Complex numbers
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, satisfying the equation i 2=−1. The number a is called the real part and the number b is called the imaginary part of the complex number.
Any real number r can be expressed as a complex number r + 0i.
The complex number a + bi can be represented visually as follows:
The phase and norm (modulus) are alternative ways to define a complex number.
Create a complex number
You create a complex number using the COMPLEX function or by entering the number directly in the form a+bi. COMPLEX(a,b) returns the number a + bi. For example, COMPLEX(2,3) returns 2+3i.
Basic mathematical operations
| Multiplication or division of a complex number with a real number | |
| A * r | both real and imaginary part of the complex number A are multiplied by the real number r. |
| A / r | both real and imaginary part of the complex number A divided by the real number r. |
| Addition or subtraction of a complex number with a real number | |
| A + r | the real number r is added to the real part of the complex number A. |
| A − r | the real number r is subtracted from the real part of the complex number A. |
| Addition or subtraction of complex numbers | |
| A + B | the corresponding parts of the two complex numbers are added. |
| A − B | the corresponding parts of the two complex numbers are subtracted. |
| Multiplication of complex numbers | |
| A * B | The multiplication of two complex numbers is defined as follows: $$ (a+bi) (c+di) = (ac-bd) + (bc+ad)i $$ |
| Division of complex numbers | |
| A / B | The division of two complex numbers is defined as follows: $$ \frac{a + bi}{c + di} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i $$ |
Complex number functions
- COMPLEX(a,b) Create complex number
- RE(A) Returns the real part of a complex number
- IM(A) Returns the imaginary part of a complex number
- NORM(A) Returns the norm (or modulus) of a complex number
- CONJ(A) Returns the complex conjugate of a complex number
- PHASE(A) Returns the phase angle (or angular component) of a complex number
- POLAR(rho,theta) Returns the complex number defined by its polar components
- CRND() Returns a random complex number.
- CSQRT(x) Square root function.
Mathematical functions on complex numbers
The following mathematical functions accept a complex number as argument.
ABS, EXP, INV, LN, LOG, POWER, POW10, SQRT, SQUARE
ACOS, ACOSH, ASIN, ASINH, ATAN, ATANH, COS, COSH, SIN, SINH, TAN, TANH
See also
External links
Recommended book
Complex Numbers from A to ... Z
Titu Andreescu
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