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


I’m too dumb to get it

In: Mathematics

Mathematicians *define* x^0 = 1 in order to make the laws of exponents work even when the exponents can no longer be thought of as repeated multiplication. For example, (x^3 ) (x^5 ) = x^8 because you can add exponents. In the same way (x^0 ) (x^2 ) should be equal to x^2 by adding exponents. But that means that x^0 must be 1 because when you multiply x^2 by it, the result is still x^2. Only x^0 = 1 makes sense here.

Essentially x^0 MUST equal one for all other exponents to work.

Edit: thank you kind strangers for your gifts of silver and gold!

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

That means that, for example,

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


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


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?

Exponents of integers is just a shorthand notation for a number multiplied by itself a number of times.

So we define the following rules

b^(1) =b

b^(n+1) = b^(n) * b

That mean that bn = b*b*b*b… (you should have n b:s on the right side.)

From that and associativity of multiplication ie that a*(b*c) =(a*b)*c =a*b*c the following line follows if m and n are positive integers.

b^(m+n) =b^(m) * b^(n)

That is the fundamental rule of positive integer exponent. If you look at what happen if the exponent is 0 you can use the face that 4=3+1=4+0 so b to the power of 4 should be identical regardless of how we write it.


b^(3+1) = b^(3) * b^(1) =(b*b*b)* b= b*b*b*b

b^(4+0)=b^(4) * b^(0) = (b*b*b*b) * b^(0)

So becaus b^(4) should be the same as b^(4+0) you get the relationship

b^(4) = b^(4+0) => b*b*b*b =(b*b*b*b) * b^(0) => b*b*b*b/(b*b*b*b) = b^(0) =>1=b^(0) So b^(0) has to be 1 for the rules to make sense. That is for any b ≠0 because we divided by b in the equation. what you define 0^(0) as depend on the context.

For the arithmetic operation we have agreed on b^(0) has to be 1 if b≠0 if not the rules do not work. Why for rules that look initially strange in math is generally because that is needed for the rules we have defined to work as smooth and simple as possible. You might say that b^(0) is not allowed but because we can define is as b^(0)=1 and it works fine with other rules we use that definition.

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.

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


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”-

*(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.))

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.

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.

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.

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!

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.

x^0 is something called an [empty product](https://en.m.wikipedia.org/wiki/Empty_product), that is, the result of multiplying zero numbers together. It is defined to be 1, the multiplicative identity (x • 1 = x), because that is the most useful and intuitive way of defining it. In any multiplication problem, let’s say (2 • 3), you can say there’s an implicit (2 • 3 • 1 • 1 • 1 • 1. . .) at the end. If you multiply zero number together, you’re left with only the (• 1 • 1 • 1 • 1. . .), so you get one.

How bout 0⁰?
0 to the power of anything is 0 but anything to the power of 0 is 1.

new problem:

why is 0! == 1?

How many times can you count a number 0 times? No times. But that set of nothing is still has nothing. Which is counter as 1. You count nothing

x^n means “Multiply 1 by x n times”

Math|Math Expanded|Words
2^4|1x2x2x2x2 = 16|Multiply 1 by 2 four times
2^0|1|Multiply 1 by 2 zero times

Don’t think of exponents as simply being a number multiplied n times itself (If n=3, then 2x2x2). Think of exponents using one extra number: 1 (allowed under Identity Property of multiplication – not important here). So 2 squared is 1x2x2=4. 2 cubed is 1x2x2x2=8. So if n=0, then 2ⁿ is 1x(zero 2s) which equals 1. Keep in mind that zero 2s means it doesn’t exist, not 2×0. So any number with an exponent of 0 is 1 times nothing else. 1.

The whole “x^m / x^n” argument is a great way of showing that it must be so, but there is a much more relatable, “real-world” explanation for x^0 = 1. As such, the “no! It’s just because we defined it that way!” response is wrong – this time. It is true that that is sometimes the answer, but not here.

How about an ELI10? This operation (powers) describes the number of possible outcomes. So, if you were to flip a coin once, how many possible outcomes are there? Two – heads or tails, and 2^1 = 2. If you were to flip it 7 times, how many possible strings of heads and tails are there? 2^7. If you flip the coin 0 times, how many possible outcomes are there? Well…1, right? You get nothing. Thus, 2^0 = 1.

This of course ignore the interesting philosophical discussion of how to identify and count nothingness, but at a real-world level it is correct.

This understanding of exponentiation is of course restricted to non-negative integers in both the base and the exponent. By extension you then define the operation for all complex numbers in the base, and then you properly have the stated result. You can also extend to all complex numbers in the exponent. If you prefer to cut back down to real numbers, you of course get weird looking rules that come with that restriction.

Lastly, always remember – don’t drink and derive!

10^3 = 10* 10* 10 = 1000

10^2 = 10*10 = 100

10^1 = 10*1 = 10

10^0 = 1

10^-1 = -10*1 = -10

10^-2 = -10*-10 = -100

10^-3 = -10* -10* -10 = -1000

See the pattern?
Some things in math are messed up.