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Algebra

The Greeks invented the geometry we know today in the period about 500 BC till the first century AD. But they did not have a very advanced use or understanding of numbers.

It was the Arabs of Persia and North Africa who invented algebra, around 500-800 AD. The word comes from Arabic and means 'restoration'. In modern algebra, symbols (typically x and y) are used in place of unknown numbers.

Maths is a universal language

We all learn arithmetic by applying operations (+, -, x, ÷) to numbers. Therefore, if someone asks: 'What is 2 and 3?', we can answer 5. We then learn to write this in a mathematical form: 2 + 3 = 5, known as an equation (because one side 'equals' the other). This is a language understood by everybody in the world. Many people do not understand 'plus', 'equals', 'times', but they understand the algebraic symbols used when writing mathematics.

The Unknown - substituting numbers with letters

But what happens if someone asks: 'What added to 2 equals 5?' Now we have an algebraic equation: 'something' + 2 = 5

Einstein's Famous Equation

Albert Einstein was a very famous physicist who lived from 1879 till 1955. He wrote a famous equation:

$$E = mc^2$$
Einstein portrait

He is saying in this equation that the mass of a particle (he is really talking about things smaller than atoms!) times the speed of light (called c because in Latin speed is 'celeritas') squared (times itself), equals the energy that mass has inside it.

The speed of light is a constant (does not change). If we know the mass (m) we can calculate the energy.

If we know the energy, we can calculate the mass by rearranging the formula:

$$m = E/{c^2}$$

Since we want to know what this 'something' is, we need to create an equation that looks like this:

'something' = 5 - 2 = 3.

Notice how the sign before the 2 changes from positive to negative when we take it over? That is one of the rules of equations: what you do to one side you must do to the other side if the equation is to remain true.

Now, writing 'something' would be ok. If you had two unknowns, you could always write 'something else' for the second one. And 'something else again' for the third one. I guess you can see that pretty soon your equation is going to look pretty strange with all those phrases in it.

To solve this problem, mathematicians decided just to use a shorthand way to name the unknown: and the easiest way is to use a letter (so as not to get confused with more numbers). We could write our equation: 2 + x = 5, so x = 5 - 2 = 3.

There is no rule about which letters we use: 'x' is a popular choice, but it could be any letter at all - and if we run out of the English alphabet, we can even switch to Greek letters!

Inequalities

We can say some algebraic expression is equal to another algebraic expression, such as ax + b = cy + d.

But we can also say something is less than something else (x < y), or less than or equal (x ≤ y).

Similarly, x > y (x is greater than y), or x ≥ y (x is greater than or equal to y).

When playing around with inequalities, we have to be careful with the signs (positive and negative). For example, if we multiply the two sides of an inequality by a negative number, we have to reverse the greater than or less than sign. For example:

Equation 1: $x - 1 < y$

Equation 2: $-x + 1 > -y$

Let's try some numbers to see if this is true: let x = 5, and y = 10. Equation 1 says: 4 < 10. This is true. Equation 2 says: -5 + 1 > -10. This is also true.

Setting up an equation

Sometimes we have enough information to be able to create a general rule or relationship, and this can often be expressed as a mathematical equation.

Example

A girl buys her lunch every day from the school canteen. On Saturday, she buys an icecream for €2.50. If she spends €20.00 every week, how much is each lunch?

Solution: let P be the total price of her food, L = price of the lunch, and I the price of the ice-cream. The equation is:

P = 5L + I

Now we can move the unknown (L) to the left of the equation:

5L = P - I

$L = {P - I}/5$

Since we know P and I, we can solve for L:

$L = {P - I}/5 = {20 - 2.50}/5 = {17.50}/5 = 3.50$

Her lunches cost €3.50 each.

Solving Equations with 2 unknowns

An unknown can be a constant (has always the same value) or a variable (can have different values).

Simultaneous Equations

If we have two unknowns, we need two equations to be able to solve for them.

2x + 3y = 18

x - 3y = -9

If we add the two equations together, we can eliminate the 'y' variable, allowing us to solve for x:

2x + x + 3y - 3y = 9

3x = 9

x = 3

To find y, we substitute this value of x = 3 into one of the equations:

2(3) + 3y = 18

3y = 18 - 6 = 12

y = 4

Remember, you can do anything to an equation, so long as you do it to both sides!

Expanding factors

$(ax + b)^2 = a^2x^2 + 2abx + b^2$

$(ax + b)(ax - b) = a^2x^2 + bx - bx - b^2 = a^2x^2 - b^2$

Content © Andrew Bone. All rights reserved. Created : September 26, 2014

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