Ahh yes your right, my bad. I was thinking the power was equal to her age but it is actually -1 her age. I was doing 2^3 when it should have been 2^2. I guess that is what I get when I don’t show my work. Dang math teacher was right.
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.
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
(2^5)/(2^3)=(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. (2^5)/(2^3)=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).
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: youtu.be/kPTp82EGjv8?feature=shared
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”
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).
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.
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)
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
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.
They are either part of the “everything is a meme” bullshit, or they don’t care whether the crap they’re posting is a meme.
Both crowds will complain that they’re not posting to other communities because they’re “dead”. Neither group sees the irony.
In the end, these idiots are looking for upvotes and have found the community of other idiots who will upvote anything in their feed without a care of where it’s posted. This group of idiots can go back to reddit.
The 2⁰ looks like 20 (for the 20yr old’s birthday) but when you calculate 2 to the power of 0 it’s equal to 1 (which is for the baby’s birthday). It’s a math joke, hope that helps
I know plenty of smart people pretending to be Winnie the Pooh while elbow deep in honey pots. Just because you weren’t fucking doesn’t mean other nerds weren’t lol.
It's been a while, but I think I remember this one. Lim 1/n =0 as n approaches infinity. Let x^0 be undefined. For any e>0 there exists an n such that |x^(1/n) -1| < e. If you desire x^(1/n) to be continuous at 0, you define x^0 as 1.
E2a: since x^(1/n)>1, you can drop the abs bars. I think you can get an inequality to pick n using logs.
Of course, your explanation is the “correct” one - why it’s possible that x^0=1. Mine is the simple version that shows how logic checks out using algebraic rules.
Ah yes. How fitting for a young new person in the world. A reminder that 2°C of warming above the pre-industrial mean would be catastrophic, but also is a good lower-limit of what to expect based on current intentions.
for anyone curious, here’s a “constructive” explanation of why a^0^ = 1. i’ll also include a “constructive” explanation of why rational exponents are defined the way they are.
anyways, the equality a^0^ = 1 is a consequence of the relation
a^m+1^ = a^m^ • a.
to make things a bit simpler, let’s say a=2. then we want to make sense of the formula
2^m+1^ = 2^m^ • 2
this makes a bit more sense when written out in words: it’s saying that if we multiply 2 by itself m+1 times, that’s the same as first multiplying 2 by itself m times, then multiplying that by 2. for example: 2^3^ = 2^2^ • 2, since these are just two different ways of writing 2 • 2 • 2.
setting 2^0^ is then what we have to do for the formula to make sense when m = 0. this is because the formula becomes
2^0+1^ = 2^0^ • 2^1^.
because 2^0+1^ = 2 and 2^1^ = 2, we can divide both sides by 2 and get 1 = 2^0^.
fractional exponents are admittedly more complicated, but here’s a (more handwavey) explanation of them. they’re basically a result of the formula
(a^m^)^n^ = a^m•n^
which is true when m and n are whole numbers. it’s a bit more difficult to give a proper explanation as to why the above formula is true, but maybe an example would be more helpful anyways. if m=2 and n=3, it’s basically saying
(a^2^)^3^ = (a • a)^3^ = (a • a) • (a • a) • (a • a) = a^2•3^.
it’s worth noting that the general case (when m and n are any whole numbers) can be treated in the same way, it’s just that the notation becomes clunkier and less transparent.
anyways, we want to define fractional exponents so that the formula
(a^r^)^s^ = a^r^ • a^s^
is true when r and s are fractional numbers. we can start out by defining the “simple” fractional exponents of the form a^1/n^, where n is a whole number. since n/n = 1, we’re then forced to define a^1/n^ so that
a = a^1/n•n^ = (a^1/n^)^n^.
what does this mean? let’s consider n = 2. then we have to define a^1/2^ so that (a^1/2^)^2^ = a. this means that a^1/2^ is the square root of a. similarly, this means that a^1/n^ is the n-th root of a.
how do we use this to define arbitrary fractional exponents? we again do it with the formula in mind! we can then just define
a^m/n^ = (a^1/n^)^m^.
the expression a^1/n^ makes sense because we’ve already defined it, and the expression (a^1/n^)^m^ makes sense because we’ve already defined what it means to take exponents by whole numbers. in words, this means that a^m/n^ is the n-th square root of a, multiplied by itself m times.
i think this kind of explanation can be helpful because they show why exponents are defined in certain ways: we’re really just defining fractional exponents so that they behave the same way as whole number exponents. this makes it easier to remember the definitions, and it also makes it easier to work with them since you can in practice treat them in the “same way” you treat whole number exponents.
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