# Math physical interpretation

1apple+0apple=2apples
1apple . 0times = 0 Apple
0apple . 2times = 0 Apple

Whats the physical interpretation of X^0=1?

2^3 = (2.2).2 then X^2=(X.X).X
then X^0=?

i know why is 1 the result because of the thory of X^a.X^b=X^a+b

But i dont understand the physical interpretation.

kezman, you are started on a very dangerous path. numbers are abstractions. operations with numbers are abstractions. you demand an intuitive straightforward equivalent for an abstract situation. there is no good reason to expect there is such a thing.

in the case of x^0, you luck out tho, there is such a thing. if i draw a point x feet away, what surface does it have ?

Its true but ± and multiplication are mathematical operations that have physical interpretation and ^ is a multiplication based operation.

Yeah, I agree with Zeno, we can explain this by intuitionistic approach, though the zero exponent rule is still context-dependent: 2 ^ 3 really means : 1 x 2 ^ 3 , meaning the starting point is always 1.

2 is the base, which is also the factor multiplied as many times as indicated by its exponent, in this case 2 is multiplied by itself 3 times times 1.

So, 2 ^ 0 means 1 x 2 ^ 0, and because exponent is zero, it further means that 2 (or a number resulting from factors of 2) does not get multiplied by 1, so answer is 1. [Actually, it becomes like this: 1 x 1 = 1, I could explain further if you’d like]

There are other intuitionistic approach by which you can explain the zero rule, I just told you the one that makes the most sense to me.

Note to Zeno: please be gentle with me. hehe.

arendt, isn’t that a bit dodgy ?

after all, we could say
2^0 is in fact 1+2^0, which is in fact 1 + (1 + 2^0)) etc

let me dwell a bit on my example.

you ask, what is a reasonable intuitive representation of the equation x^0=0 ?

i say, the question “what is the area of a point x feet from the origin ?”

now, notice how the distance from origin is irelevant for the surface of the point. just so, the value of x is irrelevant for the purpose of x^0.

you are fooled to think the expression x^0 is ulterior to the concept ^, because it seems to you to be part of a set, x^1, x^2, x^y etc. that is however not so. x^0 is anterior to the concept ^.

just like we conventionally agree that the area of a point is 0, and the distance from origin has no bearing, just so we agree that x^0=0, and the scalar value of x has no bearing. this convention allows us to then define the concept ^ so as it can be applied for x and an y not 0.

happy ?

Hi Zeno,

Soon, I will be.

No, actually we could not say that 2 ^ 0 can mean 1 + 2 ^ 0. That would mean a totally different thing. Let me start again: my explanation is not fully intuitionist, rather, semi-intuitionist ( ) since I borrowed an axiom or two and then explained away: the number 1 is a multiplicative identity, hence we can say 1 x 2 ^ 0 or just 2 ^ 0.

You mean x ^ 0 = 1, x not equal to zero. (let’s avoid calculus shivers) Anyway, go on…

Zeno, a number raised to a power, by definition, is a repetitive multiplication, itself times as many as indicated by its exponent. When evaluated, x^n is a number that has relation to 1, not to anything else. And since x^n is really another way of writing a number with a base, you can represent it in a number line and come up with x^0, x^1, x^2…x^n, though it is not necessary to do it to know what a single x^n means. By itself, or as part of a series of numbers, each is related to 1. That is why, 1 x 2^0 means 1 does not get multiply by base 2, rather 1 gets multiplied by itself, hence the answer is 1.
I don’t know why we must take x^n apart and consider its parts as concepts that could mean something. x^n is defined as a number, as one concept.

So, to answer Kezman’s question of representing x^0 = 1:

We can just say, 1 multiplied by itself is 1. Or nothing gets added to 1 (if by way of addition). You start with one apple, and if you do not multiply it, then you still have one apple. If you have 1 apple then multiply it by 4 apples (written as 2^2), then you have 4 apples.

1 multiplied by itself is 1^1=1
I think 1 multiplied no times itself is the representation of 1^0=1
and the difference with 1X0=0 is that this is 1 apple 0 times “sumed up”

the first part i think you are trying to say 1^0=1
the second part im not sure if you mean : 1X4 = 2^2

we look at x ^ n as always having a relation to the number 1. And since 1 is a multiplicative identity, we can write 1 x 2 ^ 0 = 1 OR just 2 ^ 0 = 1. But it is always implicit that we start with the number 1.

No. When I said â€œyou start with one appleâ€ I meant the number 1 in these expressions:

1 x 2 ^0 = 1-----here 1 does not get multiply by another number, or it gets multiplied by itself.

1 x 2 ^ 1 = 2

1 x 2 ^ 2 = 4 ----this is the second part you were asking.

etc.

Its true by a succesion An = 1*2^n nENo
the first number is 1.

ALthought the axiom to consider X^0=1
is
X^a*X^b=X^a+b

Example:

X^-1 *X^1=X^-1+1
X^-1 * X^1= X^0
1/X * X = X^0
1/X * X = 1
X^0 = 1

but there are different ways of representing it.

That is why I said I borrowed an axiom or two. The multiplicative identity of the number 1 in 1 x 2^0 = 1 is axiomatic. Then if you use the rules of exponents, thatâ€™s also axiomatic. In the end, when it comes to the concept of number in the form x^0 = 1, I donâ€™t think we can avoid axioms. To represent it physically or give an intuitionistic explanation, we can explain the answer 1 by saying that the number x^0 is really in relation to the number 1. And leave it at that. To attempt to explain it is again to attempt to explain using an axiom.

Yes, there are different ways, though not in many ways. Just like we can represent the number 2 as :
2 x 1 = 2

2 / 1 = 2

we can represent 2 ^ 0 = 1 as

2^0 / 2^0 = 1 ----[a number divided by itself equals 1: If I divide 4 apples 4 ways, I get 1 apple (each)]. But donâ€™t you think this is axiomatic already? It follows the rules of exponents: 2^0 / 2^0 is really 1/1. If you have: 2^2 / 2^2, you can write it as 2 raised to 2 â€“ 2, which means 2 raised to zero.

Yes that is another way of concluding 2^0=1

I’ll put it into apple notation now.

When xE|R number of apples are divided among x number of people, each person gets 1 apple.

Remember, for all x, n, aE|R (x^n) / (x^a) = x^(n-a). For all xE|R x / x = 1. So 1 = (x^n) / (x^n) = x^(n-n) = x^0 for all x, nE|R. //

Sorry if this insults someones intelligence.

No you did not insult anything. I like your notation.

Exponents ≥ 1 are powers; those < 1 are roots. For example:

5^2.3 = (5^2)(5^.3) = (25)[5^(3/10)] = (25)(tenth root of 5^3)
≈ 40.51641

Your question was about X^0, so let’s just look at exponents < 1 in the limit as the exponent goes to 0. For example:

5^.1 = 5^(1/10) = the tenth root of 5, or the number which must be multiplied times itself ten times to equal 5 ≈ 1.17642

5^.01 = 5^(1/100) = the hundredth root of 5 ≈ 1.01622

5^.001 = 5^(1/1000) = the thousandth root of 5 ≈ 1.00161

5^.0001 = the ten-thousandth root of 5 ≈ 1.000161

5^.00001 = the hundred-thousandth root of 5 ≈ 1.0000161

5^.000001 = the millionth root of 5 ≈ 1.00000161

As the exponent gets very small, the order of the root gets very large, and only a number very close to 1 can be multiplied by itself many times to equal a finite number. Slightly larger, and the product goes to infinity; slightly smaller and it goes to 0. Hence, 1 is the infinite-order root of every number. Even for a large number, as the exponent approaches 0, the root quickly approaches 1:

1000000^.00001 ≈ 1.00014

Perhaps one way to visualize X^0 = 1 is to picture each one/single object as the infinite product of all other “similar” objects –

1 grain of sand = 1 grain of sand times 1 grain of sand times 1 grain of sand times …

X^0 = 1 is the answer to the question “What number can be multiplied times itself an infinite number of times and give a finite, nonzero product?”

Physical applications of powers are easier:

4^3 means multiply (4)(4)(4). (4)(4) means, starting with 0, add 4 four times, giving 16. (16)(4) means, starting with 0, add 16 four times (or add 4 sixteen times). The result is 64 (which could be, for example, the volume of a cube with sides of magnitude 4).

Roots (fractional exponents) do not multiply the base number times itself. Rather, they determine another number – the root – which, multiplied times itself gives the base number:

X^(1/n) = the nth root of X

X^(1/n) times X^(1/n) times X^(1/n) …n times = X^1.

If 1/n = 0, then n = ∞ and

X^(1/∞) times X^(1/∞) times X^(1/∞) times …infinite times = 1.

This is another way even using limits.

Its interesting the use of infinite in mathematics as a concrete symbol.
It was a discusion in another thread.

I think I’d rather go my entire math career without ever touching that old infinity symbol again. LOL. Those who know me know I hate it.