Adding complex numbers and Argand Diagrams | Precalculus | Khan Academy

TL;DR

Complex numbers can be added by adding the real parts and imaginary parts separately, and the process can be visualized using vectors on the complex plane.

Transcript

Sal: Let's now think about what it means to add and subtract complex numbers. Let's say I have a complex number A and let's say that it's equal to three plus two I. And let's say I have complex number B that is equal to negative one minus three I. Let's think about what it means to add A and B. Let's say I have a third complex number that is C, tha... Read More

Key Insights

• 🪜 Adding complex numbers involves adding the real and imaginary parts separately.
• ✈️ Complex numbers can be visualized as vectors on the complex plane.
• ✈️ Adding vectors geometrically on the complex plane represents the addition of complex numbers.
• 🚥 The real part of a complex number corresponds to the horizontal component on the complex plane.
• 🚦 The imaginary part of a complex number corresponds to the vertical component on the complex plane.
• 🪜 Adding vectors on the complex plane follows the same principles as adding complex numbers.
• 🚥 Complex number addition is a combination of horizontal and vertical vector addition.

Questions & Answers

Q: How do we add complex numbers?

To add complex numbers, we add the real parts and imaginary parts separately. For example, if we have A = 3 + 2i and B = -1 - 3i, we add 3 and -1 to get 2 as the real part, and add 2i and -3i to get -i as the imaginary part.

Q: Can complex numbers be visualized on the complex plane?

Yes, complex numbers can be represented as vectors on the complex plane. The real part corresponds to the horizontal component, and the imaginary part corresponds to the vertical component. Addition of complex numbers can be visualized by adding the vectors geometrically.

Q: How do we represent complex numbers A and B as vectors on the complex plane?

Complex number A with the coordinates (3, 2) can be represented as a vector with its tail at the origin and its head at point (3, 2). Complex number B with coordinates (-1, -3) can be represented in a similar way.

Q: What is the result of adding complex numbers A and B?

The sum of A and B, represented as vector C, is 2 - i. This means that the real part is 2 and the imaginary part is -1.

Summary & Key Takeaways

• Complex number A is 3 + 2i, complex number B is -1 - 3i, and complex number C is the sum of A and B.

• Adding A and B involves adding the real parts (3 and -1) to get 2, and adding the imaginary parts (2i and -3i) to get -i.

• The addition of complex numbers can be visualized on the complex plane as vector addition.