So far in mathematics we have studied numbers and how they work. Letters or symbols are often used to represent numbers and you have to work out the value of the letter. On the other hand, symbols can often be used to represent potentially many different numbers or numbers that change known as variables.
This area of mathematics is known as algebra and is crucial to mathematics used in science (ever heard of E = mc squared?). This is where maths starts to get interesting.
Everything we have learnt about numbers can be applied in the exact same way to letters. The only difference is instead of writing numbers and calculating results we write letters and often cannot simplify or change the number as much as we could without letters. For example, when adding the same symbol we can do this:
$$a + a + a = 3a $$
In regular mathematics, say if we made a an actual number like 5 we could write it like this:
$$5 + 5 + 5 = 3 \times 5 = 15 $$
Here we do the same thing to the letter as we did to the actual number, but we are able to change the number whilst we can’t the letter because we don’t know it’s number.
Apart from that, we use exactly the same logic. For example the index laws (exponent laws) use exactly the same logic:
$$x^m + x^n = x^{m+n} $$ $$a^{-5} = \dfrac{1}{a^5} $$
Remember that we still need the same base (so the same symbol…) for the rules to work.
More coming soon