Page 65 - Start Up Mathematics_7

4 Rational Numbers

We have already studied about natural numbers, whole numbers, integers
and fractions. The need to extend our number system arises when we have
to divide an integer by another integer. Let’s understand this with the help
of examples. We can easily represent a hot air balloon flying at the height
5
of km above sea level. But how do we represent a whale swimming
6
3 km below sea level? Can we represent this depth as –3 km which is
5 5
neither an integer nor a fraction? We have not dealt with such numbers
till now. Therefore, to express such numbers the need to extend our number system arises. Such
numbers are categorized as rational numbers.
Rational Numbers
p
The numbers of the form , where p and q are integers and q ≠ 0, are called rational numbers.
q
2 –3 5
For example, , , , etc. are rational numbers.
3 4 7

• Every fraction is a rational number but every rational number need not be a fraction.
• All integers can be expressed as rational numbers but all rational numbers need not be integers.
• The integer zero is also a rational number.

Positive Rational Numbers
A rational number is said to be positive if its numerator and denominator Numerator Denominator Sign
+
+
+
7 8 –2
are either both positive or both negative. For example, , , and –11 – – +
12 19 –5 –23
are positive rational numbers.
Negative Rational Numbers
A rational number is said to be negative if its numerator and denominator Numerator Denominator Sign

–63 4 + – –
have opposite signs. For example, , etc. are negative rational numbers. – + –
71 –9
Equivalent Rational Numbers
p p × n
Let (q ≠ 0) be a rational number and n be a non-zero integer, then is a rational number
q q × n
p
equivalent to . In other words, a rational number remains same if we multiply the numerator and
q
denominator by a non-zero integer n.

The above mentioned result is also known as “Fundamental Law of Fractions”.