Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction, p/q where p and q are integers. The denominator q is not equal to zero (q ≠ 0). Also, the decimal expansion of an irrational number is neither terminating nor repeating.
Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p/q, where p and q are integers, q≠0. It is a contradiction of rational numbers.
Examples of Irrational numbers:
- π(pi) is an irrational number. π = 3⋅14159265… The decimal value never stops at any point. Since the value of π is closer to the fraction 22/7, we take the value of pi as 22/7 or 3.14 (Note: 22/7 is a rational number.)
- √2 is an irrational number. Consider a right-angled isosceles triangle, with the two equal sides AB and BC of length 1 unit. By the Pythagoras theorem, the hypotenuse AC will be √2. √2 = 1⋅414213⋅⋅⋅⋅
- Euler's number e is an irrational number. e = 2⋅718281⋅⋅⋅⋅
- Golden ratio, φ 1.61803398874989….
Types of Irrational numbers:
There are 2 types of irrational numbers:
1. Algebraic Irrational Numbers: An algebraic irrational number is a normal irrational number that is resulted from mathematical operations. Algebraic ones are those which have roots of the algebraic equation as the square root of 2.
2. Transcendental Irrational Numbers: A transcendental number is a number that is not algebraic, in the sense that it is not the solution of an algebraic equation with rational number coefficients. Transcendental numbers are irrational, but not all irrational numbers are transcendental. In 1844 Joseph Liouville discovered the existence of transcendental numbers. Transcendental numbers are usually the most famous – π, e, etc.
Properties of Irrational numbers:
- An irrational number is always a real number.
- An irrational number cannot be expressed as a fraction.
- An irrational number is non-repeating and non-terminating as the decimal part never ends and never repeats itself.
- The value of the square root of any prime number is an irrational number.
- The sum of a rational number and an irrational number is irrational. The product of a rational number and an irrational number is irrational. This means that any operation between a rational and an irrational number, be it addition, subtraction, multiplication or division will always result in an irrational number only.
- If r is one irrational number and s is another irrational number, then r + s and r – s may or may not be irrational numbers and rs and r/s are may or may not be irrational numbers. This means that any operation between two irrational numbers, be it addition, subtraction, multiplication or divisions will not always result in an irrational number.
- If a and b are two distinct positive irrational numbers, then \(\sqrt{ab}\) is an irrational number lying between a and b.
- For any two irrational numbers, their least common multiple (LCM) may or may not exist.
- Irrational number is simplifications of Surds. When we can’t simplify a number to remove a square root or cube root etc. then it is a surd. For example, √2 (square root of 2) can’t be simplified further so it is a surd.
Identify an Irrational number:
We know that the irrational numbers are real numbers only which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0. For example, √ 5 and √ 3, etc. are irrational numbers. On the other hand, the numbers which can be represented in the form of p/q, such that, p and q are integers and q ≠ 0, are rational numbers. Here are some tricks to identify irrational numbers:
- The numbers that are not perfect squares, perfect cubes, etc are irrational. For example √2, √3, √26, etc are irrational. But √25 (= 5), √0.04 (=0.2 = 2/10), etc are rational numbers.
- The numbers whose decimal value is non-terminating and non-repeating patterns are irrational. For example √2 = 1.4142135623730950488016887242097.... is irrational, whereas 1/7 = 0.14285714285714285714285714285714... is rational as we can observe that "142857" is keep getting repeated in the decimal portion.
