Ximpledu

Complex Number

See how to solve a complex number.
10 examples and their solutions.

Imaginary Number i

Formula

-1 = i
The number in the square root sign
cannot be (-).
So i = -1 is not a real number.
So i is called the imaginary number.

Example

-36
Solution

Powers of i

Formula

i1 = i
i2 = -1
i3 = -i
i4 = 1

i5 = i
i6 = -1
i7 = -i
i8 = 1
...
The powers of i show this pattern:
i, -1, -i, 1.
Use this property to solve the powers of i.

Example

i23
Solution

Example

i86
Solution

Example

i + i2 + i3 + i4 + ... + i99
Solution

Adding and Subtracting Complex Numbers

Complex Number

a + bi
a: real number part
+bi: imaginary number part

Example

(5 + 2i) + (8 - 3i)
Solution

Example

(3 + i) - (2 - 7i)
Solution

Multiplying Complex Numbers

Example

(3 + 2i)(1 + 6i)
Solution

Example

(4 + i)(8 - 5i)
Solution

Dividing Complex Numbers

Example

21 + 3i
Solution

Example

1 - 7i2 - i
Solution