Positive and Negative Exponents

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I know this is very basic but I’ve never been able to wrap my head around them. Please help. 😀

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Start with 1. By definition this is the zero exponent of all non zero numbers. Let’s use A to denote the base.

A^0 = 1

A^1 = 1 * A (multiply by one A)

A^2 = 1 * A * A (multiply by two A’s)

A^3 = 1 * A * A * A (and so forth)

A^(-1) = 1 / A (divide by one A)

A^(-2) = (1/A)/A = 1/(A*A) (divide by two A’s)

A^(-3) = ((1/A)/A)/A) = 1/ (A*A*A) (divide by three A)

Exponentiation is just repeated multiplication (positive) or division (negative). Eg

2^8 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256

2^-8 = 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 = 0.00390625

Positive exponents are easy. How many times you multiply the number by itself. 5^2 = 5 * 5 = 25. 5^3 = 5 * 5 * 5= 125. Negative exponents are a little trickier. The easiest way is to turn the negative symbol into 1/. 5^-2= 1/5^2 = 1/25. So 5^-3 is 1/5^3 or 1/125.

Positive exponents is the number of times you multiply something by itself. So x^1 = x, x^2 = x*x, x^3 = x*x*x, and so on.

Now you notice an interesting pattern : for example, (x^2)*(x*3) = (x*x)*(x*x*x) = x^5. And 5 = 2+3. That works for any exponent : (x^5)*(x^10) = x^15, and for any positive numbers n and m, (x^n)*(x^m) = x^(n+m).

Now you can wonder : why doing that only with positive numbers ? What if I try with zero ? Or negative numbers ? x^0 must verify the equation, for any n : (x^0)*(x^n) = x^(n+0) = x^n. If you multiply something by x^n to obtain x^n, the something must be 1. So x^0 = 1.

Now let’s try with x^(-1). Let’s take an example : (x^-1)*(x^3) = x^(3-1) = x^2. So x^(-1) = (x^2) / (x^3) = 1/x.

And you can go on with every negative exponent : x^(-4) = 1/(x^4) = 1/x/x/x/x. That’s not some arbitrary convention, that’s true because of the fundamental property of exponents : (x^n)*(x^m) = x^(n+m).

> I know this is very basic but I’ve never been able to wrap my head around them

One day mathematicians got together and said:

– We’ll define anything^0 to be equal to 1
– We’ll define anything^-something to be equal to 1 / (anything^something)

According to these definitions, 417^0 = 1. Also 5555^0 = 1. Also (7 – 25)^0 = 1.

And if you have 2^4 = 16, so 2^-4 = 1/16. Likewise 3^3 = 27, so 3^-3 = 1/27.

Maybe your difficulty is that you’re expecting negative and zero exponents to somehow “arise from nature.” And that’s not what happened. People *decided* what they mean.

– Now maybe you’re wondering, “Hmm that seems random. Why would they decide on those definitions?”

Let’s say you have 2^4 x 2^3. If you’re bad at mental math, you might punch 2^4 into your calculator and get 16, then punch 2^3 into your calculator and get 8, then use your calculator one more time to multiply 16×8 to get a final answer of 2^4 x 2^3 = 128.

But we can look at this problem another way. The first term is four twos multiplied together, and the second term is three twos multiplied together. If you multiply the two terms together, you end up with (2x2x2x2)x(2x2x2). But multiplication can happen in any order (it’s associative), so you can drop parentheses and get 2x2x2x2x2x2x2. In other words, 2^4 x 2^3 = 2^7. And this might be useful; you can punch 2^7 into your calculator, and get 128, solving the problem in only one calculator punch.

There’s nothing special about 2, 3, and 4. In general a^b x a^c = a^(b+c) for any a, b, and c. Let’s call this the *exponent addition rule*.

When the mathematicians made up the definitions for zero and negative exponents, they said:

– The exponent addition rule is very nice. We use it all the time.
– We should define zero and negative exponents in such a way that we can use the rule with them.

It turns out that, once they made the decision that zero and negative exponents should work with the exponent addition rule, the definitions at the top of this post are the only ones that work.

Pick any random number, say 5. Say you’re not sure how to define 5^0, but you want the exponent addition rule to work. Clearly 5 = 5^1. And of course 1 = 1+0, so you can substitute and get 5 = 5^(1+0). And then if the laws of exponents hold, 5^(1+0) should be equal to 5^1 x 5^0. So we have 5 = 5^1 = 5^(1+0) = 5^1 x 5^0. We don’t know what 5^0 should be, but when you multiply it by 5^1, it should give you 5. The only number that fits the bill is 1, so now you know: 5^0 should be defined to be 1.

Of course there’s nothing special about 5. This logic works for any number. And a similar thing happens for negative exponents:

Say you’re not sure how to define 2^-8. But 2^(-8 + 8) = 2^0, which we’ve already figured out is 1. But the exponent addition rule says 2^(-8 + 8) = 2^-8 x 2^8. So we know 1 = 2^0 = 2^(-8 + 8) = 2^-8 x 2^8. The main point of which is 1 = 2^-8 x 2^8, so we know that 2^-8 must be some number that, when multiplied by 2^8, gives you 1. The only number that fits the bill is 1/2^8 (which works out to 1/256, but that’s beside the point).

Of course there’s nothing special about 2 and 8. This logic works for any number.

Philosophy moment: If you think about it, the definitions of zero and negative exponents “have to be” what they are. In other words, they do sort of “arise from nature” after all! It’s an old philosophical debate:

– Is mathematics *an invention created by humans*?
– Or is mathematics *a discovery of something that already exists in nature*?

Math underpins basically all of science, engineering, technology, and seemingly the physics of our universe itself. Most people learn basic math at a very young age. But exactly what math *is*, is a very thorny question!