0 is the neutral element for addition. This is why when we have a number then 0 + number = number (0 doesnt change the value in addition) and why 0 x number = 0 (if you add a number 0 times you will have 0). (Multiplication is adding one of the numbers to itself the number of times designated by the second number)
The same way 1 is the neutral element for multiplication. This is why when you have some number then 1 * number = number. This is also why number^0 = 1 (if you never multiply by a number you are left with the neutral element. It would be weird if powering by 0 left you with 0 for example because of how negative powers work)
This is the level 1 answer.
The level 0 answer is that it is this way because all of mathematics is a construct designed to ease problem solving and all people collectively agreed that doing it this way is way more useful (because it is)
Easiest explanation I can think of using the division law for exponents:
Since we can use any number for the initial fraction, as long as the denominator is the same as the numerator, any number to the zeroth power is equal to 1. In general terms, then, for any number, x:
I like to think of it this way:
2^3 is the same as 2 x 2 x 2.
But you can arbitrarily multiply by as many 1s as you want because 1 has the identity property for multiplication.
So we can also write 2^3 as 2 x 2 x 2 x 1 x 1.
2^2 as 2 x 2 x 1 x 1.
2^1 as 2 x 1 x 1.
2^0 as 1 x 1 or just 1.
Multiplying a number by another number is the same as adding a number to itself that many times. And 0 is has the identity property for addition, so similarly:
2 x 3 = 2 + 2 + 2 + 0 + 0
2 x 2 = 2 + 2 + 0 + 0
2 x 1 = 2 + 0 + 0
2 x 0 = 0 + 0
I see other people have posted good explanations, but I think the simplest explanation has to do with how you break down numbers. Lets take a number, say, 124. We can rewrite it as 100 + 20 + 4 and we can rewrite that as 1 * 10^2 + 2 * 10^1 + 4 * 10^0 and I think you can see why anything raised to the 0th power has to equal 1. Numbers and math wouldn’t work if it didn’t.
Thanks, I couldn’t even tell what the image was about math. I thought a dirty joke was hidden somewhere involving the 0. Didn’t realize it was small and floating above on the right so people would immediately realize it’s a power lol. Many people hide clever things but I always approach them in the wrong way lol.
You can think of 1 as the “empty product” (or the “neutral element of multiplication” if you want to be fancy). 2^x means you have x factors of 2. If you have 0 factors, you have the “empty product”
2^0 isn’t multiplying by zero. Considering this law: 2^a / 2^b = 2^(a-b)
it’s obvious why 2^0 = 1
If a=b you’re dividing by the same number resulting in 1.
Unfortunately, I cannot explain/prove the first law though.
I know I’m bad at math but I don’t understand how 2x0=0 but 2^0=1
How are they different answers when they’re both essentially multiplying 2 by zero?
Someone with a bigger brain please explain this
Edit: I greatly appreciate all the explanations but all they’ve done is solidify the fact that I’ll never be good at math 😭
0 is the neutral element for addition. This is why when we have a number then 0 + number = number (0 doesnt change the value in addition) and why 0 x number = 0 (if you add a number 0 times you will have 0). (Multiplication is adding one of the numbers to itself the number of times designated by the second number)
The same way 1 is the neutral element for multiplication. This is why when you have some number then 1 * number = number. This is also why number^0 = 1 (if you never multiply by a number you are left with the neutral element. It would be weird if powering by 0 left you with 0 for example because of how negative powers work)
This is the level 1 answer.
The level 0 answer is that it is this way because all of mathematics is a construct designed to ease problem solving and all people collectively agreed that doing it this way is way more useful (because it is)
Choose which one you want
Fuck me this is the only one I understand 😭
subtracting one from Exponent means halving (when the base is two):
2⁴ = 16 2³ = 8 2² = 4 2¹ = 2 2⁰ = 1
It’s a simple continuation of the pattern and required for mathemarical rules to work.
This is confidently wrong.
3^0 is also 1. 2738394728^0 is also 1.
Edit: just saw that technically you’re correct - sure.
IF base 2, Exponent reduction equals to halving - dividing by 2.
For x^y reducing y by one is equal to dividing by x, then we have the proof it always works.
But that’s because for 3 the sequence is dividing by 3 not 2.
81, 27, 9, 3, 1, 1/3, 1/9, etc.
3^4, 3^3, 3^2, 3^1, 3^0, 3^(-1), 3^(-2), etc.
You’re not always halving, but the method is the same and it sometimes helps people understand the concept more easily.
Easiest explanation I can think of using the division law for exponents:
Since we can use any number for the initial fraction, as long as the denominator is the same as the numerator, any number to the zeroth power is equal to 1. In general terms, then, for any number, x:
I like to think of it this way:
2^3 is the same as 2 x 2 x 2.
But you can arbitrarily multiply by as many 1s as you want because 1 has the identity property for multiplication.
So we can also write 2^3 as 2 x 2 x 2 x 1 x 1.
2^2 as 2 x 2 x 1 x 1.
2^1 as 2 x 1 x 1.
2^0 as 1 x 1 or just 1.
Multiplying a number by another number is the same as adding a number to itself that many times. And 0 is has the identity property for addition, so similarly:
2 x 3 = 2 + 2 + 2 + 0 + 0
2 x 2 = 2 + 2 + 0 + 0
2 x 1 = 2 + 0 + 0
2 x 0 = 0 + 0
I see other people have posted good explanations, but I think the simplest explanation has to do with how you break down numbers. Lets take a number, say, 124. We can rewrite it as 100 + 20 + 4 and we can rewrite that as 1 * 10^2 + 2 * 10^1 + 4 * 10^0 and I think you can see why anything raised to the 0th power has to equal 1. Numbers and math wouldn’t work if it didn’t.
Thanks, I couldn’t even tell what the image was about math. I thought a dirty joke was hidden somewhere involving the 0. Didn’t realize it was small and floating above on the right so people would immediately realize it’s a power lol. Many people hide clever things but I always approach them in the wrong way lol.
You can think of 1 as the “empty product” (or the “neutral element of multiplication” if you want to be fancy). 2^x means you have x factors of 2. If you have 0 factors, you have the “empty product”
Its not the same. And theres proof, why.
2^0 isn’t multiplying by zero. Considering this law: 2^a / 2^b = 2^(a-b)
it’s obvious why 2^0 = 1
If a=b you’re dividing by the same number resulting in 1.
Unfortunately, I cannot explain/prove the first law though.
The first rule is just simple division:
(2x2x2x2) / (2x2) =
(2/2) * (2/2) * 2 * 2=
1 * 1 * 2 * 2 =
2 * 2 =
4
Writing in terms of powers:
(2^4) / (2^2) =
(2^(4-2)) =
(2^2) =
4
The two bottom 2’s “cancel out” (really they just divide into one another to make 1’s) two of the top 2’s and you’re left with two top twos.