# Expand and Simplify Linear Expressions

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# 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

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

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

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 […]

# The Sign of One

Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$, $-$, $\times$, $\div$, $($ $)$, etc by the Sign of One. Click the numbers below to see the answers. The first one is done for you.

# Inequalities using Arithmetic Mean Geometric Mean

Arithmetic Mean of $a$ and $b$ is always greater than or equal to the Geometric Mean of $a$ and $b$, for all positive real numbers with with equality if and only if $a = b$. This is also called AM-GM (Arithmetic Mean Geometric Mean) inequality. $\require{color}$  \begin{aligned} \frac{a + b}{2} \ge \sqrt{ab} \text{ or […]