Complex Numbers

Complex numbers are ordered pairs of real numbers, (a,b), where a is called the real part and b is called the imaginary part. Instead of using ordered pair notation, it is customary to combine the pairs with a plus sign and denote the resulting complex number as a+bi, where i has the property that i² = -1. In other words, i = √-1. Since there is no real number whose square is negative, i is called the imaginary unit. In addition to our complex number calculator and scientific calculator, our matrix calculator also supports operations on matrices with complex number entries.

Note: Both the real part and the imaginary part are real numbers. The imaginary part is so named because it is the coefficient of the imaginary unit i. Imaginary numbers are a specific type of complex numbers where the real part is zero, taking the form bi.

All the laws and rules for addition and multiplication used within the real number system (the field of real numbers)—including the commutative, associative, and distributive laws—are also applicable to the complex number system. This makes the basic binary operations with complex numbers straightforward.

Remark: It is worth mentioning that, unlike the real number field which is well ordered (meaning that given any two distinct real numbers, one is always bigger or smaller than the other), no order can be defined on the set of complex numbers that would make it an ordered field. This is a consequence of a mathematical fact proved in Abstract Algebra: in any ordered field, the square of every element must be "positive". This condition does not hold for the field of complex numbers because, for example, the square of the imaginary unit i is i² = -1, which is not a "positive" number.

Basic Operations with Complex Numbers

Our calculators for real and imaginary numbers use the following formulas to perform the four basic operations of addition, subtraction, multiplication and division of complex numbers, as well as parallel sum.

Complex Number Addition & Subtraction Calculator

To add or subtract complex numbers a+bi and c+di simply add or subtract the corresponding real and imaginary parts.

(a+bi)+(c+di) = (a+c)+(b+d)i

(a+bi)-(c+di) = (a-c)+(b-d)i

Complex Number Multiplication Calculator

To multiply complex numbers a+bi and c+di apply the usual laws of algebra (specifically the distributive property) and use the fact that i² = -1.

(a+bi)*(c+di) = (ac-bd)+(ad+bc)i

Complex Number Division Calculator

The calculators use the result of the following algebraic simplification to divide complex numbers a+bi and c+di, first multiply both the numerator and denominator by the conjugate (c-di), and then use the usual laws of algebra noting that i*i = -1.

(a+bi) / (c+di) = (a+bi)(c-di) / [(c+di)(c-di)]

= [(ac+bd) + (bc-ad)i] / (c²+d²)

Complex Number Parallel Sum Calculator

The calculators compute the parallel sum of a+bi and c+di as:

(a+bi) ∥ (c+di) = (a+bi)(c+di)/[(a+bi)+(c+di)]

Advanced Calculations with Complex Numbers

Our calculators use the following formulas to perform advanced calculations involving exponents, logarithms, trigonometric, hyperbolic, and other transcendental (non-algebraic) functions of complex numbers.

Complex Number Exponentiation Calculator

The value of exp(a+bi) or equivalently e^a+bi is calculated by using the celebrated Euler's formula, e^θi = cosθ+i*sinθ. Replacing θ by b, the calculators employ the following formula to perform exponentiation.

e^a+bi = e^a*e^bi

= e^a[cos(b)+i*sin(b)]

Complex Number Logarithm Calculator

To find the natural logarithm, ln(a+bi), first express the complex number in the exponential form, ln(r*e^iφ) (refer to conversion to polar forms), and use the laws of logarithm. Employing the result of the following, the calculators computes the logarithm of a complex number:

ln(a+bi) = ln(r*e^iφ)

= ln(r) + ln(e^iφ)

= ln(r) + φi

The above is called the principal natural logarithm of a+bi. Since the value of φ (argument or phasor) of a complex number is not unique (it's multivalued, differing by integer multiples of 2π), its logarithm is also not unique.

Complex Number Trigonometric Functions Calculator

The calculators evaluate trigonometric functions sin(a+bi) and cos(a+bi) by using the following formulas which contain hyperbolic functions (defined on the real numbers) sinh() (hyperbolic sine) and cosh() (hyperbolic cosine):

sin(a+bi) = sin(a)*cosh(b) + cos(a)*sinh(b)i

cos(a+bi) = cos(a)*cosh(b) + sin(a)*sinh(b)i

Since the other trigonometric functions—tangent, cotangent, secant, and cosecant— are defined in terms of the sine and cosine functions, the calculators employ the following to evaluate these functions:

tan(a+bi) = sin(a+bi)/cos(a+bi)

cot(a+bi) = cos(a+bi)/sin(a+bi)

sec(a+bi) = 1/cos(a+bi)

csc(a+bi) = 1/sin(a+bi)

Complex Number Inverse Trigonometric Functions Calculator

The inverse trigonometric functions—arc sine, arc cosine, arc tangent, arc cotangent, arc secant, and arc cosecant— are denoted by sin⁻¹() or arcsin(), abbreviated as asin(), etc. The calculators find arcsin(), arccos(), and arctan() as follows:

sin⁻¹(a+bi) = -i*ln[(-b+ai) + √(1−(a+bi)²)]

cos⁻¹(a+bi) = -i*ln[(a+bi) + √((a+bi)²-1)]

tan⁻¹(a+bi) = -i/2*ln[((i-(a+bi))/(i+(a+bi))]

For the other inverse trigonometric functions, our calculators use the following formulas:

cot⁻¹(a+bi) = tan⁻¹(1/(a+bi))

sec⁻¹(a+bi) = cos⁻¹(1/(a+bi))

csc⁻¹(a+bi) = sin⁻¹(1/(a+bi))

Complex Number Hyperbolic Functions Calculator

To find the hyperbolic functions, the calculators use the following formulas:

sinh(a+bi) = -i*sin(-b+ai)

cosh(a+bi) = cos(-b+ai)

tanh(a+bi) = sinh(a+bi)/cosh(a+bi)

coth(a+bi) = 1/tanh(a+bi)

sech(a+bi) = 1/cosh(a+bi)

csch(a+bi) = 1/sinh(a+bi)

Complex Number Inverse Hyperbolic Functions Calculator

To find the Inverse hyperbolic functions, the calculators use the following formulas:

sinh⁻¹(a+bi) = ln[a+bi + √((a+bi)^2 + 1)]

cosh⁻¹(a+bi) = ln[a+bi + √((a+bi)^2 - 1)]

tanh⁻¹(a+bi) = 1/2 * ln[(1 + (a+bi)) / (1 - (a+bi))]

coth⁻¹(a+bi) = tanh(1/(a+bi))

sech⁻¹(a+bi) = cosh(1/(a+bi))

csch⁻¹(a+bi) = sinh(1/(a+bi))

Note that these inverse hyperbolic functions are also denoted by asinh(), acosh(), atanh(), acoth(), asech(), and acsch(), respectively.