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NCERT Class 11 Mathematics Chapter 04 Complex Numbers and Quadratic Equations PDF
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 = i4m+r = (i4)m . (i)r = (1)m (i)r = ir
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
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Official NCERT Textbook PDF: Class 11 Mathematics Chapter 04 Complex Numbers and Quadratic Equations
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