Where did using Pythagorean Theorem to find the absolute value of a complex number come from?

Question

Where did the idea of the Y axis being the complex axis come from? Is this arbitrary, or substantiated? I am thinking about back when I learned about the complex plane. For example $|3+4i|$ is known to be 5. Why the Pythagorean Theorem? Is there any mathematical proof that $|3+4i| = 5$? Where did using the Y axis to represent the complex axis, where did that come from?

Answer 1

If $a + bi$ is a complex number, then the complex conjugate $a-bi$ satisfies the same algebraic properties relative to $\mathbb R$ as $a+bi$. It turns out that if you multiply a complex number with its conjugate, you get a real number: $$(a+bi)(a-bi) = a^2 -abi + abi - b^2i^2 = a^2+b^2,$$ which matches the Pythagorean identity that is mystifying you. Defining $$|a+bi| = \sqrt{(a+bi)(a-bi)}$$ is then convenient for many reasons, including the corollary that $|a+bi||c+di| = |(a+bi)(c+di)|$.

In particular, if $|a+bi| = 1$, then multiplication by $a+bi$ preserves distance (because $|(a+bi)(c+di)| = |c+di|$, and you can subtract two complex numbers to get the distance between them). This gives you a hint toward the fact that multiplying complex numbers with absolute value $1$ corresponds to rotation.

I believe that historically, the geometric connection between complex numbers and rotation was a huge step toward the acceptance of complex numbers as a meaningful notion of "number". Ethan Bolker or someone else would have to confirm or disconfirm that for me.

Answer 2

Using a "fictitious" square root of $-1$ essentially started

in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano).

Later in that wikipedia entry:

The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.