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The complex numbers

Arithmetic with complex numbers in standard form — sums, products, quotients — plus finding the flaw in a bogus proof that 1 = -1 via square roots.

Matrix Theory and Linear Algebra, The complex numbers

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3 exercises

Exercise A.1.1

Let $z=2+7i$ and let $w=3-8i$ . Compute $z+w$ , $z-2w$ , $zw$ , and $\frac{w}{z}$ .

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Exercise A.1.4

Use the properties of complex numbers to prove that if $z$ is a complex number, then there exists a complex number $w$ with $\left|w\right|=1$ and $wz=\left|z\right|$ .

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Exercise A.1.5

I claim that $1=-1$ . Here is why.

What is wrong with this argument?

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.