use digits like 0 and 1 for calculating simple problems. The number systems also help in converting one number system to another.

How are Number Systems Classified?

The number systems can be classified mainly into two categories: Positional and Non-positional number systems. For positional number systems, each digit is associated with a weight and its examples are binary, octal, decimal, etc. In non-positional number systems, the digit values are independent of their positions and its examples are gray code, cyclic code, aroma code, etc.

Why are Different Number Systems Used in Computers?

Computers cannot understand human languages, so to understand the commands and instructions given to the computers by programmers, different number systems are used such as the binary system, the octal system, the decimal system, and so on.

Numerical representations and error

Introduction.

There are many ways to represent a number within a digital computer. In this section we will examine some of the more useful and common types of these representations, as well as their strengths and weaknesses in terms of error analysis and efficiency.

and the relative error is

Floating-point representations

The most common way of representing numbers in computations is to use a floating-point representation. We will focus on the IEEE (Institute of Electrical and Electronics Engineers) double precision standard, which is currently the most often used (but some other types are important in specialized areas such as audio processing and machine learning).

A floating-point number is represented with a sign (either positive or negative), a mantissa of some number of digits in a particular base (almost always base-2 on a computer), and an exponent. In base-2 most numbers will be represented in the form

So for example the decimal number 1,073,741,824.5, which in exponential binary form is

and as it is positive its sign bit would be 0, so the 64-bit word would be

For more information on floating point numbers, the Wikipedia entry is quite good. There is also a page on "minifloats" , which use a small number of bits and are mainly used as examples, since it is awkward to do examples with 64 or 32 bits.

Special and subnormal numbers

Most numbers cannot be exactly represented in floating-point, and need to be rounded. The IEEE standard rounds them up or down to the nearest representable number, unless it is exactly halfway between the nearest two representable numbers. In that special case, the rounding is chosen so that the last digit of the mantissa will be zero. This helps avoid consistently rounding up or down for the special halfway case.

Machine epsilon

Lets compute

To make the bit representations more clear we will put in some commas between the different segments for sign, exponent, and the significand.

To see what how the number 6/5 is represented in this system, we first compute the binary representation:

This needs to be rounded to have 10 bits after the point; since the remainder is over halfway to the next largest number, we round up to

which rounds down to 1.

and we get zero.

Mathematical Representations

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• First Online: 01 January 2014
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• Gerald A. Goldin 2  

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Definitions

As most commonly interpreted in education, mathematical representations are visible or tangible productions – such as diagrams, number lines, graphs, arrangements of concrete objects or manipulatives, physical models, mathematical expressions, formulas and equations, or depictions on the screen of a computer or calculator – that encode, stand for, or embody mathematical ideas or relationships. Such a production is sometimes called an inscription when the intent is to focus on a particular instance without referring, even tacitly, to any interpretation. To call something a representation thus includes reference to some meaning or signification it is taken to have. Such representations are called external – i.e., they are external to the individual who produced them, and accessible to others for observation, discussion, interpretation, and/or manipulation.

The term representation is also used very importantly to refer to a person’s mental or cognitive constructs, concepts,...

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Home / United States / Math Classes / 4th Grade Math / The Different Forms of Representing Numbers

The Different Forms of Representing Numbers

The value of each digit in a multi-digit number is known as place value. The place value is determined by the position o f the digits in the number. Learn how to use the place value system to express numbers in the standard form, word form, and the expanded form with the help of some solved examples. ...Read More Read Less

Table of Contents

The Standard Form

The word form, the expanded form.

• Solved Examples
• Frequently Asked Questions

There are different ways of writing a number. We can represent a number in three different forms: 

Representation deals with the method that allows us to express a number to someone else, and the number should be exactly understandable to others as stated.

In our daily life, if someone says the price of a toy is twenty-five dollars, it means he said the number in word form, and this way of expressing a number could be understood. Again, if a price tag on a shirt says ‘$15’, it is said to be in the standard form which is easily understandable.

If we write a number in digits , separating the groups by commas, the number is said to be in the standard form. We often use the standard form to write the numbers in mathematics. It is also the easiest way of expressing the number. So, it is the most common or obvious way of expressing numbers.

For example, 

789,450, and so on.

If one writes or represents a standard number in words, the form is called the word form . This is the same as when we tell a number to others.

For example, 399 is stated as three hundred ninety-nine.

586,256 in the word form is five hundred eighty-six thousand, two hundred fifty-six.

If the number is written as the summation of the place value of each digit , the number is said to be  in the expanded form.

356 in the expanded form is 300 + 50 + 6.

Here, each digit is separated according to its place value. Then the ‘addition’ symbol is mentioned between them. This representation helps us analyze a number easily.

Example: If we take the number 842,062 and make a place value chart as,

This number can be written in three forms as,

The standard form: 842,062.

The word form: Eight hundred forty-two thousand, sixty-two.

The expanded form: 800000 + 40000 + 2000 + 60 + 2.

Solved Examples on Number Representation

Example 1: Write the number in three different forms.

The standard form: 8,563.

The word form: Eight thousand, five hundred sixty-three.

The expanded form: 8,000 + 500 + 60 + 3 . 

Example 2: Write the number in the standard and expanded forms:

Eighty-six thousand, seven hundred and three.

The word form is eighty-six thousand, seven hundred three.

The standard form:

The standard form: 86,703.

The expanded form: 80,000 + 6,000 + 700 + 3 .

Example 3: Write in the standard and word forms.

100,000 + 30,000 + 6,000 + 500 + 50  

The expanded form is,

100,000 + 30,000 + 6,000 + 500 + 50

The standard form: 136,550

The word form: One hundred thirty-six thousand, five hundred fifty.

Example 4: Use the number 700,000 + 5,000 + 20 + 1 to complete the check.

700,000 + 5,000 + 20 + 1

The standard form: 705,021

The word form: Seven hundred five thousand, twenty one.

Example 5: Ava asks Liam and Mia to write “ninety-nine thousand, three hundred thirty-four” in the standard form. Who wrote the correct number? What mistake did the other make?

The word form is ninety-nine thousand, three hundred thirty- four.

The standard form : 99,334

Therefore, Liam wrote the correct answer.

Mia has made a mistake in the tens place where instead of writing 3 she has written as 0.

Write the three forms for representing a number.

The three forms to represent a number are:

The standard form: For example, 78,562.

The word form: For example, Six hundred fifty six.

The expanded form: For example, 500 + 20 + 3.

If a number has a 0 in the middle, what is done with that place value?

If there is a zero in between a number while writing the number in the expanded form or in the word form, the place is skipped.

What is the use of the expanded form of a number?

The expanded form of a number helps us understand the place value of digits clearly. It also helps while comparing two or more numbers.

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A to Z Mathematics Glossary

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Understanding math terms is important because mathematics is often referred to as the language of science and the universe, and it's not just about numbers. It encapsulates a vast array of concepts, principles, and terminology—from the foundational basics of counting to the complexities of calculus and beyond.

In this A to Z glossary, you'll find fundamental math concepts ranging from absolute value to zero slope. There's also a bit of history, with terms named after famous mathematicians.

A to Z Glossary of Math Terms

Abacus : An early counting tool used for basic arithmetic.

Absolute Value : Always a positive number, absolute value refers to the distance of a number from 0.

Acute Angle : An angle whose measure is between zero degrees and 90 degrees, or with less than 90-degree radians.

Addend : A number involved in an addition problem; numbers being added are called addends.

Algebra : The branch of mathematics that substitutes letters for numbers to solve for unknown values.

Algorithm : A procedure or set of steps used to solve a mathematical computation.

Angle : Two rays sharing the same endpoint (called the angle vertex).

Angle Bisector : The line dividing an angle into two equal angles.

Area : The two-dimensional space taken up by an object or shape, given in square units.

Array : A set of numbers or objects that follow a specific pattern.

Attribute : A characteristic or feature of an object—such as size, shape, color, etc.—that allows it to be grouped.

Average : The average is the same as the mean. Add up a series of numbers and divide the sum by the total number of values to find the average.

Base : The bottom of a shape or three-dimensional object, what an object rests on.

Base 10 : Number system that assigns place value to numbers.

Bar Graph : A graph that represents data visually using bars of different heights or lengths.

BEDMAS or PEMDAS Definition : An acronym used to help people remember the correct order of operations for solving algebraic equations. BEDMAS stands for "Brackets, Exponents, Division, Multiplication, Addition, and Subtraction" and PEMDAS stands for "Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction".

Bell Curve : The bell shape created when a line is plotted using data points for an item that meets the criteria of normal distribution. The center of a bell curve contains the highest value points.

Binomial : A polynomial equation with two terms usually joined by a plus or minus sign.

Box and Whisker Plot/Chart : A graphical representation of data that shows differences in distributions and plots data set ranges.

Calculus : The branch of mathematics involving derivatives and integrals, Calculus is the study of motion in which changing values are studied.

Capacity : The volume of substance that a container will hold.

Centimeter : A metric unit of measurement for length, abbreviated as cm. 2.5 cm is approximately equal to an inch.

Circumference : The complete distance around a circle or a square.

Chord : A segment joining two points on a circle.

Coefficient : A letter or number representing a numerical quantity attached to a term (usually at the beginning). For example, x is the coefficient in the expression x (a + b) and 3 is the coefficient in the term 3 y.

Common Factors : A factor shared by two or more numbers, common factors are numbers that divide exactly into two different numbers.

Complementary Angles: Two angles that together equal 90 degrees.

Composite Number : A positive integer with at least one factor aside from its own. Composite numbers cannot be prime because they can be divided exactly.

Cone : A three-dimensional shape with only one vertex and a circular base.

Conic Section : The section formed by the intersection of a plane and cone.

Constant : A value that does not change.

Coordinate : The ordered pair that gives a precise location or position on a coordinate plane.

Congruent : Objects and figures that have the same size and shape. Congruent shapes can be turned into one another with a flip, rotation, or turn.

Cosine : In a right triangle, cosine is a ratio that represents the length of a side adjacent to an acute angle to the length of the hypotenuse.

Cylinder : A three-dimensional shape featuring two circle bases connected by a curved tube.

Decagon : A polygon or shape with ten angles and ten straight lines.

Decimal : A real number on the base ten standard numbering system.

Denominator : The bottom number of a fraction. The denominator is the total number of equal parts into which the numerator is being divided.

Degree : The unit of an angle's measure represented with the symbol °.

Diagonal : A line segment that connects two vertices in a polygon.

Diameter : A line that passes through the center of a circle and divides it in half.

Difference : The difference is the answer to a subtraction problem, in which one number is taken away from another.

Digit : Digits are the numerals 0-9 found in all numbers. 176 is a 3-digit number featuring the digits 1, 7, and 6.

Dividend : A number divided into equal parts (inside the bracket in long division).

Divisor : A number that divides another number into equal parts (outside of the bracket in long division).

Edge : A line is where two faces meet in a three-dimensional structure.

Ellipse : An ellipse looks like a slightly flattened circle and is also known as a plane curve. Planetary orbits take the form of ellipses.

End Point : The "point" at which a line or curve ends.

Equilateral : A term used to describe a shape whose sides are all of equal length.

Equation : A statement that shows the equality of two expressions by joining them with an equals sign.

Even Number : A number that can be divided or is divisible by 2.

Event : This term often refers to an outcome of probability; it may answer questions about the probability of one scenario happening over another.

Evaluate : This word means "to calculate the numerical value".

Exponent : The number that denotes repeated multiplication of a term, shown as a superscript above that term. The exponent of 3 4 is 4.

Expressions : Symbols that represent numbers or operations between numbers.

Face : The flat surfaces on a three-dimensional object.

Factor : A number that divides into another number exactly. The factors of 10 are 1, 2, 5, and 10 (1 x 10, 2 x 5, 5 x 2, 10 x 1).

Factoring : The process of breaking numbers down into all of their factors.

Factorial Notation : Often used in combinatorics, factorial notations require that you multiply a number by every number smaller than it. The symbol used in factorial notation is ! When you see x !, the factorial of x is needed.

Factor Tree : A graphical representation showing the factors of a specific number.

Fibonacci Sequence : Named after Italian number theorist Leonardo Pisano Fibonacci, it's a sequence beginning with a 0 and 1 whereby each number is the sum of the two numbers preceding it. For example, "0, 1, 1, 2, 3, 5, 8, 13, 21, 34..." is a Fibonacci sequence.

Figure : Two-dimensional shapes.

Finite : Not infinite; has an end.

Flip : A reflection or mirror image of a two-dimensional shape.

Formula : A rule that numerically describes the relationship between two or more variables.

Fraction : A quantity that is not whole that contains a numerator and denominator. The fraction representing half of 1 is written as 1/2.

Frequency : The number of times an event can happen in a given period of time; often used in probability calculations.

Furlong : A unit of measurement representing the side length of one square acre. One furlong is approximately 1/8 of a mile, 201.17 meters, or 220 yards.

Geometry : The study of lines, angles, shapes, and their properties. Geometry studies physical shapes and object dimensions.

Graphing Calculator : A calculator with an advanced screen capable of showing and drawing graphs and other functions.

Graph Theory : A branch of mathematics focused on the properties of graphs.

Greatest Common Factor : The largest number common to each set of factors that divides both numbers exactly. The greatest common factor of 10 and 20 is 10.

Hexagon : A six-sided and six-angled polygon.

Histogram : A graph that uses bars that equal ranges of values.

Hyperbola : A type of conic section or symmetrical open curve. The hyperbola is the set of all points in a plane, the difference of whose distance from two fixed points in the plane is a positive constant.

Hypotenuse : The longest side of a right-angled triangle, always opposite to the right angle itself.

Identity : An equation that is true for variables of any value.

Improper Fraction : A fraction whose numerator is equal to or greater than the denominator, such as 6/4.

Inequality : A mathematical equation expressing inequality and containing a greater than (>), less than (<), or not equal to (≠) symbol.

Integers : All whole numbers, positive or negative, including zero.

Irrational : A number that cannot be represented as a decimal or fraction. A number like pi is irrational because it contains an infinite number of digits that keep repeating. Many square roots are also irrational numbers.

Isosceles : A polygon with two sides of equal length.

Kilometer : A unit of measure equal to 1000 meters.

Knot : A closed three-dimensional circle that is embedded and cannot be untangled.

Like Terms : Terms with the same variable and same exponents/powers.

Like Fractions : Fractions with the same denominator.

Line : A straight infinite path joining an infinite number of points in both directions.

Line Segment : A straight path that has two endpoints, a beginning, and an end.

Linear Equation : An equation that contains two variables and can be plotted on a graph as a straight line.

Line of Symmetry : A line that divides a figure into two equal shapes.

Logic : Sound reasoning and the formal laws of reasoning.

Logarithm : The power to which a base must be raised to produce a given number. If nx = a , the logarithm of a , with n as the base, is x . Logarithm is the opposite of exponentiation.

Mean : The mean is the same as the average. Add up a series of numbers and divide the sum by the total number of values to find the mean.

Median : The median is the middle value in a series of numbers ordered from least to greatest. When the total number of values in a list is odd, the median is the middle entry. When the total number of values in a list is even, the median is equal to the sum of the two middle numbers divided by two.

Midpoint : A point that is exactly halfway between two locations.

Mixed Numbers : Mixed numbers refer to whole numbers combined with fractions or decimals. Example 3 1 / 2 or 3.5.

Mode : The mode in a list of numbers are the values that occur most frequently.

Modular Arithmetic : A system of arithmetic for integers where numbers "wrap around" upon reaching a certain value of the modulus.

Monomial : An algebraic expression made up of one term.

Multiple : The multiple of a number is the product of that number and any other whole number. 2, 4, 6, and 8 are multiples of 2.

Multiplication : Multiplication is the repeated addition of the same number denoted with the symbol x. 4 x 3 is equal to 3 + 3 + 3 + 3.

Multiplicand : A quantity multiplied by another. A product is obtained by multiplying two or more multiplicands.

Natural Numbers : Regular counting numbers.

Negative Number : A number less than zero denoted with the symbol -. Negative 3 = -3.

Net : A two-dimensional shape that can be turned into a two-dimensional object by gluing/taping and folding.

Nth Root : The n th root of a number is how many times a number needs to be multiplied by itself to achieve the value specified. Example: the 4th root of 3 is 81 because 3 x 3 x 3 x 3 = 81.

Norm : The mean or average; an established pattern or form.

Normal Distribution : Also known as Gaussian distribution, normal distribution refers to a probability distribution that is reflected across the mean or center of a bell curve.

Numerator : The top number in a fraction. The numerator is divided into equal parts by the denominator.

Number Line : A line whose points correspond to numbers.

Numeral : A written symbol denoting a number value.

Obtuse Angle : An angle measuring between 90° and 180°.

Obtuse Triangle : A triangle with at least one obtuse angle.

Octagon : A polygon with eight sides.

Odds : The ratio or likelihood of a probability event happening. The odds of flipping a coin and having it land on heads are one in two.

Odd Number : A whole number that is not divisible by 2.

Operation : Refers to addition, subtraction, multiplication, or division.

Ordinal : Ordinal numbers give relative positions in a set: first, second, third, etc.

Order of Operations : A set of rules used to solve mathematical problems in the correct order. This is often remembered with acronyms BEDMAS and PEMDAS.

Outcome : Used in probability to refer to the result of an event.

Parallelogram : A quadrilateral with two sets of opposite sides that are parallel.

Parabola : An open curve whose points are equidistant from a fixed point called the focus and a fixed straight line called the directrix.

Pentagon : A five-sided polygon. Regular pentagons have five equal sides and five equal angles.

Percent : A ratio or fraction with the denominator 100.

Perimeter : The total distance around the outside of a polygon. This distance is obtained by adding together the units of measure from each side.

Perpendicular : Two lines or line segments intersecting to form a right angle.

Pi : Pi is used to represent the ratio of the circumference of a circle to its diameter, denoted with the Greek symbol π.

Plane : When a set of points join together to form a flat surface that extends in all directions, this is called a plane.

Polynomial : The sum of two or more monomials.

Polygon : Line segments joined together to form a closed figure. Rectangles, squares, and pentagons are just a few examples of polygons.

Prime Numbers : Prime numbers are integers greater than one that are only divisible by themselves and 1.

Probability : The likelihood of an event happening.

Product : The sum obtained through the multiplication of two or more numbers.

Proper Fraction : A fraction whose denominator is greater than its numerator.

Protractor : A semi-circle device used for measuring angles. The edge of a protractor is subdivided into degrees.

Quadrant : One quarter ( qua) of the plane on the Cartesian coordinate system. The plane is divided into 4 sections, each called a quadrant.

Quadratic Equation : An equation that can be written with one side equal to 0. Quadratic equations ask you to find the quadratic polynomial that is equal to zero.

Quadrilateral : A four-sided polygon.

Quadruple : To multiply or to be multiplied by 4.

Qualitative : Properties that must be described using qualities rather than numbers.

Quartic : A polynomial having a degree of 4.

Quintic : A polynomial having a degree of 5.

Quotient : The solution to a division problem.

Radius : A distance found by measuring a line segment extending from the center of a circle to any point on the circle; the line extending from the center of a sphere to any point on the outside edge of the sphere.

Ratio : The relationship between two quantities. Ratios can be expressed in words, fractions, decimals, or percentages. Example: the ratio given when a team wins 4 out of 6 games is 4/6, 4:6, four out of six, or ~67%.

Ray : A straight line with only one endpoint that extends infinitely.

Range : The difference between the maximum and minimum in a set of data.

Rectangle : A parallelogram with four right angles.

Repeating Decimal : A decimal with endlessly repeating digits. Example: 88 divided by 33 equals 2.6666666666666... ("2.6 repeating").

Reflection : The mirror image of a shape or object, obtained from flipping the shape on an axis.

Remainder : The number left over when a quantity cannot be divided evenly. A remainder can be expressed as an integer, fraction, or decimal.

Right Angle : An angle equal to 90 degrees.

Right Triangle : A triangle with one right angle.

Rhombus : A parallelogram with four sides of equal length and no right angles.

Scalene Triangle : A triangle with three unequal sides.

Sector : The area between an arc and two radii of a circle, sometimes referred to as a wedge.

Slope : Slope shows the steepness or incline of a line and is determined by comparing the positions of two points on the line (usually on a graph).

Square Root : A number squared is multiplied by itself; the square root of a number is whatever integer gives the original number when multiplied by itself. For instance, 12 x 12 or 12 squared is 144, so the square root of 144 is 12.

Stem and Leaf : A graphic organizer used to organize and compare data. Similar to a histogram, stem and leaf graphs organize intervals or groups of data.

Subtraction : The operation of finding the difference between two numbers or quantities by "taking away" one from the other.

Supplementary Angles : Two angles are supplementary if their sum is equal to 180°.

Symmetry : Two halves that match perfectly and are identical across an axis.

Tangent : A straight line touching a curve from only one point.

Term : Piece of an algebraic equation; a number in a sequence or series; a product of real numbers and/or variables.

Tessellation : Congruent plane figures/shapes that cover a plane completely without overlapping.

Translation : A translation, also called a slide, is a geometrical movement in which a figure or shape is moved from each of its points the same distance and in the same direction.

Transversal : A line that crosses/intersects two or more lines.

Trapezoid : A quadrilateral with exactly two parallel sides.

Tree Diagram : Used in probability to show all possible outcomes or combinations of an event.

Triangle : A three-sided polygon.

Trinomial : A polynomial with three terms.

Unit : A standard quantity used in measurement. Inches and centimeters are units of length, pounds, and kilograms are units of weight, and square meters and acres are units of area.

Uniform : Term meaning "all the same". It can be used to describe size, texture, color, design, and more.

Variable : A letter used to represent a numerical value in equations and expressions. Example: in the expression 3 x + y , both y and x are the variables.

Venn Diagram : A Venn diagram is usually shown as two overlapping circles and is used to compare two sets. The overlapping section contains information that is true of both sides or sets and the non-overlapping portions each represent a set and contain information that is only true of their set.

Volume : A unit of measure describing how much space a substance occupies or the capacity of a container, provided in cubic units.

Vertex : The point of intersection between two or more rays, often called a corner. A vertex is where two-dimensional sides or three-dimensional edges meet.

Weight : The measure of how heavy something is.

Whole Number : A whole number is a positive integer.

X-Axis : The horizontal axis in a coordinate plane.

X-Intercept : The value of x where a line or curve intersects the x-axis.

X : The Roman numeral for 10.

x : A symbol used to represent an unknown quantity in an equation or expression.

Y-Axis : The vertical axis in a coordinate plane.

Y-Intercept : The value of y where a line or curve intersects the y-axis.

Yard : A unit of measure that is equal to approximately 91.5 centimeters or 3 feet.

Zero Slope: The slope of a horizontal line. Its slope is zero because a horizontal line has no incline.

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• Ninth Grade Math: Core Curriculum
• What Is the Gamma Function?
• Math Formulas for Geometric Shapes
• 11th Grade Math: Core Curriculum and Courses
• The Typical 10th Grade Math Curriculum
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Number representations: An evidence-based math strategy

By Brendan Hodnett, MA

Expert reviewed by Sarah R. Powell, PhD

What you’ll learn

Download: printable number puzzles.

• Read: How to use this number representation strategy

Understand: Why this strategy works

Connect: link school to home, research behind this strategy.

Many students start kindergarten already knowing how to count. They can say the numbers “one, two, three…” and so on up to 10. But they don’t always understand that each number they’re saying is a quantity .

On top of that, they may be confused by the fact that the same number can be represented in different ways — by its number name (“one”) and by its numeral (“1”). Confusion about different number representations can make it hard for students to learn addition and subtraction. 

Practicing the different ways to show a number can help students understand each number as a quantity. With this strategy, you’ll use explicit instruction to teach three different representations for the numbers 1–10: the number name, the numeral, and a picture or set of objects showing the quantity.

Students will move through a series of collaborative practice activities with these representations. The strategy ends with an assessment of learning as well as options for continued practice.

Number puzzles PDF - 56.9 KB

Read: How to use this number representation strategy 

Objective: Students will identify a number by its written numeral, its number name, and a picture of repeated objects showing the quantity.

Grade levels (with standards): 

K (Common Core K.CC.A.3: Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20, with 0 representing a count of no objects.)

K (Common Core K.CC.B.4.A: When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.)

K (Common Core K.CC.B.4.B: Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.)

K (Common Core Math Practice MP2: Reason abstractly and quantitatively.) 

Best used for instruction with: 

Whole class

Small groups

How to prepare:

Gather materials. Print a number puzzle for each group of three to five students. Consider cutting the puzzle pieces out ahead of time. You can keep them clipped together or bundled in small bags. Printing on card stock or laminating the puzzle pieces can make it easier for students to work with them.

Prepare a set of index cards for the numerals 1–10. Write one numeral on each card. Prepare a matching set for the number names. 

Organize small objects like straws, blocks, erasers, or counters into groups of one to 10. You can group the objects in small plastic bags or cups, like five straws in one bag or six erasers in a cup.

Teaching tip: Make a word wall to support English language learners (ELLs) and other students who benefit from visual supports. Having the number names alongside the written numerals will help students learn them as sight words. If a word wall is not an option, make an anchor chart large enough for all students to see.

How to teach:

1. Warm up by reviewing number identification. Write a number between 1 and 10 on the board, or point to a number on a word wall or anchor chart. Say the number aloud. Ask students to repeat the number. Then ask students to show and count the number of fingers to match the number. Model the correct answer. For example, for 3, you would model the answer by holding up three fingers and saying, “That’s right. This is 3.” Then count each finger aloud. ”One, two, three.” 

After several rounds, explain to the students that there are many ways to represent numbers, and using fingers is just one of them. Ask, “What is another way we can show the number 3?” Take different responses from students. Model the responses by writing, drawing, and using objects depending on what students say.

2. Model representing numbers in different ways. Tell students the goal of today’s lesson. “Today we’re going to learn three different ways you can show a number. You can use a numeral. You can use a number name, which is the word for the number. Or you can use a picture or a set of objects.” Have students repeat the concept. Say, “So, you can use a numeral. Say numeral with me. You can use a number name. Say number name with me. Or you can use a picture or set of objects. Say picture or set of objects with me.”

Show them what these representations look like. For example, hold up a card with the numeral 5 on it and another with the number name “five.” Then say, “The number 5 can be written as a numeral like this. But it can also be written as the word five . F-i-v-e. Five. Five is the number name. And it can be shown by five of the same object.” Draw five squares on the board counting each one until you get to five. “One, two, three, four, five. Five squares. All of these mean the same thing: 5 (point to the written numeral), 5 (point to the number name), and 5 (point to the squares).”

“Let’s try it again.” Repeat the steps with another number from the set of 1–10.

Use objects around the room to represent the quantity. You can use blocks, counters, bean bags, or anything large and bright enough for students to see from their seats. 

3. Practice matching numerals and number names to counted objects. Before moving on to working with all the numbers 1–10 and their representations, have students practice with just a few numbers. Place a set of objects in front of each group of students, varying the number of objects from group to group. These objects can be counters, blocks, straws, etc. One student should be in charge of displaying the objects for the other members to see.

Next, give students four index cards: two with numerals and two with matching number names. For example, if a group has six objects in front of them, they might get the following four cards: 1, 6, one, six. As a group, have the students decide which index cards (one numeral and one number name) go with the set of objects. Let the students discuss it and come to an agreement. Check in with each group and offer support as needed. Once all the groups have made their decisions, have them share out. As they hold up the cards, prompt them to say the number and to spell the number name: “s-i-x...six.”

Rotate the set of objects from one group to the next and pass out a new set of cards to each group. Repeat the same steps. After two rounds, if each group has had a successful match, move onto the number puzzles activity. 

4. Practice with number puzzles. Pass out a set of number puzzles to each group (pre-cut or with scissors if students are making the sets). Give the directions: “Now we’ll use number puzzles to practice with the different ways we can show numbers. Sort all the puzzle pieces into three piles: numerals, number names, and pictures.” 

Once students have made three piles, have them shuffle each pile so its pieces are in no particular order. Say, “Now each student will take a turn selecting one piece from the numeral pile. Tell your group members the number you have selected. Together the group will search for a matching number name and a matching picture card from the other two piles to go with the number. Continue taking turns and matching your pieces together. When all of the pieces have been matched into groups of three (numeral, number name, and picture), raise your hands to have your work checked.” Model how to make a matching set before students begin. 

5. Assessment of learning. Have students meet with you one-on-one at a table with objects on it. Give each student a numeral (an index card or from the puzzle piece set). Ask students to show you the matching quantity using objects on the table. Then, ask them to pick the matching number name from the set of index cards or number puzzle pieces. Have students count out the objects and say the number name to show their understanding that all three represent the same quantity. Students who have difficulty with this task should be given additional small group instruction focused on a limited set of numbers like 1–5 and then 6–10. 

6. Continued practice. Give students opportunities to keep practicing representing numbers. Try these as follow-up activities to reinforce learning: 

Make a set of puzzles into necklaces by punching a hole on either side of each piece and adding string. Give each student a necklace. Ask them to get up and find the other two classmates who are wearing the same “number” but in a different form.

Use a word wall as a way to check in with small groups of students or individuals. Have them identify a number, spell out the number name, and count the objects.

Show a number in any form that you choose. Give students a moment to think about what number it is. Ask them to find examples of that number around the room. Then, ask students where else they might see forms of this number in their lives. 

Students who struggle with math in the early elementary grades, especially with number sense , often have a hard time switching between different number representations. They have trouble understanding that the quantity is the same. By directly teaching the three representations of the same quantity, you can help students avoid this confusion. 

Using physical objects and pictures to represent quantities helps students practice and apply the concepts of stable order and one-to-one correspondence . These concepts are essential to understanding addition, subtraction, and comparison of quantities.

The cooperative learning activities in this number puzzle strategy give students practice with showing and communicating their understanding. Cooperative learning is especially beneficial for ELLs because they may find peers more accessible and easier to understand than teachers. Students who may not be ready to complete the task independently also benefit from working with peers. At times peers may offer corrective feedback that other students can learn from. 

Give suggestions for ways that families can practice number representation at home. For example, families can give students a number name written on a small piece of paper and ask them to find that quantity of objects. Or families can ask students to label a set of items in the house with a numeral or number name on a piece of paper to show the quantity. 

Playing games is also a great way to practice numbers at home. Board games , like Five Little Monkeys and Hi Ho Cherry-O, represent quantities in more than one way. 

“Better Learning Through Structured Teaching: A Framework for the Gradual Release of Responsibility (2nd ed.),” by Douglas Fisher and Nancy Frey

“Experimental evaluation of the effects of cooperative learning on kindergarten children's mathematics ability,” from International Journal of Educational Research

“The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge,” from Journal of Educational Psychology

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Decimal is a numerical representation that uses a dot, which we call a decimal point, to separate the whole number part from its fractional part. The decimal numeral system is used as the standard system that is used to distinguish integer and non-integer numbers.

In this article, we will understand what decimals are, the place value of decimals, and how to round decimals along with some solved examples based on it.

Table of Content

What are Decimals?

Decimals place value chart, decimals properties, types of decimals, arithmetic operations on decimals, rounding decimals, comparing decimals, decimals to fraction, decimal conversion examples.

Decimals are numerical representations that extend our understanding of whole numbers by introducing a fractional component. In the decimal system, numbers are expressed in units, tenths, hundredths, and so forth, with each digit’s position indicating a specific decimal place value. This system allows for precise representation of values lying between two whole numbers, facilitating more accurate measurements and calculations in various fields.

Reading Decimals

A decimal point is read as,

Decimals Definition

Decimals are used to represent numbers in smaller parts than a whole using a system where each digit’s position indicates a specific decimal place. This allows for a more accurate representation of values between whole numbers.

Reading and Writing Decimals

Reading and writing decimals involves understanding the place value of digits after the decimal point. Here are the steps:

Step 1. Read Whole Number Part: Identify the digits to the left of the decimal point as the whole number part.

Step 2. Read Decimal Part: State the digits to the right of the decimal point as individual digits. For example, in the decimal 38.75, read “thirty eight point seven five.”

Step 3. Write Decimal in Words: Express the decimal numerically and verbally. For example, 4.2 can be written as “four point two”.

In the chart given below, you will see that from the tenth place, there is only 1 digit after decimal. Still, we count it as the tenth-place value. This is because decimals also take a place value.

What is Place Value in Decimals?

Place value in decimals refers to the position of a digit in relation to the decimal point. Each position represents a power of 10, determining the digit’s value in the number.

For example, in the decimal 0.75:

• The 7 is in the “tenths” place because it’s one-tenth of the whole.
• The 5 is in the “hundredths” place because it’s one-hundredth of the whole.

In 0.75, you have 7 tenths and 5 hundredths.

Expanded Form of Decimals

Expanded form of decimals shows the value of each digit based on its place value. For example, in the decimal 0.75, the expanded form is 0.70 + 0.05, indicating the value of each digit.

Here, 0.7 represents 7 is at tenth place and 0.05 represents 5 is at hundredth place. This expanded form can also be written as 7/10 + 5/100.

Properties of decimals under multiplication and division operations are as follows:

• Product of any two decimal numbers remains the same regardless of the order of multiplication.
• Multiplying a whole number by a decimal number in any order yields the same product.
• When a decimal fraction is multiplied by 1, the result is the decimal fraction itself.
• Multiplying a decimal fraction by 0 results in a product of zero (0).
• Dividing a decimal number by 1 yields the decimal number as the quotient.
• Dividing a decimal number by itself results in a quotient of 1.
• Division of a decimal number by 0 is not possible, as the reciprocal of 0 does not exist.

There are three basic types of decimals in maths. These are:

Recurring Decimals

Non-recurring decimals, decimal fractions.

Non-recurring decimals are numbers where the decimal expansion does not repeat. The digits after the decimal point do not form a recurring pattern. An example is 0.274, where the digits 2, 7, and 4 do not repeat in a predictable manner.

Decimal fractions are numbers that fall between two consecutive integers on the number line and are expressed in decimal form. These numbers have a finite number of digits after the decimal point. For example, in the decimal fraction 0.75, the digits 7 and 5 represent the fractional part, and there is no recurring or infinite pattern.

Arithmetic operations on decimals involve addition , subtraction , multiplication , and division , following similar principles as whole numbers .

Addition and Subtraction of Decimal Numbers

• To add or subtract decimals, align them based on the decimal point.
• Perform the operation as usual, treating the decimals as if they were whole numbers.
• Place the decimal point in the result directly below the aligned decimals.

Multiplication of Decimal Numbers

• Multiply decimals as if they were whole numbers, disregarding the decimal points.
• Count the total decimal places in the factors.
• Place the decimal point in the product, counting from the right, equal to the total decimal places.

Division of Decimal Numbers

• Similar to multiplication, perform division as if the decimals were whole numbers.
• Count the total decimal places in both the dividend and divisor.
• Move the decimal point in the quotient to make it a whole number, then adjust accordingly.

Rounding decimals means approximating a decimal number to a specified place value. For example, rounding 3.78 to the nearest tenth results in 3.8, as it is closer to 3.8 than 3.7.

Rule of Rounding Decimals

The rule for rounding decimals is to identify the desired decimal place, look at the digit immediately to its right, and round up if that digit is 5 or greater, rounding down if it is 4 or less. For instance, rounding 3.78 to the nearest tenth gives 3.8, as the digit in the hundredths place (8) is greater than 5.

Rounding Decimals to Nearest Tenth

Rounding decimals to the nearest tenth means you’re making the number simpler by keeping only one digit after the decimal point. For example:

In the decimal 4.72. To round it to the nearest tenth, look at the digit in the hundredths place, which is 2. Since 2 is less than 5, you round down the digit in the tenths place. So, 4.72 rounded to the nearest tenth is 4.7

Rounding Decimals Examples

Rounding decimals involves simplifying them to a specified place value. Here are examples and techniques:

Rounding to Nearest Whole Number:

• For Example 4.78 rounded to the nearest whole number is 5, as the digit in the tenths place is 8 (greater than 5).

Rounding to Nearest Tenth:

• For example 3.46 rounded to the nearest tenth is 3.5, considering the digit in the hundredths place, which is 6(greater than 5).

Rounding to Nearest Hundredth:

• For example 2.934 rounded to the nearest hundredth is 2.93, with the digit in the thousandths place being 4(less than 5).

Comparing decimals involves determining which decimal is greater or smaller. Follow these steps:

Step 1: Compare Whole Numbers

Start by comparing the whole number parts of the decimals. The decimal with the greater whole number is larger. If the whole numbers are the same, move to the decimal places.

Step 2: Compare Decimal Places

Compare the digits in the decimal places from left to right. The first digit where the decimals differ determines the larger number.

Note: If one decimal has fewer decimal places, consider the missing places as zeros when making comparisons.

Example: Compare 3.25 and 3.15

Whole numbers are same (3) Compare tenths place (2 vs. 1) 3.25 is greater than 3.15

The conversion of decimal to fraction or vice versa can be performed easily.

Decimal to Fraction Conversion

We can easily convert decimal to fraction by following the given steps:

Step 1: Identify Decimal: Begin by identifying the decimal you want to convert to a fraction.

Step 2: Write Decimal as a Fraction with a Denominator of 1: Express the decimal as a fraction with a denominator of 1. For example, if the decimal is 0.5, write it as 0.5/1

Step 3: Multiply to Eliminate Decimal Places: Multiply both the numerator and the denominator by 10, 100, 1000, or any power of 10 sufficient to eliminate the decimal places. For instance, for the decimal 0.5, multiply both numerator and denominator by 10 to get 5/10

Step 4: Simplify Fraction: If possible, simplify the fraction by finding the greatest common factor (GCF) and dividing both the numerator and denominator by it. In the example, 5/10​ can be simplified to 1/2​ by dividing both by 5.

Fraction to Decimal Conversion

To convert a fraction into decimal, it needs simple division of numerator by denominator.

For Example: To convert 3/4 into decimal we need to divide 3 by 4, this will give us 0.75.

Few examples of Decimal Conversion are as follow:

1. Convert 1/4 to decimal.

To convert the fraction 1/4 into a decimal, you can divide 1 by 4: 1/4 =0.25 Therefore, 1/4 as a decimal is 0.25

2. Express the percentage 25% as a decimal.

25% can be written as 25/100 in fraction on solving we get 1/2 1/2= 0.5

Convert into whole fraction as [(4 × 2) + 1]/4 = 9/4 Now divide 9 by 4 we get = 2.25

4. Represent the repeating decimal 1/3 as a fraction.

To convert the fraction 1/3 into a decimal, you can divide 1 by 3: 1/3 =0.3333…. Therefore, 1/3 as a decimal is 0.3333……, this is a non terminating repeating decimal representation.

Solved Examples on Decimals

Example 1: Compare 4.67 and 4.678. Which decimal is greater, and by how much?

To compare 4.67 and 4.678: Both decimals have the same whole number part (4), so we move to the decimal place. In the decimal place, 678 is greater than 67. Therefore, 4.678 is greater than 4.67. The difference between them is 0.008.

Example 2: Convert the fraction 3/5 into a decimal.

To convert fraction 3/5 into a decimal, you can divide 3 by 5: 3/5 =0.6 Therefore, 3/5 as a decimal is 0.6.

Example 3: Express the decimal 4.267 in expanded form.

4.267= 4×10 0 +2×10 −1 +6×10 −2 +7×10 −3

Example 4: Add 0.25, 1.6, and 4.75

0.25 + 1.6 + 4.75 = 0.25 + 1.60 + 4.75 = 6.60

Practice Questions on Decimals

Q1. Compare 0.325 and 0.53. Determine the relationship between these decimals and express it using the symbols “>” or “<“.

Q2. Express the ratio 7:9 as a decimal.

Q3. Write the expanded form of the decimal 0.825.

Q4. Add the decimals 2.34 and 1.89.

Q5. Subtract 0.56 from 3.72.

Q6. Multiply 0.25 by 4.

Decimals Frequently Asked Questions

What does decimal 0.5 mean.

The decimal 0.5 represents the half or 1/2 in fraction.

What is 1 Decimal Place?

1 decimal place represents the tenth place in a number, it is the first digit after decimal point.

How to Convert 1/4 into Decimals?

To convert 1/4 to decimals, divide 1 by 4. The result is 0.25. In decimal form, 1/4 is equal to 0.25.

What is Difference Between Decimals and Fractions?

Decimals and fractions both represent parts of a whole, but decimals are expressed in base-10 notation, while fractions are ratios of integers. Decimals use a decimal point, while fractions use a numerator and denominator.

How to Round Decimals in Mathematical Calculations?

To round decimals, determine the desired decimal place and look at the digit to its right. If the digit is 5 or greater, round up; if less than 5, round down.

What are Examples of Non-Terminating Decimals?

Examples of non-terminating decimals include the recurring decimals like 1/3 (0.333…) and irrational numbers like the square root of 2 (1.41421356…).

How are Decimals Used in Our Daily Lives?

Decimals are used in various aspects of daily life, such as shopping (pricing), cooking (measurement), and financial transactions (currency). They provide a precise way to represent quantities that fall between whole numbers.

What Decimal Means?

Decimal is a fraction written in a special form, using a decimal point. For example: 1/4 in decimal form can be wriiten as 0.25.

What is Decimal Place in Math?

In mathematics, a decimal place refers to the position of a digit to the right of the decimal point in a decimal number. Each place represents a power of 10, and the value of the digit is determined by its position. For example, in the number 23.456, the digit 2 is in the tenths place.

What are Types of Decimals?

There are primarily three types of decimals – decimal fractions, terminating decimals and non-terminating decimals. Terminating decimals have a finite number of digits after the decimal point, while non-terminating decimals continue indefinitely without repeating. Non-terminating decimals can further be categorized as repeating or non-repeating, based on whether certain digits repeat in a pattern or not.

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