# Subtracting complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy | Summary and Q&A

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February 13, 2014
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Subtracting complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy

## TL;DR

Learn how to subtract complex numbers and visually represent the result on an Argand diagram.

## Key Insights

• 🥳 Complex number subtraction involves subtracting the real parts and imaginary parts separately.
• #️⃣ Visualizing complex number subtraction on an Argand diagram involves adding the negative of the second complex number to the first complex number.
• #️⃣ The resulting complex number can be represented as a vector on the Argand diagram, with the tail of the negative of the second complex number positioned at the head of the first complex number.
• 🤕 The position of the new head of the vector represents the complex number obtained from the subtraction.
• 🥳 Manipulating complex number expressions requires adding the real parts to the real parts and the imaginary parts to the imaginary parts.
• #️⃣ Subtracting a complex number is equivalent to adding the negative of that complex number.
• 🐬 The negative of a complex number is obtained by flipping the vector representation over the origin on an Argand diagram.

## Transcript

Voiceover:So we've plotted two complex numbers right over here on this argand diagram. And what I want to think about is what if we defined a third complex number, C, as being equal to the complex number, A, minus the complex number, B. What is C going to look like if we were to write it as just a sum of its real and imaginary parts and what would ... Read More

### Q: How can complex numbers be subtracted?

Complex numbers can be subtracted by subtracting their real parts and imaginary parts separately. For example, to subtract (6 + 2i) from (2 - i), we subtract the real parts (2 - 6 = -4) and the imaginary parts (-1 - 2i = -3i), resulting in the complex number -4 - 3i.

### Q: How can complex number subtraction be visualized on an Argand diagram?

To visualize complex number subtraction on an Argand diagram, first plot the original complex numbers A and B as vectors. Then, add the negative of the second complex number (-B) to the first complex number (A). This is equivalent to subtracting B from A. The new head of the resulting vector C represents the complex number obtained from the subtraction.

### Q: What does the negative of a complex number look like on an Argand diagram?

The negative of a complex number can be visualized by flipping the complex number vector over the origin. For example, if the complex number B is (2 - i), its negative (-B) would be (-2 + i), and when plotted on the Argand diagram, it would be reflected across the origin.

### Q: How can the complex number subtraction be represented on an Argand diagram?

The complex number obtained from subtraction, C, can be represented as a vector on the Argand diagram. The tail of C is positioned at the origin, the tail of -B is placed at the head of A, and the resulting vector C extends from the tail of -B to a new head, which represents the complex number C.

## Summary & Key Takeaways

• The video teaches how to subtract complex numbers by determining the difference between their real parts and imaginary parts.

• It explains how to visualize complex number subtraction on an Argand diagram by adding the negative of the second complex number to the first complex number.

• The resulting complex number can be represented as a vector on the diagram.