# Algebra

# Algebraic Factorisation with Exponents (Indices)

# Algebraic Expansion with Exponents (Indices)

# 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 he $\textit{Method on Exhaustion}$: 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 […]

# Digital Cubes

The number $153$ is equal to the sum of the cubes of its digits: $$1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153$$ In general, $$a^3+b^3+c^3=100a+10b+c$$ There are three other $3$-digit numbers (Digital Cubes) with the same property, excluding numbers like $001$ with a leading zero. Do you want to try […]

# Multiplication using Exponents (Indices)

# Proof by Contradiction

$\textbf{Introduction to Proof by Contradiction}$ The basic idea of $\textit{Proof by Contradiction}$ is to assume that the statement that we want to prove is $\textit{false}$, and then show this assumption leads to nonsense. We then conclude that it was wrong to assume the statement was $\textit{false}$, so the statement must be $\textit{true}$. As an example […]

# 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.

# Number Curiosity

Are these numbers beautiful? \begin{align} \displaystyle 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 &= 98 \\ 123 \times 8 + 3 &= 987 \\ 1234 \times 8 + 4 &= 9876 \\ 12345 \times 8 + 5 &= 98765 \\ 123456 \times 8 + 6 &= 987654 \\ 1234567 […]

# Nicomachus Theorem

$\textbf{Nicomachus}$ discovered “Nicomachus Theorem” interesting number patterns involving cubes and sums of odd numbers. Nicomachus was born in Roman Syria (now, Jerash, Jordan) around 100 AD. He wrote in Greek was a Pythagorean. $\textbf{Nicomachus Theorem: Cubes and Sums of Odd numbers}$ \begin{eqnarray*} 1 &=& 1^3 \\ 3 + 5 &=& 8 = 2^3 \\ 7 […]

# 10 Deadly Common Algebra Mistakes

Share0 Share +10 Tweet0 Common Patterns of Algebra Mistakes Students often make Common Algebra Mistakes due to confusions such as expand and simplify rules, fractions, indices and equations which that lead the students to the wrong answer. Also these error patterns are very basic and quite easily rectified. Check yourself whether you make similar mistakes, […]

# 4 important types of absolute value equation

Share0 Share +10 Tweet0 Absolute Value Equation There are 4 main type of absolute value equation regarding whether there are; absolute value and a static value absolute value and an expression involving unknown pronumerals two absolute values and a value two absolute values in both sides Type 1 of Absolute Value Equation Solve \( | […]

# Harder Factorise

Share0 Share +10 Tweet0 Algebra – Harder Factorise or Factoring can be done by collecting common factors and using sums and products of factors. $$x^2 – (a+b)x + ab = (x-a)(x-b)$$ Question 1 Factorise \( (x^2-3x)^2 -2x^2 + 6x – 8 \). Question 2 Factorise \( (x-1)(x-3)(x+2)(x+4) + 24 \). Question 3 Factorise \( x^4 […]

# Solving Radical Equations

Share0 Share +10 Tweet0 Solving radical equations are required to isolate the radicals or surds to one side of the equations. Then square both sides. Here we have two important checkpoints. Checkpoint 1 for Solving radical equations Make sure the whole both sides are to be squared, but not squaring individual terms. This is what […]

# Inequality with Variables in the Denominator

Share0 Share +10 Tweet0 Solving Inequality with Variables in the Denominator requires special cares due to the direction of the inequalities. Let’s have a look at the following key points. Key Point 1 \( \begin{aligned} \displaystyle \require{color} \frac{1}{x} &\ge 2 \\ \frac{1}{x} \times x &\ge 2 \times x &\color{green} \text{Many of you may think this […]

# Inequality Proofs

Share0 Share +10 Tweet0 Inequality Proofs can be done many ways. For proving \(A \ge B \), one of the easiest ways is to show \(A – B \ge 0 \). Worked Example of Inequality Proofs If \(a, b\) and \(c\) are positive real numbers and \(a+b\ge c \), prove that \( \displaystyle \frac{a}{1+a} + […]

# Rationalising Denominators

Share0 Share +10 Tweet0 Rationalising Denominators simplifies the fractions by multiplying the conjugates of the denominators. Worked Example of Rationalising Denominators Simplify \( \displaystyle\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} \).

# Absolute Value Inequalities

Share0 Share +10 Tweet0 Absolute Value Inequalities are usually proved by the absolute value of a certain value is greater than or equal to it. The square of the value is equal to the square of its absolute value. Proof of Absolute Value Inequalities Prove \(|a| + |b| \ge |a+b|\).