Exponents, Negatives, and Division
In the previous post I set up the rules of arithmetic for addition, subtraction, multiplication, and addition. If you paid really close attention you would have seen one thing pop out that wasn't ever described, which was an exponent. It just shows up all of a sudden in my explanation of the multiplicative inverse. In the post before that, I gave a very basic explanation of what an exponent is, but then said I'd be covering it in more depth. This is that, at long last.
I: Couple Bits of Terminology
We label mathematical statements with some terminology in order to more accurately communicate what we're talking about.
In an addition problem, like 3 + 5 = 8, 3 and 5 are the addends or the summends (some people differentiate the numbers as: 3 is the addend and 5 is the summend, because 5 is "being added to" 3 like it was an action). Generally people will just refer to both numbers as the addends. The result, 8, is the sum.
In a subtraction problem, like 8 - 5 = 3, 8 is the minuend, 5 is the subtrahend, and the result, 3, is the difference.
In a multiplication problem, like 8 ⋅ 5 = 40, 8 and 5 are the multiplicands or the factors (again, most people are gonna use just one of those words, and the word most commonly chosen is factor because, let's face it, multiplicand is a mouthful; so, from now on IMPORTANTLY!!! we will only use the word factor to mean "a number we're multiplying with another number"), and the result, 40, is the product.
In a division problem, like 40 ÷ 5 = 8, 40 is the dividend, 5 is the divisor, and 8 is the quotient.
And finally, in an exponential problem, like 2³ = 8, 2 is the base, 3 is the exponent, and 8 is the product -- just as it is with multiplication, because exponentiation is multiplication:
I'm aware that this is going to be repeating myself a bit, but I think it'll be good to read through anyway!
In the same way that multiplication is repeated addition, exponentiation (raising a number to an exponent or power) is, at first glance, repeated multiplication. We'll say it that way because there's some complication later and even then it's basically true, but it's absolutely good enough for your first exposure.
We write exponents as a superscript number, meaning we are writing it smaller and above where we normally would write, kind of at the top-right corner of the thing before it: in 2³, the 3 is the exponent. The number it follows, 2, is the base that the exponent is attached to.
The exponent tells the base that there are that many factors of itself in this term; in this case, 2³, there are three factors of 2, like so: 2 ⋅ 2 ⋅ 2
There can be more than one base in a term, and not every base will have a visible exponent. If there is a base with no exponent attached, assume its exponent is 1, meaning there is just that one factor of that base in the term. Some examples:
5⋅2³ has one factor of 5 and three factors of 2, making it 5⋅2⋅2⋅2
7³⋅5 has three factors of 7, and one factor of 5, making it 7⋅5⋅5⋅5
2⋅3⁴⋅5² has one factor of 2, four factors of 3, and two factors of 5, making it 2⋅3⋅3⋅3⋅3⋅5⋅5
IIa: How to Read Exponents
So how do you say 7³ -- you know, like, out loud, so other people can understand what you're telling them?
First, we can say "7 raised to the exponent of 3." But that's a bit stuffy. We can replace that word "exponent" with "power," as we saw briefly above.
So now we can say "7 raised to the power of 3." Still stuffy. Change the ordering a little and we can then say that it is "7 raised to the 3rd power."
But if we're only ever "raising" a number to a "power," do we have to say that it's being "raised" every time? No. So, we can then say it is "seven to the third power."
But when else do we ever say something is a power? Never! So, finally, we can reduce this to "seven to the third."
Also, we have a couple of shorthands for the specific 2nd and 3rd powers, which is useful since these are the exponents you will end up using the most often!
When something is raised to the second power, like 9², we can just say that 9 is being squared -- "nine squared." This is because if you find the area of a square whose sides are a length of 9, you would do 9⋅9, which is 9².
Likewise, if something is raised to the third power, like 5³, we can just say that 5 is being cubed -- "five cubed." This is because if you find the volume of a cube whose sides are all a length of 5, you would do 5⋅5⋅5, which is 5³.
IIb: Raising to the Power of Zero
This is actually one of my favorite lessons to teach!
So, when we have 5³, that means we have 5⋅5⋅5, which is 125. In the following, think of the left side as the exponential form of the term, the middle side as a demonstration of the factors in the term, and the right side as the value of the term.
If we take one factor of 5 away, we now have two factors of 5, which makes it 5².
Now we'll take another factor of 5 away, so we now have one factor of five, which makes it 5¹.
Something is funny here. If you didn't notice, our middle section, our demonstration of the factors in the term, has a subtle issue. And that's the word "factors." What are factors? Remember from earlier in I: Couple Bits of Terminology, I said, "IMPORTANTLY!!, that factors are numbers that multiply with other numbers." So, uhhhhhhh.... what number is 5 being multiplied with?
Well, remember way way back in the previous post when I was talking about the Properties of Arithmetic? There was one there that was kind of important: the Identity Property. This states that any number multiplied by the multiplicative identity remains unchanged -- anything times 1 is itself. And any equality in mathematics that works in one direction, must work in the other. This means that if you have a number, it is secretly being multiplied by one.
Every number. All the time. Even the ones you're multiplying by secretly, are secretly multiplied by one.
We just don't write them because, wow, what a hassle it would be to write literally infinite multiplications of 1 every time we want to do math. We'd (literally) never get anything done.
So that 5? Is actually 1⋅5. And, to be honest, the 5¹ is also 1⋅5¹
And this holds true for every row above as well!
This important bit of setup will help us understand what's coming.
Because what if we take another factor of 5 away? Well, we now have zero factors of five: 5⁰
And if we take that factor of 5 away from our demonstration of the factors in the term, we just lose the five from our 1⋅5 leaving us with... 1.
Every number (except for zero -- math nerds fight me! we'll see more about this later) raised to the power of zero is 1. This is the zero property of exponents:
IIc: Raising to a Negative Exponent (Pt. 1)
What if we take another factor of 5 out? Well, there aren't any other factors of 5 in the term, so how would we do that?
If we look at the results of our experiment so far, from one row to the next (top to bottom), we might see that removing a factor of 5 is effectively the same as dividing by 5:
125 ÷ 5 = 25, 25 ÷ 5, = 5, and 5 ÷ 5 = 1.
So the process of removing a factor is the same as dividing by that factor. What's more, we know that every time we remove a factor, the exponent reduces by 1. So, from 5⁰ to the next row where we remove a factor, we'll get 5⁻¹, and we'll have to divide by 5 to get what that is!
So we get the fraction one-fifth as the result. We'll talk more about why this becomes a fraction in a moment, but first! Notice that this means that 5⁻¹ = 1/5, whereas 5¹ = 5. 5 is actually a fraction whose denominator is 1: 5¹ = 5/1. Comparing the results from 5¹ and 5⁻¹, which are 5/1 and 1/5, we see that these are fractions that have the same numbers but are sort of upside down.
When you take a fraction and flip it over like that, you have found its reciprocal. So:
A negative exponent is the reciprocal of the result you get from a positive exponent!
Okay, so, why would any of that be? Well...
III. Some Facts About Division
Fractions are division problems. Whenever you see a fraction, you are actually looking at a division problem. Reading 6/2 as "six halves" or "six divided by two" makes no mathematical difference; these two things have the same value, which is 3.
You ever wonder why the division sign looks like that? I mean. Probably not, right? Why does the addition sign look like that? We made it up! But it turns out, there's a specific reason for the division sign's appearance!
It turns out the division sign, ÷, is just a way of formatting a fraction on a single line! We can do that too with the slash, like in 1/5, but 1÷5 is in fact the same thing! It turns out the dot on the top of the horizontal line in the division sign is the numerator of a fraction, and the dot on bottom is the denominator! So it's telling you to take the number to the left and put it on top, and the number on right and put it on the bottom!
So, yeah, you can think of the fraction line (the "vinculum") as the actual division sign, and the commonly-used division sign as a formatting shorthand to make it easier to write on, for example, a printing press or typewriter.
IIIb. The Word "Negative"
What does the word "negative" mean?
The first instinct of most people is, "less than zero." This is not correct. That is the definition of a "negative number." But that's not the definition of "negative."
The root of "negative" is "negate." To negate something is to reverse the effect of that thing; to cancel, undo, reverse, or revert. A negative stands in opposition to a positive. Simply put:
"Negative" means "opposite"
A negative number is the opposite of that number. What do we mean by that? We mean that it stands on the opposite side of the number line, with zero being at the center. 6 and -6 are both 6 away from zero, but in opposite directions from zero.
Subtraction is negative addition -- the opposite of addition. To subtract is to add a negative number. 9 - 5 is the same as 9 + -5. Here we can see the concept of "negative" can affect operations, and the negative operation is always the inverse operation. The inverse of addition is subtraction.
Likewise, the inverse of multiplication is... division! But in what way is it negative? It's not that the numbers we're dividing become negative. Remember we have described negative exponents as "removing factors." This is the thing that is becoming negative: the exponent on the divisor!
6 ÷ 2 = 6 ⋅ 2⁻¹ = 6 ⋅ (1/2)
So this is where we get to finish our "table" from earlier:
1⋅5⁻³ = 1÷(5⋅5⋅5) = 1/125
