Complex Numbers Calculator
Use this complex numbers calculator to add, subtract, multiply, divide, find the conjugate, calculate the modulus, find the argument, convert to polar form, raise a complex number to a power, or find complex roots. A complex number has the standard form \(z=a+bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i^2=-1\). Enter the real and imaginary parts below, choose the operation, and the calculator will show the result with formula-based steps.
Calculate Complex Numbers
Enter the first complex number as \(z_1=a+bi\). For operations that need a second complex number, enter \(z_2=c+di\). For powers and roots, enter the integer value \(n\).
Result
The calculator displays rectangular form \(a+bi\) when possible. Polar form uses \(r(\cos\theta+i\sin\theta)\) and \(re^{i\theta}\).
Complex Number Formula
A complex number is a number with a real part and an imaginary part. The standard rectangular form is:
Where \(a\) is the real part, \(b\) is the imaginary coefficient, and \(i\) is the imaginary unit. The imaginary unit is defined by:
If \(z=3+4i\), then the real part is \(3\) and the imaginary part is \(4\). If \(z=-5-2i\), then the real part is \(-5\) and the imaginary part is \(-2\).
Rectangular form
The form \(a+bi\) is called rectangular form or standard form. It is best for addition, subtraction, and identifying real and imaginary parts.
Polar form
The form \(r(\cos\theta+i\sin\theta)\) or \(re^{i\theta}\) is best for powers, roots, rotation, and geometric interpretation.
How to Use the Complex Numbers Calculator
- Choose the operation you want: addition, subtraction, multiplication, division, conjugate, modulus, argument, polar form, power, or roots.
- Enter the first complex number as real part \(a\) and imaginary part \(b\).
- For operations with two complex numbers, enter the second complex number as real part \(c\) and imaginary part \(d\).
- For powers and roots, enter the integer value \(n\).
- Choose degrees or radians for angle-based results such as argument, polar form, powers, and roots.
- Click Calculate Complex Number and read the answer with the displayed formula steps.
Complex Number Operations and Formulas
The most important complex number formulas are listed below. These are the same rules used by the calculator.
| Operation | Formula | Meaning |
|---|---|---|
| Addition | \((a+bi)+(c+di)=(a+c)+(b+d)i\) | Add the real parts and add the imaginary parts. |
| Subtraction | \((a+bi)-(c+di)=(a-c)+(b-d)i\) | Subtract real parts and subtract imaginary parts. |
| Multiplication | \((a+bi)(c+di)=(ac-bd)+(ad+bc)i\) | Use expansion and the fact that \(i^2=-1\). |
| Division | \(\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{c^2+d^2}\) | Multiply numerator and denominator by the conjugate of the denominator. |
| Conjugate | \(\overline{z}=\overline{a+bi}=a-bi\) | Change the sign of the imaginary part. |
| Modulus | \(|z|=\sqrt{a^2+b^2}\) | Distance from the origin to the point \((a,b)\). |
| Argument | \(\theta=\arg(z)=\tan^{-1}\left(\frac{b}{a}\right)\) | Angle the complex number makes with the positive real axis, adjusted by quadrant. |
| Polar form | \(z=r(\cos\theta+i\sin\theta)=re^{i\theta}\) | Represent the complex number using distance and angle. |
| Power | \(z^n=r^n(\cos n\theta+i\sin n\theta)\) | Use De Moivre’s Theorem. |
| Roots | \(w_k=r^{1/n}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)\) | Find all \(n\)th roots, where \(k=0,1,\ldots,n-1\). |
Worked Examples
Example 1: Add complex numbers
Add \(z_1=3+4i\) and \(z_2=1-2i\).
The answer is \(4+2i\). Addition is simple because real parts combine with real parts and imaginary parts combine with imaginary parts.
Example 2: Multiply complex numbers
Multiply \(z_1=3+4i\) and \(z_2=1-2i\).
The answer is \(11-2i\). The key step is replacing \(i^2\) with \(-1\).
Example 3: Divide complex numbers
Divide \(3+4i\) by \(1-2i\).
The answer is \(-1+2i\). Complex division uses the conjugate of the denominator to remove the imaginary term from the denominator.
Example 4: Find the modulus
Find the modulus of \(z=3+4i\).
The modulus is \(5\). Geometrically, \(3+4i\) is the point \((3,4)\), and its distance from the origin is \(5\).
Example 5: Convert to polar form
Convert \(z=3+4i\) to polar form.
The polar form is approximately \(5(\cos53.13^\circ+i\sin53.13^\circ)\), or equivalently \(5e^{i53.13^\circ}\) if the angle is written in degrees for display.
Example 6: Use De Moivre’s Theorem
If \(z=r(\cos\theta+i\sin\theta)\), then:
For \(z=3+4i\), we have \(r=5\) and \(\theta\approx53.13^\circ\). So \(z^2=25(\cos106.26^\circ+i\sin106.26^\circ)\), which equals \(-7+24i\) in rectangular form.
Complete Guide to Complex Numbers
Complex numbers extend the real number system by adding the imaginary unit \(i\), where \(i^2=-1\). This single definition makes it possible to solve equations that have no real solutions, such as \(x^2+1=0\). In the real number system, no number squared equals \(-1\), because the square of a positive real number is positive and the square of a negative real number is also positive. Complex numbers solve this limitation by introducing \(i=\sqrt{-1}\), so the solutions of \(x^2+1=0\) become \(x=i\) and \(x=-i\).
The standard form of a complex number is \(a+bi\). The part \(a\) is called the real part, and the part \(b\) is called the imaginary part. The imaginary part is the coefficient of \(i\), not the whole term \(bi\). For example, in \(7-5i\), the real part is \(7\) and the imaginary part is \(-5\). When \(b=0\), the complex number is purely real. When \(a=0\), the number is purely imaginary.
Complex numbers are not just abstract symbols. They can be represented as points on a plane called the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. The complex number \(a+bi\) corresponds to the point \((a,b)\). This geometric interpretation is extremely useful because addition, subtraction, modulus, argument, polar form, multiplication, and powers all have visual meaning. A complex number is both an algebraic object and a geometric object.
Real and imaginary parts
The real part and imaginary part are written using the notation \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\). If \(z=a+bi\), then:
This notation is especially useful in higher mathematics, engineering, and physics because complex quantities often represent two related components. In electrical engineering, for example, complex numbers can represent phase and amplitude. In signal processing, real and imaginary parts appear in Fourier transforms. In algebra, real and imaginary components help solve equations by comparing matching parts.
Adding and subtracting complex numbers
Adding complex numbers is similar to adding vectors. You add the horizontal components and vertical components separately. If \(z_1=a+bi\) and \(z_2=c+di\), then:
Subtraction works the same way but subtracts each component:
This is why rectangular form is the best form for addition and subtraction. The real and imaginary components are already separated, so the operation is direct. Geometrically, adding complex numbers corresponds to vector addition in the complex plane. Subtracting complex numbers corresponds to finding the displacement from one point to another.
Multiplying complex numbers
Complex multiplication uses ordinary algebra plus the rule \(i^2=-1\). If \(z_1=a+bi\) and \(z_2=c+di\), then:
This formula is important because it shows how the real and imaginary parts mix. Unlike addition, multiplication does not simply multiply real parts and imaginary parts separately. The term \(bd i^2\) becomes \(-bd\), so it contributes to the real part. This is the most common place where students make errors in complex arithmetic.
Geometrically, complex multiplication has a beautiful interpretation. If two complex numbers are written in polar form, multiplying them multiplies their moduli and adds their arguments. In other words, multiplication combines scaling and rotation. If one complex number has magnitude \(r_1\) and angle \(\theta_1\), and the other has magnitude \(r_2\) and angle \(\theta_2\), then their product has magnitude \(r_1r_2\) and angle \(\theta_1+\theta_2\).
Dividing complex numbers
Complex division is usually performed by multiplying by the conjugate of the denominator. If the denominator is \(c+di\), its conjugate is \(c-di\). Multiplying the denominator by its conjugate gives:
The imaginary terms cancel, leaving a real denominator. The full division formula is:
This formula is valid as long as \(c+di\ne0\). In other words, the denominator cannot be \(0+0i\). Division by zero is undefined in the complex number system just as it is in the real number system.
Complex conjugate
The conjugate of a complex number changes the sign of the imaginary part:
If \(z=5-7i\), then \(\overline{z}=5+7i\). The conjugate is useful for simplifying division, finding moduli, and proving identities. A key property is:
For \(z=a+bi\), this becomes \((a+bi)(a-bi)=a^2+b^2\). That expression is real and nonnegative, which is why conjugates are so useful in complex division.
Modulus of a complex number
The modulus of a complex number is its distance from the origin in the complex plane. If \(z=a+bi\), then:
This is the distance formula applied to the point \((a,b)\). For example, \(|3+4i|=5\). The modulus is always nonnegative. It equals zero only when the complex number is \(0+0i\). The modulus is also called the absolute value or magnitude of a complex number.
The modulus follows useful rules. For complex numbers \(z_1\) and \(z_2\), we have \(|z_1z_2|=|z_1||z_2|\). Also, if \(z_2\ne0\), then \(\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\). These rules match the geometric idea that multiplication scales distances.
Argument of a complex number
The argument of a complex number is the angle it makes with the positive real axis. If \(z=a+bi\), then the argument is often written as \(\arg(z)\). A basic formula is:
However, this formula must be interpreted carefully because the quadrant matters. A point in Quadrant I and a point in Quadrant III can have the same ratio \(\frac{b}{a}\), but their angles differ by \(\pi\) radians or \(180^\circ\). A calculator normally uses the two-argument arctangent function, often called \(\operatorname{atan2}(b,a)\), to place the angle in the correct quadrant.
The argument is not unique because adding a full turn gives the same direction. If \(\theta\) is an argument of \(z\), then \(\theta+2\pi k\) is also an argument for any integer \(k\). The principal argument usually chooses one standard angle, often in the interval \((-\pi,\pi]\) or \([0,2\pi)\), depending on the convention.
Polar form
Polar form represents a complex number using its modulus and argument. If \(r=|z|\) and \(\theta=\arg(z)\), then:
Euler’s formula connects this to exponential form:
So the complex number can also be written as:
Polar form is especially powerful for multiplication, division, powers, and roots. In rectangular form, powers can become algebraically messy. In polar form, powers are straightforward because the modulus is raised to a power and the angle is multiplied by that power.
Converting rectangular form to polar form
To convert from \(a+bi\) to polar form, calculate:
Then write \(z=r(\cos\theta+i\sin\theta)\). For example, \(3+4i\) has \(r=5\) and \(\theta\approx53.13^\circ\). So \(3+4i=5(\cos53.13^\circ+i\sin53.13^\circ)\).
Converting polar form to rectangular form
To convert from polar form to rectangular form, use:
Then write \(z=a+bi\). This conversion is useful when an answer is easier to find in polar form but must be reported in standard rectangular form. The calculator above displays both forms for polar-related operations whenever possible.
De Moivre’s Theorem
De Moivre’s Theorem is one of the most important formulas for powers of complex numbers. If:
Then for an integer \(n\):
This theorem turns a difficult algebraic expansion into a simple polar operation. Instead of repeatedly multiplying \(a+bi\) by itself, you raise the modulus to the power and multiply the argument by the power. The result can then be converted back to rectangular form using cosine and sine.
Roots of complex numbers
Complex roots are found using polar form. If \(z=r(\cos\theta+i\sin\theta)\), then the \(n\)th roots are:
This formula gives \(n\) different roots for a nonzero complex number. Geometrically, the roots are equally spaced around a circle centered at the origin. The radius of that circle is \(r^{1/n}\), and the angles are separated by \(\frac{2\pi}{n}\). This beautiful structure is one reason complex numbers are so important in algebra and geometry.
The complex plane
The complex plane gives a visual interpretation of complex numbers. The real part determines the horizontal position, and the imaginary part determines the vertical position. The number \(4+2i\) is plotted at \((4,2)\). The number \(-3+5i\) is plotted at \((-3,5)\). The origin represents \(0+0i\). The horizontal real axis contains all real numbers, while the vertical imaginary axis contains pure imaginary numbers.
This plane turns complex arithmetic into geometry. Addition moves points like vectors. Multiplication scales and rotates. Conjugation reflects across the real axis. Modulus measures distance from the origin. Argument measures direction from the positive real axis. Polar form describes a point by distance and angle instead of horizontal and vertical components.
Why complex numbers are useful
Complex numbers are useful because many mathematical and scientific systems naturally involve both magnitude and direction, oscillation and phase, or two linked components. In electrical engineering, alternating current is often modeled using complex numbers because voltage and current can have phase shifts. In signal processing, the Fourier transform uses complex exponentials to represent waves. In quantum mechanics, wave functions are complex-valued. In control systems, complex roots determine stability and oscillation behavior. In algebra, complex numbers guarantee that every nonconstant polynomial has roots in the complex number system.
This last fact is known as the Fundamental Theorem of Algebra. It says that every polynomial of degree \(n\) with complex coefficients has exactly \(n\) complex roots, counting multiplicity. Without complex numbers, equations such as \(x^2+1=0\) would have no solution. With complex numbers, algebra becomes more complete and consistent.
Common mistakes with complex numbers
A common mistake is forgetting that \(i^2=-1\). For example, when multiplying \((2+3i)(4+5i)\), the term \(15i^2\) becomes \(-15\), not \(15i\). Another mistake is treating the imaginary part as always positive. In \(6-8i\), the imaginary part is \(-8\), not \(8\). A third mistake is forgetting to multiply by the conjugate when dividing complex numbers.
Students also confuse modulus and argument. The modulus is the distance from the origin, while the argument is the angle. They describe different features of the same complex number. Another common error is using \(\tan^{-1}(b/a)\) without checking the quadrant. For example, \(1+i\) and \(-1-i\) have the same ratio \(b/a\), but they point in opposite directions. Correct angle calculation must account for signs of both \(a\) and \(b\).
Choosing the best form
Rectangular form \(a+bi\) is usually best for addition and subtraction. It is also good for identifying real and imaginary parts. Polar form \(r(\cos\theta+i\sin\theta)\) is usually best for multiplication, division, powers, and roots. Exponential form \(re^{i\theta}\) is especially useful in advanced mathematics, differential equations, signal processing, and physics. A strong complex number skill is knowing when to switch forms.
For example, multiplying two complex numbers in rectangular form requires expansion. Multiplying in polar form only requires multiplying moduli and adding angles. Finding roots in rectangular form is difficult, but in polar form it becomes a direct formula. This is why the calculator includes rectangular operations and polar operations in one tool.
Complex numbers in equations
Complex numbers often appear when solving quadratic equations. The quadratic formula is:
If the discriminant \(b^2-4ac\) is negative, the square root is imaginary. For example, the equation \(x^2+4=0\) gives \(x^2=-4\), so \(x=\pm2i\). In real-number algebra, this equation has no real solution. In complex-number algebra, it has two solutions.
Complex numbers and rotation
Multiplication by a complex number can rotate points in the plane. Multiplying by \(i\) rotates a point by \(90^\circ\) counterclockwise. For example:
The point \((a,b)\) becomes \((-b,a)\), which is a \(90^\circ\) counterclockwise rotation. This geometric interpretation connects complex numbers to transformations, graphics, robotics, and physics.
Summary
Complex numbers are numbers of the form \(a+bi\), where \(i^2=-1\). They can be added, subtracted, multiplied, divided, graphed, converted to polar form, raised to powers, and used to find roots. Rectangular form is best for component-based operations, while polar form is best for angle and magnitude operations. The calculator above gives quick results while showing the formulas, helping students understand both the arithmetic and the geometry behind complex numbers.
Quick Reference: Complex Number Forms
| Form | Expression | Best use |
|---|---|---|
| Rectangular form | \(z=a+bi\) | Addition, subtraction, real and imaginary parts |
| Coordinate form | \(z\leftrightarrow(a,b)\) | Graphing on the complex plane |
| Polar form | \(z=r(\cos\theta+i\sin\theta)\) | Geometry, powers, roots, multiplication, division |
| Exponential form | \(z=re^{i\theta}\) | Advanced algebra, calculus, differential equations, signals |
Related Calculators and Study Tools
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Complex Numbers Calculator FAQs
What is a complex number?
A complex number is a number of the form \(a+bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i^2=-1\).
How do you add complex numbers?
Add real parts together and imaginary parts together: \((a+bi)+(c+di)=(a+c)+(b+d)i\).
How do you multiply complex numbers?
Expand like algebra and use \(i^2=-1\). The formula is \((a+bi)(c+di)=(ac-bd)+(ad+bc)i\).
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. For \(\frac{a+bi}{c+di}\), use the conjugate \(c-di\).
What is the conjugate of a complex number?
The conjugate of \(a+bi\) is \(a-bi\). It changes the sign of the imaginary part.
What is the modulus of a complex number?
The modulus is the distance from the origin in the complex plane. For \(z=a+bi\), the modulus is \(|z|=\sqrt{a^2+b^2}\).
What is the argument of a complex number?
The argument is the angle the complex number makes with the positive real axis. It is commonly calculated using \(\operatorname{atan2}(b,a)\) so the correct quadrant is used.
What is polar form of a complex number?
Polar form writes a complex number as \(z=r(\cos\theta+i\sin\theta)\), where \(r=|z|\) and \(\theta=\arg(z)\).
What is De Moivre’s Theorem?
De Moivre’s Theorem says \([r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta)\). It is used for powers of complex numbers.
How many nth roots does a complex number have?
A nonzero complex number has \(n\) distinct \(n\)th roots. These roots are equally spaced around a circle in the complex plane.
