Imaginary Numbers
Imaginary numbers are used to represent the even roots of negative
numbers. They use the quantity i, where i =
v-1. For example:
Square roots of negative numbers are called imaginary numbers because they do
not lie on the real number line.
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Complex Numbers
A complex number is the sum of a real number and an imaginary number. A
complex number is written in the form a + bi, where a and
b are real numbers, and i = v-1.
There are two things you need to be able to do with complex numbers:
- Know the powers of i
- Know how to do operations, like addition, subtraction, and multiplication, on
complex numbers
The Powers of i
The powers of
i are easy to work with. For example:
Evaluate
i2 ×
i9.
The trick is to remember that the powers of
i work in cycles of four:
- i1 = i
- i2 = v-1 × v-1 = (< v-1)2
= –1
- i3 = v-1 × v-1 × v-1 = v-1 × (v-1 )2
= –i
- i4 = v-1 × v-1 × v-1 × v-1 = (v-1 )4
= 1
This way, the expression i2 × i9 becomes (–1)(i) = –i. If you know these
cycles, you can reduce any exponent of i to a much more manageable size.
The expression i2 × i9
becomes (–1)(i) = –i.
Operations on Complex Numbers
Algebraic manipulation of complex numbers is exactly like dealing with real
numbers. See for yourself:
Simplify the expression (3
x +
i)(
x – 2
i).
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Mathematics Practice Questions
Video Lessons and 10 Fully Explained Grand Tests
Large number of solved practice MCQ with explanations. Video Lessons and 10 Fully explained Grand/Full Tests.