Why every number to the power of 0 is equal to 1?

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I’m too dumb to get it

In: Mathematics

31 Answers

Anonymous 0 Comments

To anyone who wrote on this thread – thank you all for your answers! They are all really helpful. Unfortunately, I don’t have time to answer to every one of you, but I really do appreciate your help! Thank you!

Anonymous 0 Comments

Here’s another deeper angle I haven’t seen explored yet:

Consider that “fractional powers” can mean square/cube/etc roots, so 2^(1/2) = about 1.41. 2^(1/3) = about 1.26. Even when you take the 10th root of 2, it’s 1.07. So if you take the general case, N^(1/M), you can think of it as if you’re taking the M^(th) root of a number N. 1/infinity is 0, so it’s describing what happens if you take the “infinite root” of a number, and it just so works out that it’s always 1, even with fractions.. the 10th root of 0.1 is 0.99. It’s not something that makes practical sense, per se, so that’s probably why we’re just given the rule without much explanation.

Anonymous 0 Comments

Increasing the power by 1 will add another multiplicator of the value of the base. So while 2^2 is 2*2, 2^3 is 2*2*2. So again, if you have the base 2 and increase the power by one you just add a *2. If you reduce the power by 1 it is taking away a “*2” which is the same as divinding by the base.

So if 2^1 is just 2. And you reduce the power by one then you get 2^0 which is the just dividing the base value by itself which always results in a 1.

Anonymous 0 Comments

You have three sheep, which is 1+1+1, and you ask what happens as you remove them. You go 3, then 2, then 1 and then, when you remove the last one you know that for additions 3 is actually 1+1+1+0+0+0+0, etc.

But exponents are one floor above sums. At that floor, 8 is 2·2·2. You start removing twos and ask yourself why am I not left with a zero?
The answer is that 8 isn’t 2·2·2·0·0·0, etc. It’s actually 2·2·2·1·1·1·1·1. Because exponents are in the multiplication floor, and in that floor “nothing” is a bunch of ones.

Anonymous 0 Comments

Go to [graphsketch.com](https://graphsketch.com), and let it draw a few functions such as 1^x, 2^x, 3^x, 0.5^x, 1.1^x. You’ll notice that every line goes through the point at (0,1).

It’s not an explanation… but at least it visualizes how exponents work.

Anonymous 0 Comments

*(This is a bit like those* **”think of a number”** *tricks, where it doesn’t matter what number you start with, the answer always ends up the same. And I’m deliberately putting this into words rather than algebraic notation.)*

* A number to the power 1 is the number itself (by definition).

* Each time you multiply a power of the number by the number itself, you add 1 to the power (again, by definition).

* *This is the first important bit.* Turn that previous statement on its head. To reduce the power by 1, you divide by the number.

* *This is the second important bit.* So. Power 0? The obvious way to get there is to (A) start with power 1, and (B) reduce the power by 1 – because you already know both of those. To reduce the power from 1 to 0, you have to divide by the number.

* But any number to power 1 is just the number itself. And any number divided by itself is simply 1 (except when the number is zero, because dividing by zero is undefined).

* **So any number (except zero) raised to the power 0 is 1.**

((Put a little deeper – this is about the maths having meaning and consistency. Once we’ve defined the concept of positive integer powers *(1, 2, 3, etc.)*, then if the concept of non-positive powers *(0, -1, -2, etc.)* is going to have any meaning *and* give consistent results, those powers have to obey the same rules as the positive ones. And it follows that, to do that, the power 0 always has value 1, for all numbers except zero, as I’ve shown above.))

Anonymous 0 Comments

It’s just a continuation of the pattern exponents make

2^3 is 2x2x2

2^2 is 2×2, or you could say (2x2x2)/2

2^1 is 2, or (2×2)/2

2^0 is 1, or (2)/2

then going further

2^-1 is 0.5, or (1)/2

2^-2 is 0.25 or (0.5)/2

Basically, it just makes things more consistent and workable when you extend exponents past what they are intuitively representing.

Anonymous 0 Comments

x^n / x^m = x^(n-m) , right?

That means that, for example,

2^2 / 2^2 = 2^(2-2) = 2^0

Since

2^2 / 2^2 = 4 / 4 = 1

Then

2^0 = 1

Another way to look at it is to go backwards.

2^4 = 16

2^3 = 8, or 16 / 2

2^2 = 4, or 8 / 2

2^1 = 2, or 4 / 2

2^0 = 1, or 2 / 2

Does that help?

Anonymous 0 Comments

Take

4^(5) = 4*4*4*4*4

4^(4) = 4*4*4*4

4^(3) = 4*4*4

4^(2) = 4*4

4^(1) = 4

4^(0) = 1

Going up in power means you multiply

4^(4) = 4^(3) * 4

Going down in power means you divide

4^(3) = 4^(4) / 4

So you can even extend this to negative numbers

4^(0) = 1

4^(-1) = 4^(0) /4 = 1/4

4^(-2) = 4^(-1) /4 = [ 1/4 ]/4 = 1/(4*4)

4^(-3) = 4^(-2) /4 = [ 1/(4*4) ]/4 = 1/(4*4*4)

4^(-4) = 4^(-3) /4 = [ 1/(4*4*4) ]/4 = 1/(4*4*4*4)

4^(-5) = 4^(-4) /4 = [ 1/(4*4*4*4) ]/4 = 1/(4*4*4*4*4)

Keep in mind dividing a fraction:

[ 1/2 ]/2

is the same as multiplying the denominator

[ 1/2 ]/2 = 1/(2*2) = 1/4

-“A half of a half is a fourth”-

Anonymous 0 Comments

Let’s use 2 as the base of exponent for simplicity.

Exponents are naturally defined only for positive integers. Exponent tells you how many 2’s you multiply together. 2^3 for example is 2 multiplied by itself 3 times, 2 * 2 * 2.

If you take two numbers like 2^3 and 2^2, and multiply them together, you have 2 multiplied by itself 3 times, and then 2 times. (2x2x2)x(2×2) = 2x2x2x2x2 = 2^5. We can just add 2 and 3 together to know how many 2’s we are multiplying together.

In math speak, this means 2^n * 2^m = 2^(n+m)

Now, we are almost done. You see, exponents are supposed to be positive, but actually this equation seems to work just fine for all integers. So let’s try putting in 0.

2^3 * 2^0

According to our rule, our cool formula, this should equal 2^(3+0) = 2^3. But we don’t really know yet what this 2^0 means. Let’s study it. We know what 2^3 is, it’s 2 * 2 * 2 = 8. So let’s try to solve for 2^0

8 * 2^0 = 8

Divide both sides by 8:

1 * 2^0 = 1

2^0 = 1

Well that was easy. We can use 2^0 when using our cool formula above, and it tells us that the value we should assign to 2^0 is 1. So we can just do that.

In a very similar way, you can figure out what values we should use for negative or fractional exponents.

So in short, we noticed that the natural definition, “how many 2’s we multiply together” obeyed this really neat equation. But then we noticed that we can actually use that equation for values that don’t seem to make sense. And as it turns out, it works.

In mathematics this kind of thing is very common. We notice a pattern, and then we start to apply this pattern onto new things, things where it might not make sense. And if it works out, well, that’s pretty cool. We can start using it for new things and find new patterns, and so on.

In some cases, it’s also possible to find out that everything breaks if you try some extensions or other such ideas.

There’s this very strong sense among mathematicians that some extensions are “natural”. They flow out of the initial definitions almost by themselves. While one could try to be more rigorous about it, I think it’s helpful to try to see these things through the lens of aesthetics. Does this thing look pretty, does it feel right? You eventually want to prove that things actually work and all that, but it often starts with this feeling of something being natural