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Complex numbers: FAQ

Frequently asked questions about complex numbers

What is the imaginary unit i?

The imaginary unit i is a special number that, when squared, gives us minus, 1. It's a key part of complex numbers, which we'll learn about in this unit.

What are complex numbers?

Complex numbers are numbers that have both a "real" part and an "imaginary" part. We can write them as a, plus, b, i, where a is the real part and b, i is the imaginary part.

What are some real-world applications of complex numbers?

Complex numbers are used in a wide variety of fields, such as engineering, physics, and mathematics. They are especially useful for modeling certain types of electrical circuits and for analyzing signals in electronics.

What is the complex plane?

The complex plane is a way of visualizing complex numbers. We plot the real part of the number on the horizontal axis, and the imaginary part on the vertical axis.

How do we add and subtract complex numbers?

To add or subtract two complex numbers, we combine their real parts and their imaginary parts separately. For example, left parenthesis, 2, plus, 3, i, right parenthesis, plus, left parenthesis, 4, minus, 2, i, right parenthesis, equals, 6, plus, i.

How do we multiply complex numbers?

To multiply two complex numbers, we use the distributive property to multiply all the terms together. For example:
(2+3i)(42i)=84i+12i6i2=84i+12i6(1)=8+8i+6=14+8i\begin{aligned} (2 + 3i)(4 - 2i) &= 8 - 4i + 12i - 6i^2 \\\\ &= 8 - 4i + 12i - 6(-1) \\\\ &= 8 + 8i + 6 \\\\ &= 14 + 8i \end{aligned}

Why do we need complex numbers to solve some quadratic equations?

Some quadratic equations don't have any "real" solutions, but they do have complex solutions. For example, x, squared, plus, 1, equals, 0 doesn't have any real solutions, but it does have the two complex solutions x, equals, plus minus, i.

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