Schritt-für-Schritt-Anleitung

Add Complex Numbers

To add complex numbers, simply add the real parts together and the imaginary parts together. The formula is: (a + bi) + (c + di) = (a + c) + (b + d)i. For example, (3 + 4i) + (2 + 5i) = (3 + 2) + (4 + 5)i = 5 + 9i.

Subtract Complex Numbers

To subtract complex numbers, subtract the real parts and the imaginary parts separately. The formula is: (a + bi) - (c + di) = (a - c) + (b - d)i. For example, (5 + 9i) - (2 + 3i) = (5 - 2) + (9 - 3)i = 3 + 6i.

Multiply Complex Numbers

To multiply complex numbers, use the distributive property (also known as FOIL) and the fact that i^2 = -1. The formula is: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. For example, (2 + 3i)(4 + 5i) = (2*4 - 3*5) + (2*5 + 3*4)i = (8 - 15) + (10 + 12)i = -7 + 22i.

Divide Complex Numbers

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a + bi is a - bi. The formula is: (a + bi) / (c + di) = ((a + bi)(c - di)) / ((c + di)(c - di)). For example, (3 + 4i) / (2 + 5i) = ((3 + 4i)(2 - 5i)) / ((2 + 5i)(2 - 5i)) = ((6 - 20i + 8i - 20i^2)) / (4 - 25i^2) = ((6 - 12i + 20)) / (4 + 25) = (26 - 12i) / 29.

Convert to Polar Form

To convert a complex number to polar form, use the formula: r = sqrt(a^2 + b^2) and θ = atan2(b, a). For example, for the complex number 3 + 4i, r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 and θ = atan2(4, 3) = 53.13 degrees.

Use a Calculator for Convenience

While it's good to know how to perform complex arithmetic manually, it's often more convenient to use a calculator. You can enter the real and imaginary parts and see the results in rectangular and polar form, along with an Argand diagram. This can save you time and help you avoid common mistakes, such as forgetting to multiply by the conjugate when dividing or mistaking the sign of the imaginary part.

Introduction to Complex Arithmetic

Complex numbers have a real and imaginary part. They can be added, subtracted, multiplied, and divided just like regular numbers. In this guide, we will walk you through how to perform these operations manually.

Understanding Complex Numbers

A complex number is represented as z = a + bi, where a is the real part and b is the imaginary part. The imaginary unit i is defined as the square root of -1.

Rectangular and Polar Form

Complex numbers can be represented in rectangular form (z = a + bi) and polar form (z = r(cosθ + isinθ)). The polar form is often more convenient for multiplication and division.

How to Calculate Complex Arithmetic: Step-by-Step Guide