One of the most powerful ideas in mathematics is that we can use letters — like \(a, b, n\) or \(x\) — to stand for numbers, shapes or other things. When we do this, we can reason about all such possible objects at once, without knowing exactly which number or shape we are really dealing with.
For example, the statement \[\text{``Let } a,b \text{ be numbers. Then } a+b=b+a."\] is true no matter what numbers \(a\) and \(b\) are. It tells us all of the following at the same time: \[3+5=5+3, \qquad (-10)+(-2)=(-2)+(-10), \qquad 7+0=0+7,\] and many more.
The rule \(a+b=b+a\) does not depend on the “three-ness” of \(3\) or the “five-ness” of \(5\) — it works for any numbers. Of course, we could not let \(a\) be a triangle and \(b\) be a tiger, because we do not know what it means to add a triangle to a tiger! Our rules only apply to objects for which the operations make sense.
This way of using symbols to express rules and patterns is what we call algebra. As long as we follow the rules that numbers follow, our reasoning will stay true. Today we will practise using these symbols to work with the algebra of numbers — it may take effort, but it is an important skill that will help you a lot in your mathematical journey.