Measuring

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We begin to explore Ancient Greek thinking about geometry.

Let’s say one segment $AB$ measures another segment $CD$ is we can find a whole number $N$ so that $CD$ is composed of exactly $N$ copies of $AB$ .

Draw $AB$ of length $3$ cm and $CD$ of length $12$ cm, and show in your drawing that $AB$ measures $CD$ .

Let’s say that two segments $AB$ and $CD$ are commensurable if we can find another segment $EF$ so that $EF$ measures both $AB$ and $CD$ . Otherwise, we will say that $AB$ and $CD$ are incommensurable

Find a pair of line segments which are commensurable. Explain why the Pythagoreans could have concluded that every pair of line segments must be commensurable.

Find a pair of line segments which are incommensurable. Explain why the Pythagoreans were distressed by such a discovery.

From Euclid’s Elements A straight line (segment) is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

What is this definition saying? Express the ratio using modern mathematical symbols.

[Picture]

Can you find any examples of extreme and mean ratio in the diagram of a regular pentagon? Prove your claim!

[Picture]

2025-01-06 15:52:26

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norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
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floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
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Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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