All numbers, including whole numbers, integers, fractions and decimal numbers, can be written in the Numerator Denominator form.
A rational number is a number that can be written in the form p/q, where p and q are integers and q ≠ 0. The denominator of a rational number can never be zero.
e.g. 911 , 58 , 712
A rational number is positive if its numerator and denominator are both either positive integers or negative integers.
e.g. 25 , 34 , -7-10 , -5-11
If either the numerator or the denominator of a rational number is a negative integer, then it is a negative rational number.
e.g. -25 , 3-4 , 7-10 , 5-11
Representation of rational numbers on the number line
The rational number zero is neither negative nor positive. Positive rational numbers are represented to the right of zero on the number line. Negative rational numbers are represented to the left of zero on the number line.
A rational number obtained by multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer, is said to be the equivalent form of the given rational number.
Rational numbers in Standard form
A rational number is said to be in its standard form if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer.
To reduce a rational number to its standard form, divide its numerator and denominator by their highest common factor (HCF). To find the standard form of a rational number with a negative integer as the denominator, divide its numerator and denominator by their HCF with a minus sign.
Counting numbers 1, 2, 3, 4, 5, …….etc. are called Natural numbers. Set of natural numbers is generally denoted by N.
All the natural numbers together with zero are called Whole numbers. The numbers 0, 1, 2, 3, 4, 5, ……. etc. are called Whole numbers. Set of Whole numbers is generally denoted by W. Every Natural number is a Whole number.
All natural numbers, zero and negatives of the natural numbers are called Integers, i.e. ……– 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5,………..etc. are integers. Set of Integers is generally denoted by I or Z. Every Whole number is an Integer.
Numbers that can be written in the form of p q , where p and q are integers and q ≠ 0 are called Rational numbers. The collection of Rational numbers is denoted by Q. Between any two rational numbers there exists infinitely many rational numbers.
Numbers which cannot be expressed in the form of p q , where p and q are integers and q ≠ 0.
The set of irrational numbers is denoted by Q _ . 2 , 3, 7 are the examples of irrational numbers.
The ratio of the length of circumference of a circle to the length of its diameter is always constant. It is an irrational number and denoted by π. Decimal expansion of π is non-terminating and non-repeating. Value of π = 3.14159265……… Approximate value of π is 22 7 , but not equal to the exact value.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.