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Complex Number Formula

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

Complex number extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

Here are formulas for Complex Number

Equality of Complex Numbers Formula

\[\LARGE  a+bi=c+di\Leftrightarrow a=c\:\:and\:\:b=d\]

Addition of Complex Numbers

\[\LARGE  (a+bi)+(c+di)=(a+c)+(b+d)i\]

Subtraction of Complex Numbers

\[\LARGE  (a+bi)-(c+di)=(a-c)+(b-d)i\]

Multiplication of Complex Numbers

\[\LARGE  (a+bi)\times(c+di)=(ac-bd)+(ad+bc)i\]

Multiplication Conjugates

\[\LARGE  (a+bi)(a+bi)=a^{2}+b^{2}\]

Division of Complex Numbers

\[\LARGE  \frac{(a+bi)}{(c+di)}=\frac{a+bi}{c+di}\times\frac{c-di}{c-di}=\frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i\]

Powers of Complex Numbers

1.  i^{n} = i, if n = 4a+1, i.e. one more than the multiple of 4.

Example – $\large i^{1}=i\:;\:i^{5}=i\:;\:i^{9}=i\:; i^{4a+1}\:;$

2. i^{n}= -1, if n = 4a+2, i.e. two more than the multiple of 4.

Example – $\large i^{2}=-1\:;\:i^{6}=-1\:;\:i^{10}=-1\:; i^{4a+2}\:;$

3. i^{n}= -i, if n = 4a+3, i.e. three more than the multiple of 4.

Example – $\large i^{3}=-i\:;\:i^{7}=-i\:;\:i^{11}=-i\:;i^{4a+3}\:;$

4. i^{n}= 1, if n = 4a, i.e. the multiple of 4.

Example – $\large i^{4}=1\:;\:i^{8}=1\:;\:i^{12}=1\:;i^{4a}\:;$

More topics in Complex Number Formula
Complex Number Division Formula Complex Number Power Formula
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