NCERT Class 11 Maths Complex Numbers And Quadratic Equations Questions

5.1 Overview
We know that the square of a real number is always non-negative e.g. (4)2 = 16 and (– 4)2 = 16. Therefore, square root of 16 is ± 4. What about the square root of a negative number? It is clear that a negative number can not have a real square root. So we need to extend the system of real numbers to a system in which we can find out the square roots of negative numbers. Euler (1707 - 1783) was the first mathematician to introduce the symbol i (iota) for positive square root of – 1 i.e., i = √−1 .

5.1.1 Imaginary numbers

Square root of a negative number is called an imaginary number., for example,

√−9 = √−1√ 9 = i3, √−7 = √−1 7 =i √7

5.1.2 Integral powers of i

       i = √−1 , i ² = – 1, i ³ = i ² i = – i , i ⁴ = (i ²)² = (–1)² = 1.

To compute in for n > 4, we divide n by 4 and write it in the form n = 4m + r, where m is quotient and r is remainder (0 ≤ r ≤ 4)

Hence iⁿ = i^4m+r = (i⁴)^m . (i)^r = (1)^m (i)^r = i^r

For example, (i)³⁹ = i ^{4 × 9 + 3} = (i⁴)⁹ . (i)³ = i³ = – i

and (i)^–435 = i ^{– (4 × 108 + 3}) = (i)^{– (4 × 108)} . (i)^{– 3}

Please refer to attached file for NCERT Class 11 Maths Complex Numbers And Quadratic Equations Questions