What are Fractions? – Definitions and Properties of Fractions

What is a Fraction?

A fraction represents a part of a whole or, more generally, any number of equal parts of a value. When we are working with fractions we assume about the relationship between the part of any number or a whole sum number. Something we do not even realize that we are working on a fraction because fractions are everywhere. A common or simple fraction consists of two parts, an integer numerator displayed above a line, and a non-zero integer denominator, displayed below that line. It tells us how many parts a whole is divided into. For example:

5/7 = means we have 5 parts out of the whole of 7.

5 is the numerator, it tells us how many parts we have.

7 is the denominator, it tells us how many parts the whole is divided into.

Properties Obeyed by the Fractions

Fraction obeys the commutative, associative, and distributive laws, and the rule against division by zero, like whole numbers.

Commutative Property

In mathematics, a binary operation is commutative if changing the sequence of the operands in an operation do not change the result of it. It is a fundamental property for many binary operations, and many mathematical proofs depend on this property.

For example:

Suppose that there are two fractions A and B and they are multiplied together. Then the order of A and B during multiplication cannot change the result. Same goes for addition too.

A × B = B × A

A + B = B + A

Associative Property

In mathematics, the associative property is a property of binary operations. In propositional logic, associativity is a valid rule of replacement for expressions with logical proofs. Within an operation containing two or more operands in a row of some associative operator, the way in which the operations are performed does not matter as long as the sequence of the operands in the operation remains the same. It means that rearranging the parentheses in such an expression will not change its result. Consider the following equations:

(4 + 5) + 6 = 4 + (5 + 6) = 15

(4 × 5) × 6 = 4 × (5 × 6) = 120

Distributive Property

The distributive property of binary operations is widely applicable in the distributive law. Distribution refers to two valid rules of replacement. The rules allow one to put conjunctions and disjunctions differently within logical proofs. By observing the example given below, you can understand the distributive property easily.

If there are three operands A, B, and C are in such an operation where A × (B + C) then it must be equal to A × B + A × C.

Division by Zero

In mathematics, division by zero is division where the denominator is zero. In ordinary mathematics, this expression has no meaning, as there is no number which, when multiplied by 0, gives another number, and so division by zero is undefined.

Equivalent Fraction

In mathematics, equivalent fraction can be defined as the fractions with different numerators and denominators that represent the same value or proportion of the whole. Here are some examples of equivalent fraction, like1/2, 2/4, 8/16. They all contain the different values in their numerators and denominators, but at last after evaluation they all give the same value 0.5 as the answer. Equivalent fractions are those types of fractions which give us the same value at last, even though they may look different.

Equivalent fractions represents the same amount of distance or points on a number line to one another.

All equivalent fractions are reduced to the same fractions in their simplest forms by dividing both numerator and denominator by their greatest common factor.

How to Find Equivalent Fractions:

By multiplying the numerator and denominator of a fraction by the same non-zero number, we can change the fraction into equivalent of the original fraction. This is true because for any non-zero number a, the fraction a/a = 1. n n = 1 Therefore, multiplying by a/a is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number.

For Example:

To find the equivalent fraction of 2/5we multiply both the numerator and denominator by 2, then we get equivalent fraction 4/10.

How to Simplify the Equivalent Fractions?

Dividing an equivalent fractions numerator and denominator by the same non-zero number will also yields an equivalent fraction. This is called simplifying or reducing the fraction. A simple fraction in which the numerator and denominator both are prime number is said to be irreducible.

For Example:

To find the equivalent fraction of 6/18 we divide both the numerator and denominator by their greatest common factor, then we get equivalent fraction 1/3.

How to Check Two Fractions are Equivalent Fraction:

To check whether the fractions are equivalent or not, just simplify all the fractions. If they reduce into the same fraction, then the fractions are equivalent fractions.

For example:

We will check whether the fractions 5/10 and 12/36 are equivalent or not. We will simplify both the fractions –

510=1525=12

510=1∗52∗5=12

1236=43433=13

1236=4∗34∗3∗3=13

The fractions ½ and ⅓ are not the same, hence the two fractions are not Equivalent fractions.

Equivalent fractions are an important tool when adding or subtracting fractions with different denominators. Let’s look at an example:

John bought one half of a cake. He didn’t know that while he was out, his wife, Linda, bought one fourth of a cake. When they got home, how much cake did they have altogether?

When we are adding or subtracting fractions, they must have the same denominator. But in the above case, the denominator are two and four (2 and 4). As we read earlier by multiplying the numerator and denominator of a fraction, we get an equivalent fraction. So in this case, for making the denominator equal we have to perform this operation:

1222=24

1222=24

Now we have got the same denominator and we can add them like:

Converting Between Decimals and Fractions

Decimal numbers are more useful to work with when performing calculations, but sometimes it lacks in precision that common fractions have, because sometimes an infinite repeating decimal is required to reach the exact precision. Thus, it is useful to convert repeating decimals into fractions. To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator and round the answer to the desired accuracy. For example, to change ¼ to a decimal, divide 1 1.00 {\displaystyle 1.00} by 4 to obtain 0.25. To change 1/3 to decimal, divide 1 by 3 and stop when the desired accuracy is obtained, e.g. 4 digits after the decimal point. To change a decimal into a fraction, write 1 in the denominator 1 followed by as many zeroes as there are digits to the right of the decimal point, and write all the digits of the original decimal in the numerator, excluding the decimal point. Thus 12.346 = 12346/1000.

Summary:

• You can make equivalent fraction just by multiplying or dividing both top and bottom numerator and denominator) by the same value.

• You can only multiply or divide, never add or subtract, to get an equivalent fraction because the fraction we get from addition and subtraction will not be equivalent to the value we have.

• You can divide if and only if the top and bottom stay as whole numbers