# Calculator cis form complex numbers

**Rectangular form (standard form):**

*z*= 3.5355339+3.5355339

**i**

**Angle notation (phasor):**

*z*= 5 ∠ 45°

**Polar form:**

*z*= 5 × (cos 45° +

**i**sin 45°)

**Exponential form:**

*z*= 5 ×

**e**

^{i 0.7853982}= 5 ×

**e**

^{i π/4}

**Polar coordinates:**

r = |

*z*| = 5 ... magnitude (modulus, absolute value)

θ = arg

*z*= 0.7853982 rad = 45° = 0.25π = π/4 rad ... angle (argument or phase)

**Cartesian coordinates:**

Cartesian form of imaginary number:

*z*= 3.5355339+3.5355339

**i**

Real part: x = Re

*z*= 3.536

Imaginary part: y = Im

*z*= 3.53553391

### Calculation steps

- cis φ = cos φ +i*sin φ = e
^{iφ}: cis(45°) = 0.7071068+0.7071068**i**

We assume trigonometric angle argument in degrees. - Multiple: 5 * the result of step No. 1 = 5 * (0.7071068+0.7071068
**i**) = 5 * 0.70710678118655 + 5 * 0.70710678118655**i**= 3.53553391+3.53553391**i**= 3.53553391 +**i**(3.53553391) = 3.5355339+3.5355339**i**

This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers.
As an imaginary unit, use

**i**or**j**(in electrical engineering), which satisfies the basic equation**i**or^{2}= −1**j**. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2^{2}= −1
Complex numbers in the angle notation or phasor (

Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

**polar coordinates**r, θ) may you write as**rLθ**where**r**is magnitude/amplitude/radius, and**θ**is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°).Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

## Basic operations with complex numbers

We hope that working with the complex number is quite easy because you can work with imaginary unit**i**as a variable. And use the definition

**i**to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

^{2}= -1### Addition

It is very simple: add up the real parts (without i) and add up the imaginary parts (with i):This is equal to use rule: (a+b

**i**)+(c+d

**i**) = (a+c) + (b+d)

**i**

(1+i) + (6-5i) = 7-4

**i**

12 + 6-5i = 18-5

**i**

(10-5i) + (-5+5i) = 5

### Subtraction

Again it is very simple: subtract the real parts and subtract the imaginary parts (with i):This is equal to use rule: (a+b

**i**)+(c+d

**i**) = (a-c) + (b-d)

**i**

(1+i) - (3-5i) = -2+6

**i**

-1/2 - (6-5i) = -6.5+5

**i**

(10-5i) - (-5+5i) = 15-10

**i**

### Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and apply**i**.

^{2}= -1This is equal to use rule: (a+b

**i**)(c+d

**i**) = (ac-bd) + (ad+bc)

**i**

(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8

**i**

-1/2 * (6-5i) = -3+2.5

**i**

(10-5i) * (-5+5i) = -25+75

**i**

### Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the denominator's complex conjugate. This approach avoids imaginary unit**i**from the denominator. If the denominator is c+d

**i**, to make it without i (or make it real), multiply with conjugate c-d

**i**:

(c+d

**i**)(c-d

**i**) = c

^{2}+d

^{2}

$c+dia+bi =(c+di)(c−di)(a+bi)(c−di) =c_{2}+d_{2}ac+bd+i(bc−ad) =c_{2}+d_{2}ac+bd +c_{2}+d_{2}bc−ad i$

(10-5i) / (1+i) = 2.5-7.5

**i**

-3 / (2-i) = -1.2-0.6

**i**

6i / (4+3i) = 0.72+0.96

**i**

### Absolute value or modulus

The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5|1-i| = 1.4142136

|6i| = 6

abs(2+5i) = 5.3851648

### Square root

The square root of a complex number (a+bi) is z, if z^{2}= (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is to use De Moivre's formula. Our calculator is on edge because the square root is not a well-defined function on a complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.1213203+2.1213203

**i**

sqrt(10-6i) = 3.2910412-0.9115656

**i**

pow(-32,1/5)/5 = -0.4

pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225

**i**

pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303

**i**

### Square, power, complex exponentiation

Our calculator can power any complex number to an integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...Famous example:

$i_{i}=e_{−π/2}$

i^2 = -1i^61 =

**i**

(6-2i)^6 = -22528-59904

**i**

(6-i)^4.5 = 2486.1377428-2284.5557378

**i**

(6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224

**i**

i^i = 0.2078795764

pow(1+i,3) = -2+2

**i**

### Functions

- sqrt
- Square Root of a value or expression.
- sin
- the sine of a value or expression. Autodetect radians/degrees.
- cos
- the cosine of a value or expression. Autodetect radians/degrees.
- tan
- tangent of a value or expression. Autodetect radians/degrees.
- exp
- e (the Euler Constant) raised to the power of a value or expression
- pow
- Power one complex number to another integer/real/complex number
- ln
- The natural logarithm of a value or expression
- log
- The base-10 logarithm of a value or expression
- abs or |1+i|
- The absolute value of a value or expression
- phase
- Phase (angle) of a complex number
- cis
- is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+
**i** - conj
- the conjugate of a complex number - example: conj(4i+5) = 5-4
**i**

## Complex numbers in word problems:

- The expression 2

The expression (3+i)(1+2i) can be written in the form a+bi, where a and b are integers. What are the values of a and b? - Mistake in expression

While attempting to multiply the expression (2 - 5i)(5 + 2i), a student made a mistake. (2 - 5i)(5 + 2i) = 10 + 4i - 25i - 10i2 = 10 + 4(-1) - 25(-1) - 10(1) = 10 - 4 + 25 - 10 = 21 Complete the explanation and correct the error. Hint: The student incorre - Suppose 5

Suppose z5=2+3i and z6=6+9i are complex numbers and 3 z5 + 7 z6= m+in. What is the value of m and n? - Cis notation

Evaluate multiplication of two complex numbers in cis notation: (6 cis 120°)(4 cis 30°) Write the result in cis and Re-Im notation. - Let z1=x1+y1i

Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2) - Complex expr with fractions

Find 1½ of 16 ÷2⅓+(2¼ of ⅑). Use the correct order of operations. The order is PEMDAS: Parentheses, Exponents, Multiplication, Division (from left to right), addition, and Subtraction (from left to right). - Linear combination of complex

If z1=5+3i and z2=4-2i, write the following in the form a+bi a) 4z1+6z2 b) z1*z2

more math problems »