Tag Archives: Algebra

Perfect Numbers

Perfect Numbers

Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers with a bizarre property that they can be formed by adding up all the smaller numbers that make up their divisors. The number $6$ is the first perfect number, because it may be divided by […]

Method of Exhaustion

Method of Exhaustion

Eudoxus, Greek mathematician 408-355 BC, developed the idea of seeking mathematical solutions using the $\textit{Method on Exhaustion}$: at this stage, you achieve a more accurate figure. To find an increasingly accurate solution to $\sqrt{2}$, for example, he produced a ladder of numbers. Starting with $1$ and $1$ as the first row, he then added those […]

The Golden Ratio

The Golden Ratio

Since the days of old, artists, as well as mathematicians, have known that there is a special, aesthetically pleasing rectangle with width $1$, length $x$, and the following property: When a square of side $1$ is removed, the rectangle that remains has the same proportions as the original rectangle. Since the new rectangle has a […]

Index Notation

Index Notation

A convenient way to write a product of $\textit{identical factors}$ is to use $\textbf{exponential}$ or $\textbf{index notation}$.Rather than writing $5 \times 5 \times 5 \times 5$, we can write this product as $5^4$. The small $4$ is called the $\textbf{exponent}$ or $\textbf{index}$, and the $5$ is called the $\textbf{base}$.If $n$ is a positive integer, then […]