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Integers and Fractions

The arithmetic is an important branch of mathematics. Elementary arithmetic includes the study of numbers and basic operations over numbers. Arithmetic operations include addition, multiplication, subtraction and division. The study of arithmetic also includes different word problems about time, distance, speed and other concepts on numbers.

We know that the numbers have great importance in our day to day life. A number is defined as a mathematical object which is utilized for the counting, labelling and measuring purposes. They are also known as the numerals.

The detailed study of numbers and their properties are done in an important and one of the oldest branch of mathematics, called the number theory. There are different kinds of numbers in mathematics, such as - natural numbers, whole numbers, integers, real numbers, fractional numbers etc.

In this article, we shall understand about integers and fractions. Let us go ahead and learn in detail about integers and fractions, operations on integers and fractions and various problems based on those.

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Integer Definition

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In mathematics, an integer is defined as a type of number that can be written as a whole. It does not have a fractional portion in it. Thus, a number having no fractional component is known as an integer.
The set of integers include the following:

i) All counting numbers such as 1, 2, 3, 4, ...

ii) Negative of all counting numbers such as -1, -2, -3, -4 ...

iii) Zero i.e. 0

For Example:

50, -64, 0, -5, -134, 4985645, -7840 are all integers. While, $\sqrt{7}$, 34.7, -0.09 are not integers.

The set of integers is represented by $\mathbb{Z}$. i.e.

$\mathbb{Z}$ = {..., -4 -3, -2, -1, 0, 1, 2, 3, 4, ...}

The integers are countably infinite. These are represented in a line known as number line. It proceeds forever in both directions shown by by arrows. This is indicated by a line with numbers written on an equal interval.
The number line has zero in the middle and on the right side of zero, it has positive integers. On the left side of zero, there are negative integers. The zero is neither positive or negative, its neutral. 

The number line is demonstrated in the following picture:

Number Line Representation
Two integers staying at the same distance in both directions from zero, are known as the opposites, such as - +9 and -9, +5 and -5, +70 and -70 are the pair of opposites.

Fraction Definition

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The word "fraction" is taken from Latin language word "fractus" which means "broken". A fraction is defined as a part of the whole. It illustrates how many parts of a particular object there are. 

A simple fraction consists of a two numbers one above another separated by a line or bar. The number located above the line is called numerator and the number below the line is termed as denominator. 

For Example: one-half ($\frac{1}{2}$), one-quarter ($\frac{1}{4}$), three-quarters ($\frac{3}{4}$), eight-fifths ($\frac{8}{5}$) etc.

The fractions are shown in the following diagram:

Fraction Example

There are different types of fractions which are defined below:

Proper Fractions:
The fraction in which the denominator is bigger than the numerator (numerator and denominator both being positive), is known as the proper fraction. i.e. absolute value of denominator is greater than numerator. Such fraction is also called a bottom-heavy fraction. 

For Example: $\frac{3}{4}$, $\frac{53}{99}$.

Improper Fractions:

A fraction having its denominator smaller than the numerator (given that numerator and denominator are both positive), is termed as an improper fraction. i.e. the absolute value of denominator is less than numerator. 

For Example:
 $\frac{7}{5}$, $\frac{134}{79}$. It is also called top-heavy fraction.

Mixed Fraction:
A mixed fraction is defined as combination of a whole number and a proper fraction. Mixed fraction is written in the following form.

a $\frac{b}{c}$

There is no visible addition (+) operator. It implies:

a $\frac{b}{c}$ = a + $\frac{b}{c}$

For Example: 4 $\frac{1}{5}$ means 4 whole objects and a one-fifth part of whole. i.e.

$4 \frac{1}{5}$ = 4 + $\frac{1}{5}$

= $\frac{21}{5}$

Writing Integer as Fraction

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Every integer can be represented in the form of a fraction. Integer has no fractional or decimal part. But still an integer can be written as a fraction. It can be done by simply dividing by 1. By doing so, the value of the integer does not change. We can say that the integer can be made a fraction by assuming the integers itself the numerator and 1 as the denominator. i.e.

a can be written as $\frac{a}{1}$

For Example:

4 = $\frac{4}{1}$

-17 = -$\frac{17}{1}$

Multiplying and Dividing Fractions by Integers

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Multiplication
While multiplying a fraction and an integer, check the denominator of the fraction and integer, if they have any common divisor. If yes, then the common divisor should be cancelled out from both and result is obtained after multiplication.

For Example:

4 x $\frac{9}{2}$

= 2 x 9

= 18

Division
In order to divide a fraction by an integer, the integer is to be written in the form of fraction. In the next step, it should be reciprocated and division sign should be changed into multiplication. Now multiplication operation (as discussed above) is performed.

For Example:

$\frac{3}{4}$ $\div 6$

= $\frac{3}{4}$ $\div$  $\frac{6}{1}$

= $\frac{3}{4}$ $\times$ $\frac{1}{6}$

= $\frac{1}{4}$ $\times$ $\frac{1}{2}$

= $\frac{1}{8}$

Same process is followed while dividing an integer by a fraction, but this time reciprocation is performed on fraction.

Examples

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Have a look at the following examples:

Example 1: Write 5 in the form of four fractions.

Solution: 5 = $\frac{5}{1}$

$\frac{5}{1}$ = $\frac{5 \times 2}{1 \times 2}$ = $\frac{10}{2}$

$\frac{5}{1}$ = $\frac{5 \times 3}{1 \times 3}$ = $\frac{15}{3}$

$\frac{5}{1}$ = $\frac{5 \times 4}{1 \times 4}$ = $\frac{20}{4}$

Thus, 5 can be written as

5 = $\frac{5}{1}$ = $\frac{10}{2}$ = $\frac{15}{3}$ = $\frac{20}{4}$

Example 2: Multiply $\frac{4}{11}$ and 33.

Solution: $\frac{4}{11}$ x 33

= 4 x 3 = 12

Example 3: Divide the following:

i) $\frac{7}{3}$ $\div 28$

ii) -5 $\div$ $\frac{30}{13}$

Solution: i) $\frac{7}{3}$ $\div 28$

= $\frac{7}{3}$ $\times$ $\frac{1}{28}$

= $\frac{1}{3}$ $\times$ $\frac{1}{4}$

= $\frac{1}{12}$

ii) -5 $\div$ $\frac{30}{13}$

= -5 $\times$ $\frac{13}{30}$

= -1 $\times$ $\frac{13}{6}$

= -$\frac{13}{6}$ 
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