What is Rational Number - Definition, Properties, & Examples

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What is Rational Number - Definition, Properties, & Examples

What is a Rational Number?

While rationality meaning is ‘reason’ in common usages, however, in mathematics the meaning of rational comes from word ‘ratio’ i.e. rational number is a ratio between integers. If we attempt to define rational numbers further it will be:

Any number that can be written in the form fraction i.e. x/y where x and y must be integers but y≠0.Example:

6/8 can be called a rational number because both 6 and 8 are integers -17/19 is also a rational number because both -17 and 19 are integers 4.5 can be written as 45/10=9/2 where both 9 and 2 are integers

1. Rational numbers also include all those decimal numbers that can be expressed in the form of fractions.

Example:

1. -17.65 can be written as -1765/100=-353/20 where both -353 and 20 are integers.

Rational Numbers Examples:

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Rational Numbers as part of number system:

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When we talk about the number system it can be thought of as a large set that has different kinds of numbers, as depicted in the diagram below:

1. Natural Numbers (N) – 1,2,3,……..are all those numbers that you use for counting objects around you, and i.e. why these are also known as counting numbers.
2. Whole Numbers (W) – 0,1,2,3,4…….i.e 0 added to the set of natural numbers gives you whole numbers
3. Integers (Z) - …..-5,-4,-3,-2,-1,0,1,2,3,4,5……..i.e negative values of all natural numbers together with whole numbers form integers.
4. Rational Numbers (Q): All those real numbers that include not just the natural numbers, whole numbers, integers but also all those numbers that can be written in the form of fraction eg. 9/8,-3/5 or in the form of decimal eg. 5.367666,-4.65, etc.
5. Irrational numbers (P): All those real numbers which cannot be expressed as a fraction eg. values of π(pi), √2 (square root of 2), etc.

Properties of rational numbers:

1. Closure properties:

a. Rational numbers are closed under addition i.e. for an equation p+q=r. If p and q are rational numbers then their sum i.e. r is also a rational number.

Example:

Since 3/2 and 2/9 are rational numbers, their sum i.e. 3/2 + 2/9 = 31/18 is also a rational number.

b. Rational numbers are closed under subtraction i.e. for an equation p-q=r. If p and q are rational numbers then their difference i.e. r is also a rational number.

Example:

Since 4/5 and 11/5 are rational numbers, their difference i.e. 4/5 – 11/5 = -7/5 is also a rational number.

c. Rational numbers are closed under multiplication i.e. for an equation p * q = r. If p and q are rational numbers then their product i.e. r is also a rational number.

Example:

Since 0/8 and -14/28 are rational numbers, their product i.e. 0/8 * -14/28 = 0 is also a rational number.

d. However, rational numbers are not closed under division i.e. division of 2 rational numbers may not always yield a rational number.

Example:

Both 5 and 0 are rational numbers, but their division i.e. 5/0 is not defined.

2. Commutativity property:

a. Addition and multiplication are commutative for rational numbers i.e.

x + y = y + x and x * y = y * x

Examples:

If x = -2/5 and y = 11/5 then,

-2/5 + 11/5 = 9/5 = 11/5 + (– 2/5)

and -2/5 * 11/5 = -22/25 = 11/5 * -2/5

2. Subtraction and division are not commutative for rational numbers i.e.

a – b ≠ b – a and a ÷ b ≠ b ÷ a

Examples:

If a = 2/3 and b = 5/4 then, 2/3 - 5/4 = -7/12 but, 5/4 -2/3 = 7/12

2/3 ÷ 5/4 = 8/15 but, 5/4 ÷ 2/3 = 15/8

3. Associativity:

a. Addition and multiplication are associative for rational numbers i.e.

(x + y) + z = x + (y + z) and (x * y) * z = x * (y * z)

Examples:

If x = -2/3, y = 3/5 and z = -5/6 then,

(-2/3 + 3/5) + (-5/6) = -9/10 = -2/3 + [3/5 + (-5/6)]

(-2/3 * 3/5) * (-5/6) = 1/3 = -2/3 * [3/5 * (-5/6)]

b. Subtraction and division are not associative for rational numbers i.e.

(a – b) - c ≠ a – (b – c) and (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

Examples:

If a = 3/5, b = -17/5, c = 20/5 then,

[3/5 – (-17/5)] – 20/5 = 0 but, 3/5 – [(-17/5) – 20/5] = 8

[3/5 ÷ (-17/5)] ÷ 20/5 = -3/68 but, 3/5 ÷ [(-17/5) ÷20/5] = -12/17

What is known as the standard form of rational numbers?

If p/q is a rational number such that: p is any integer and q is a positive integer and there is no common factor between them other than 1. Then this p/q is said to be a rational number in its standard form.

Examples: -9/7, 7/2, 16/15, -10/7