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:
You can see this in the example above but perhaps it’s better to use different powers to make things a bit clearer.
2^5=2x2x2x2x2
2^3=2x2x2
(25)/(23)=(2x2x2x2x2)/(2x2x2)
You can cancel 3 of the 2s from the top and bottom of the fraction to be left with 2x2, or 2^2.
I.e. (25)/(23)=2^2
The quicker way to calculate this is doing 2^(5-3) which when you resolve the bracket is obviously just 2^2 or 2x2.
If both numbers in the bracket are the same the bracket will always resolve to 0, which is the same as saying a number divided by itself, any number divided by itself is one so it follows that any number to the power 0 is also 1 (because it’s essentially exactly the same calculation).
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”
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.
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
The simplest way I think of it is by the properties of exponentials:
2^3 / 2^2 = (2 * 2 * 2) / (2 * 2) = 2 = 2^(3-2)
Dividing two exponentials with the same base (in this case 2) is the same as that same base (2) to the power of the difference between the exponent in the numerator minus the exponent in the denominator (3 and 2 in this case).
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.
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.
Well looks like some people already answered your question but let me show you quick proof video that may help you understand how powers work: https://youtu.be/kPTp82EGjv8?feature=shared
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 😭
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.
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 😭
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:
This isn’t strictly speaking a proof, but it did help me to accept it as it demonstrates the function that makes it 1.
2^3 = 2x2x2
2^2 = 2x2
(23)/(22) = (2x2x2)/(2x2) = 2
= 2^(3-2)
In general terms:
(xa)/(xb) = x^(a-b)
If a and b are the same number this is x^0 and obviously (xa)/(xa) is one because anything divided by itself is 1.
Hope that helps
Yes, of course, obviously…JFC, what??
2^(a-b) = (2a)/(2b)
You can see this in the example above but perhaps it’s better to use different powers to make things a bit clearer.
2^5=2x2x2x2x2
2^3=2x2x2
(25)/(23)=(2x2x2x2x2)/(2x2x2)
You can cancel 3 of the 2s from the top and bottom of the fraction to be left with 2x2, or 2^2.
I.e. (25)/(23)=2^2
The quicker way to calculate this is doing 2^(5-3) which when you resolve the bracket is obviously just 2^2 or 2x2.
If both numbers in the bracket are the same the bracket will always resolve to 0, which is the same as saying a number divided by itself, any number divided by itself is one so it follows that any number to the power 0 is also 1 (because it’s essentially exactly the same calculation).
Rule = #^0 = # x 1
Don’t ask why…got it.
No not quite, #^0 = 1
Wait, so 5^0 = 1??
Yup
5^0 can be rewritten as 5^(2-2)
5^(2-2) = (52)/(52)
This is a number divided by itself so cancels to 1 every time, regardless of #.
That was pretty complicated, here is a simpler answer I hsve come up with:
1=(2x2x2)/(2x2x2)=2³/2³=2³⁻³＝2⁰
If that makes sense to you…
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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”
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.
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
In addition to the explanation others have mentioned, here it is in graph form. See the where the graph of 2^x intersects the y axis (when x=0):
https://people.richland.edu/james/lecture/m116/logs/exponential.html
This also has some additional verbal explanations:
http://scienceline.ucsb.edu/getkey.php?key=2626
The simplest way I think of it is by the properties of exponentials:
2^3 / 2^2 = (2 * 2 * 2) / (2 * 2) = 2 = 2^(3-2)
Dividing two exponentials with the same base (in this case 2) is the same as that same base (2) to the power of the difference between the exponent in the numerator minus the exponent in the denominator (3 and 2 in this case).
Now lets make both exponents the same:
2^3 / 2^3 = 8/8 = 1
2^3 / 2^3 = 2^(3-3) = 2^0 = 1
Its not the same. And theres proof, why.
This dude is great at explaining math, including this: https://yewtu.be/watch?v=r0_mi8ngNnM
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.
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.
Well looks like some people already answered your question but let me show you quick proof video that may help you understand how powers work: https://youtu.be/kPTp82EGjv8?feature=shared
Here is an alternative Piped link(s):
https://piped.video/kPTp82EGjv8?feature=shared
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I’m open-source; check me out at GitHub.