Lesson Seven: Multiplication and Addition

Now we know how to combine two numbers with multiplication, and we know how to combine two numbers with addition, but how do multiplication and addition interact?

Multiplication First:

It is an unfortunate truth that multiplication and addition do not get along as well with each other as addition gets along with addition and multiplication gets along with multiplication. Since both are associative 2+3+4 is unambiguously 9, and 2*3*4 is unambiguously 24. However 2+3*4 takes different values depending on if you interpret it as (2+3)*4, which is 20, or 2+(3*4), which is 14. So, to resolve this ambiguity, we decree that multiplication occurs before addition. This is a common convention upon which we agree, rather than a mathematical truth which we discover, but we should stick with it nonetheless. So, things within parenthesis happen before things outside of them and multiplication happens before addition, these are conventions by which we must agree to live.

Distribution

The other interesting way in which addition and multiplication interact is called distribution. Consider 2(3+4), which is 2*7 or 14. This is the same value as 2*3+2*4 which, since we know to multiply before we add, is 6+8 or still 14. This is not a coincidence, when we switch from doing addition first to doing multiplication first, we must make sure that we multiply both values in the sum by 2. Consider if you had two bank accounts, into one you deposited a certain amount of money each month to save up for a vacation, and into the other you deposited a different amount of money to save for a rainy day. If, after three months, you wished to know how much money was in the accounts total, you could find out in two different ways. You could first figure out how much you deposited each month, by adding the amounts going into each account together, then multiply this monthly deposit size by 3, this way you add first. You could also figure out how much money was in each account individually after three months, by multiplying the deposit to each account by 3 separately, then add these account totals together to obtain the overall total, this way you multiply first. Notice that in the second case, the amount of money you put into the first account must be multiplied by 3 and the amount you put into the second account must be multiplied by 3, we say that the 3 is distributed to each of them.

Lesson Three: In Which Nothing Happens

Suppose people are arriving at school in the morning, and we are keeping track of how many arrive each ten minutes. For a while life is good, people arrive and we are adding them to the total number of people at school. After a while however, school starts and the arrivals stop. We eventually need to add nothing to the number of people at school!

Zero:

If we want to add nothing to our total, it would be helpful to have a numerical representation of nothing. This is of course 0. Anything plus 0 remains the same. This means that 0+1=1, so, since adding one gives us the next number, we can think about 0 as being the number before 1.

Optional Tangent:

For some rather advanced reasons, which I may get to later, 0 is actually a more philosophically sound place to start our numbers than 1. However, 1 is the starting place that I chose for three reasons: the argument for 0 is advanced, but not complicated, and I want to keep things simple for now; if I remember correctly, math education starts with the positive numbers then introduces 0 and I think that is worthy of emulation and; it makes sense historically, giving nothing its own symbol is something that came after the other numbers.

A Note on Equality

I am slightly tired of writing 3+4 is 7, so I think it is time to introduce the equal sign. The symbol '=' means that the values on either side of it are actually the same. As another of my colleagues pointed out a while back, due to its use on a calculator, students sometimes think of '=' as having the same meaning as the Enter key, in that it signifies that a computation need be performed. However '=' is simpler than that, it just indicates that the left side and right side are quantitatively the same. So 3+4=4+3=5+2=1+1+1+1+1+1+1.

Lesson Two: In Which We Add

So far, whenever we have changed whole numbers we have done so by the smallest amount possible, incrementing them 1. However, it is not entirely without precedent that two largish groups of whole numbers should collide and need to be combined. When this happens it would be convenient if we didn't have to do so each 1 at a time. This is the problem which we create addition to solve.

Addition and the Next Number Generator:

Given what we already know, the easiest to define addition is to use the Next Number Generator. Any number plus one is the next number, so 1+1 is 2 and 2+1 is 3. Notice that since 1+1 and 2 are the same thing and 2+1 and 3 are the same thing, 3 must also be 1+1+1. Using the Next Number Generator in this way, any whole number can be considered as an addition of units. Thinking about whole numbers this way means that we already know how to add them together.

Addition:

Consider 3+4, while you may know that it is 7, we do not yet know exactly what 3+4 means. However, we do know that 3 means 1+1+1, and that 4 means 1+1+1+1, so 3+4 must also be 1+1+1+1+1+1+1. Aha, that we know the meaning of! Start with the unit, then find the next number six times, so 3+4 must be 7. Hmm, but if you look at 4+3, converting it to 1's you also get 1+1+1+1+1+1+1 so it looks like 4+3 is the same as 3+4. In fact, adding together any two numbers will be the same, no matter which order you add them in, because of this we say that addition is commutative.

Math is Beautiful

Since it has come up so much recently, I have decided to attempt to explain math in a way closer to how I think it were taught. I am going to try to emphasize the interconnectedness of the concepts and the almost organic way math grows from simple ideas to more complicated ones. In doing so, it is my hope that math not only makes more sense, but also that you come to find some beauty in its intricate, interlocking logic.

My intended audience is not the mathematical novice, but rather the person who, while having been exposed to math, finds it unintuitive and unappealing. My reason for this is simple, while I wish math were taught more like this, I do not feel confident enough to want to be someone's first experience in math. I think that elementary mathematics educators are dealt a bad hand, both having their love of math stifled by the dull methods that they were taught and being trained to teach in the same vein, but it is a testament to their care for their students that any of us take our first tentative steps in the realm of mathematics.

Because my thesis is that math makes more sense when approached from a holistic, interconnected point of view, I advise against skipping posts. Although you may be both familiar and comfortable with the very fundamentals, I may have a new perspective on what is occurring that ties in with later concepts. I do heartily appreciate feedback on what is insufficiently clear, or any other subject, and will endeavor to make revisions incorporating said feedback.

Let us wish each other good luck on our journey!