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Complex Number Calculator

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Complex number are the composition of a real number and an imaginary number. The basic form is (a + bi), where, a and b are the real numbers and i is the imaginary unit i.e., i = $\sqrt{-1}$. Real numbers can be anything integers, rational and irrational numbers.

Complex Number

A Complex number is denoted by "z" i.e., z = a + bi, where "a" is the real part and "b" is the imaginary part of the complex number. To calculate the complex number plug the real and imaginary numbers and the respected arithmetic operations into this online complex number calculator.
 

Steps

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Step 1 : Observe the complex numbers and check the basic operation whether it is addition or subtraction, multiplication or division.
Operations on complex numbers are,
Addition: (m + ni) + (s + ti) = (m + s) + i (n + t)

Subtraction: (m + ni) - (s + ti) = (m - s) + i(n - t)


Multiplication: (m + ni) (s + ti) = ms + mti + nsi + $i^{2}$nt = (ms - nt) + i (mt + ns)
$i^{2}$ = -1Division: $\frac{(m + ni)}{(s + ti)} = \frac{(ms + nt)}{(s^2 + t^2)} + \frac{(ns − mt)}{(s^2+t^2)i}$

Step 2 : Perform the basic operations and find the real and imaginary part of the complex number.

Problems

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Based on complex number some problems has given below.

Solved Examples

Question 1: Simplify (10 - 2i) - (-8 - i).
Solution:
Step 1 : Given : (10 - 2i) - (-8 - i)
We need to perform subtraction of complex numbers.
Subtraction: (a + bi) - (c + di) = (a - c) + i(b - d)

Step 2 : Subtrating the complex numbers, we get
z = (10 - 2i) - (-8 - i) 
  = (10 + 8) + i(-2 + 1)
  = 18 - i

The value of Complex number (10 - 2i) - (-8 - i) is 18 - i.


Question 2: Simplify (8 + i)(4 + 3i).
Solution:
Step 1 : Given : (8 + i)(4 + 3i)
We need to perform multiplication of complex numbers.
Multiplication: (a + bi) (c + di) = ac + adi + bci + $i^{2}$bd = (ac - bd) + i (ad + bc)

Step 2 : Multiplying the complex numbers, we get
z = (8 + i)(4 + 3i)
  = (8 $\times$ 4 - 1 $\times$ 3) + i(8 $\times$ 3 + 1 $\times$ 4)
  = 29 + 28i

The value of Complex number (8 + i)(4 + 3i) is 29 + 28i.