Elementary Algebra

How do we de­scribe re­la­tions be­tween differ­ent things? How can we figure out new true things from true things we already know? How can we find and think about pat­terns we no­tice with num­bers? Alge­bra is a unified frame­work for think­ing about these ques­tions, and gives us lots of tools to help us an­swer them, which work in an ex­tremely wide va­ri­ety of situ­a­tions.

Equations

Alge­bra is based on the ar­ith­metic of num­bers, and the re­la­tions be­tween them. Let’s look at a ba­sic equa­tion:

$$2 + 2 = 4$$
The ‘equals’ sign (=) tells us that both sides of the equa­tion are ac­tu­ally the same. If we have two, and add two more to it, then we’ll have four.

Some other re­la­tions are ‘less than’ (<), for ex­am­ple \(2 < 4\). or ‘greater than’ (>), for ex­am­ple \(5 > 1\).

The right order

In equa­tions, paren­the­sis tell us the right or­der to do things—things in­side of paren­the­sis have to be done be­fore the rest. This is im­por­tant be­cause do­ing things in differ­ent or­ders can give us differ­ent an­swers!

$$2 + (3 \times 4) = 14$$
$$(2 + 3) \times 4 = 20$$

It’s an­noy­ing to have to use paren­the­ses all the time (though it might be helpful if you find your­self get­ting con­fused about some­thing). It would be nice if we could just write \(2 + 3 \times 4\) and have ev­ery­one know that we meant \(2 + (3 \times 4)\) . There’s a stan­dard or­der of op­er­a­tions that ev­ery­one uses so that we don’t have to use too many paren­the­ses.

Balanc­ing the truth

If we know one equa­tion, what are some ways we could get new equa­tions from it, that will still be true? We could make a change to one side, but then the equa­tion would stop be­ing true… un­less we did the same change to the other side also! For ex­am­ple, we know \(2+2=4\). What if we add three to both sides? If we check \((2 + 2) + 3 = 4 + 3\), we can see that it’s still true!

Substitution

One way of get­ting new true things is re­ally im­por­tant. If we know two things are the same, we can always sub­sti­tute one of them for the other, and this au­to­mat­i­cally will give us an equal­ity re­la­tion be­tween the two things!

This is re­ally helpful for break­ing down calcu­la­tions into man­age­able pieces. For ex­am­ple, if we want to calcu­late \(2^3 \times 2^4\), we can first ex­pand \(2^3 = 2 \times 2 \times 2\), and then calcu­late \(2 \times 2 \times 2 = 8\). Now, we com­bine these last two equa­tions to see that \(2^3 = 8\). Often, peo­ple will do both of these in one step, so if you ever are hav­ing a hard time figur­ing out how some­one got an equa­tion, you can try break­ing it down like this to see if that helps. Next, we can do the same thing to see that \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). We can then sub­sti­tute both of these back into the origi­nal ex­pres­sion, to get \(2^3 \times 2^4 = 8 \times 16\). One fi­nal calcu­la­tion lets us see that \(2^3\times 2^4 = 128\).

Nam­ing numbers

What are all those let­ters do­ing in math, any­way?

When you first learn some­one’s name, do you know ev­ery­thing there is to know about them? Some­times, we know that there’s some num­ber that fits in an equa­tion, but we don’t know what par­tic­u­lar num­ber it is yet. It’s re­ally helpful to be able to talk about the num­ber any­way, in fact, giv­ing it a name is of­ten the first step to figur­ing out what it re­ally is.

This kind of name is called a vari­able.

Do­ing lots of things at once

Another way names are use­ful is if we want to say some­thing about lots of num­bers at once! For ex­am­ple, you might no­tice that \(0 \times 3 = 0\), \(0 \times -4 = 0\), \(0 \times 1224 = 0\) and so on. In fact, it’s true that zero times any num­ber is zero. We could write this as \(0 \times \text{any number} = 0\). Or if we need to keep refer­ring to that num­ber, we could give it a shorter name, while men­tion­ing it could be any num­ber: \(0 \times x = 0\), where \(x\) can be any num­ber. This is re­ally use­ful be­cause it al­lows us to ex­press pat­terns much more eas­ily!

Im­por­tant patterns

Th­ese pat­terns hold for all nat­u­ral num­bers, in­te­gers, and ra­tio­nal num­bers, in­clud­ing for vari­ables that are known to be one of these types of num­bers, and much more! For an in-depth ex­plo­ra­tion of these pat­terns and their con­se­quences, see the page on rings.

Commutativity

$$ a + b = b + a$$
$$ a \times b = b\times a$$

Identity

$$ 0 + a = a$$
$$ 1 \times a = a$$

Associativity

$$ (a + b) + c = a + (b + c)$$
$$ (a \times b ) \times c = a \times (b\times c)$$

Distributivity

$$ a \times (b + c) = a\times b + a\times c$$

Ad­di­tive inverse

$$ a + (-a) = a - a = 0 $$

Next steps

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.