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# A Complex Mistake?

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Age 14 to 18

*This article is part of our Adventures with Complex Numbers collection*

At the end of the first century AD, the Greek mathematician Heron of Alexandria missed the chance to explore a brand new area of mathematics. He was interested in *frustums* - truncated pyramids with a square base and top, joined by straight sloping sides. In his book *Stereometria* he asked the question: If
the length of the sides of the square base is *a*, the length of the sides of the square top is *b*, and the length of the sloping edges running from top to bottom is *c*, how high is your frustum?

*This frustum is drawn with the values a=10, b=2, c=9 and h=7.*

He came up with this tidy formula (you can see how here) for calculating the height, $h$, of a frustum: $$ h = \sqrt{c^2 - \frac{(a-b)^2}{2}}. $$ He demonstrated his formula first with the example of a frustum with a base with sides of length $a=10$, top with sides of length $b=2$ and sloped sides of length $c=9$. He could easily calculate that the height, $h$, of such a shape was 7, using his formula: $$ h = \sqrt{c^2 - \frac{(a-b)^2}{2}} = \sqrt{81 - \frac{64}{2}} = \sqrt{49} = 7. $$

Heron then tried this approach for a different example, with a base of side length $a=28$, top of side length $b=4$ and sloped edge of length $c=15$. This time his formula gives the height as: $$ h = \sqrt{c^2 - \frac{(a-b)^2}{2}} = \sqrt{225 - 288} = \sqrt{-63}. $$

Heron had managed to come up with an impossible example: no frustum with these measurements could exist. But he had also come up with the first recorded example we have of a calculation involving the square root of a negative number, a *complex number*. The answer was recorded in his book, however, as $\sqrt{63}$. We don't know if he deliberately ignored the strange result or simply
made a mistake and didn't notice. But he definitely missed his complex opportunity.

*You can explore complex numbers for yourself in Open the Door, and discover more about the many applications of complex numbers with this collection of articles on Plus*.