This online Complex Numbers Calculator performs basic arithmetic operations with two complex numbers. When typing the imaginary part of a complex number in the appropriate field of the calculator, make sure that the symbol ‘i‘, representing the imaginary unit, is adjacent to the numeric part without space.
Arithmetic of complex numbers
A complex number is a number that can be expressed in the form \(x + yi\), where \(x\) (called the real part) and \(y\) (called the imaginary part) are real numbers, and \(i\) represents the imaginary unit, satisfying the equation \(i\)2 = −1.
Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. Therefore they are rather viewed as being elements in the complex plane, since points in a plane also lack a natural ordering.
Complex numbers are fundamental in many aspects of the scientific description of the world and also widely used in many applied sciences and engineering.
Common arithmetic operations apply to complex numbers. The easiest way to perform these operations is to use our Complex Numbers Calculator.
• Addition and subtraction
Addition or subtraction of two complex numbers \(a\) and \(b\) are performed separately with their real and imaginary parts:
$$ a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i,$$
and
$$ a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.$$
• Multiplication
Since the real part, the imaginary part, and the imaginary unit \(i\) in a complex number are all considered as numbers in themselves, two complex numbers, given as \(a=x+yi\) and \(b=u+vi\) are multiplied under the rules of the distributive property, the commutative properties and the defining property \(i\)2 = −1 in the following way:
$$ ab=(x+yi)(u+vi)=(xu-yv)+(xv+yu)i.$$
• Division
Using the conjugation, the reciprocal of a nonzero complex number \(a = x + yi\) can be expressed the following way:
$$\frac { 1 }{ a } =\frac { (x-yi) }{ (x+yi)(x-yi) } =\frac { (x-yi) }{ { x }^{ 2 }+{ y }^{ 2 } } =\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } } -\frac { y }{ { x }^{ 2 }+{ y }^{ 2 } } i,$$
since non-zero \(a\) implies that \(x^{2}+y^{2}\) is greater than zero.
This can be used to express a division of an arbitrary complex number \(b=u+vi\) by a non-zero complex number \(a\) as
$$\frac {b}{a}= b \frac {1}{a}= (u+vi) \left(\frac{x}{x^2+y^2} -\frac{y}{x^2+y^2}i\right)= \frac{1}{x^2+y^2}\left((ux+vy)+(vx-uy)i\right).$$
• Power
Computing of complex powers of nonzero complex numbers is not a trivial problem. To calculate the principal value of such a power we use the following formula:
$$ { a }^{ b }={ (x+yi) }^{(u+vi)} = exp((u+vi) ln(x+yi)) = {({ x }^{ 2 }+{ y }^{ 2 }) }^{ (u+vi)/2 } exp(i(u+vi) atan2(y,x))$$
where \(atan2(y,x)\) is the 2-argument arctangent.
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