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 of Rational Numbers

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

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:

h
Rational Numbers as Part of Number System:
what is rational number

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

(b.) 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 Number?

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

Read More Here About What Is Circle

18 thoughts on “What is Rational Number – Definition, Properties, & Examples”

  1. Does your blog have a contact page? I’m having a tough time
    locating it but, I’d like to send you an email. I’ve
    got some creative ideas for your blog you might be interested in hearing.
    Either way, great website and I look forward to seeing it grow over time.

    Reply
  2. Great post. I was checking continuously this blog and I am impressed!
    Extremely useful info particularly the last part 🙂 I care for such info much.

    I was seeking this particular information for a long time.

    Thank you and best of luck.

    Reply
  3. Do you have a spam issue on this website; I also am a blogger, and
    I was curious about your situation; many of us have created some
    nice methods and we are looking to swap techniques with others, please shoot me an e-mail if interested.

    Reply
  4. Howdy! This is kind of off topic but I need some guidance from an established blog.
    Is it hard to set up your own blog? I’m not very techincal but I can figure things out
    pretty quick. I’m thinking about creating my own but I’m not sure where to begin. Do you have any points or suggestions?
    Many thanks

    Reply
  5. Hi, i think that i saw you visited my site so i came to “return the favor”.I am attempting
    to find things to improve my website!I suppose its ok to use some of
    your ideas!!

    Reply

Leave a Comment

Open chat