Algebra: The Beginnings

As mentioned in a previous post, our math abilities are a by-product of our language skills. Indeed, mathematics can be thought of as another language, limited in its subject matter but powerful in developing its own sentences (equations). Maths has its own set of words (symbols), its own dialects (scientists use a different set of symbols and equations than do engineers), and its own syntax and grammar. Algebra is the mathematical version of syntax and grammar, which is why it is such and important subject. It underpins all that is mathematics.

You are already familiar with many of the symbols (words) used in maths:

+ means “plus” or “positive” depending on context

− means “subtract” or “negative” depending on context

× means “multiplied by” or “times”

÷ means “divided by”

= means “equals”

So you already can read math sentences (equations) like 2 + 3 = 5 and you already know that there are rules (syntax) to be followed when writing an equation. For example, 2 3 + 5 = uses the same symbols, but it doesn’t make sense.

Other equations are 3 = 3, 4 = 4, or 2 = 1. Just as in language, maths sentences can be true or false. But let’s start with an obviously true sentence: 3 = 3. Now if I add 1 to both sides of the “=” sign, I get:

3 + 1 = 3 + 1 or 4 = 4, another true equation. Now let’s divide both sides by 2:

4 ÷2 = 4 ÷2 or 2 = 2, Another true equation. I can do this all day, but let’s just do one more:

2 – 2 = 2 – 2 or 0 = 0. In each step, notice that I had to do the same thing on both sides. If there is only one algebra rule you can remember, it’s this one: you can create a new true equation from a true equation by doing the same arithmetic on both sides of the equation. We will see some caveats (cautions) later regarding this, but this is the most important rule in algebra.

Tomorrow, I will introduce the idea of equations with unknown numbers.