Today real numbers are most often represented by applying (elementary) functions to (decimal) integers. Throughout history, though, arithmetic and propositions involving (positive) real numbers were often considered from a purely geometrical point of view. Real numbers were identified by the length of some line segment and, e.g., the product of two numbers was identified by [...]
Posts tagged visualization
Visualizing the Pythagorean Theorem
Most people are familiar with the Pythagorean theorem: In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. As the name of the theorem implies, it is attributed to Pythagoras, a Greek mathematician who lived around 500 B.C. The theorem is also included [...]
Remembering Trigonometric Addition Formulas
The addition formulas for sine and cosine look like this:
\begin{aligned}
\cos(\alpha + \beta) &= \cos \alpha \cos \beta – \sin \alpha \sin \beta, \\
\sin(\alpha + \beta) &= \cos \alpha \sin \beta + \sin \alpha \cos \beta. \\
\end{aligned}
\cos(\alpha + \beta) &= \cos \alpha \cos \beta – \sin \alpha \sin \beta, \\
\sin(\alpha + \beta) &= \cos \alpha \sin \beta + \sin \alpha \cos \beta. \\
\end{aligned}
I can never remember them [...]
Nice Proof of a Geometric Progression Sum
Consider the geometric series,
s_r = \sum_{k=0}^\infty r^k = 1 + r + r^2 + r^3 + \ldots,
for 0 < r < 1. The goal is to find a closed-form expression for s_r [...]
