why 0**0=1?

Question

i thought 0**0=0**(1-1)=0**1/(0**(-1))=0/0=nulity,but in python,0**0=1.

Answer 1

When you raise any number to the power of 0 the output will always be 1....

Answer 2

Good Question! Your question is already answered by Wikipedia through the binomial theorem :
https://en.m.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero

Answer 3

Good question indeed. There is quite the amount of argument regarding 0 to the power of 0.

http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/

ATOW, if you type 0^0 into the calculator provided by Google (web and calculator app), the answer given will be 1. However, my fx-570MS disagrees.

Answer 4

you guys just hurt my brain. 1am isn't the time to be reading these types of things. I like it though. Just ow

Answer 5

There's a few reasons why, however it's purely mathematical and is not related to computers.

So why is a^0 = 1? When a is any number? (Including 0)!

Lets take for example this function.
f(x) = a^x

If f(x) is continuous:
The lim f(x) = lim f(x) = lim f(x)
x -> 0 x-> 0- x -> 0+

lim f(x) = 1 = lim f(x) = 1
x -> 0- x -> 0+

This is easy to see. For example,
2 ^0.1 = 1.07
2 ^0.01 = 1.007
So, we are approching 1 as we get closer to 0, from both sides.

Therefore the lim f(x) = 1,
x -> 0
(if f(x) is continuous)

So is f(x) continuous?
* f(c) is defined as a^c.
* a^x should be in the domain of f.
* The limit exists

But is f(0) = 1?

According to pascals triangle,
(1-1)^ 0 = (0-0)^0 = 1.

According to definition,
x^a is x * x * x... a amount of times.
x^0, x within itself 0 times, is 1 time. = 1.
The power rule would also only work if this is true.

According to mapping there is 1 way to map an element onto an empty set.

Therefore we define a^x to be continuous for all x, and therefore limits prove that a^0 = 1.
So if you see the graph a^x, you will notice it crosses 1 when x = 0 and is continuous.

Answer 6

Power of Any number or variable is zero so Answer is 1 because of
we know that
x=x
x^1=x^1
x^1=x^1×1 // any number multiply by one don't change value
so...
x^1/x^1=1
x^1-1=1
x^0=1 ...i Hope you Understand

Answer 7

https://code.sololearn.com/WTzQUWKXcoxT/?ref=app

Answer 8

Also...Python doesn't​ even bother. It ships "pow" off to libm (c++/the OS...at link 1):
1472 : comment: let libm do it
1475 : external call: pow()

*** openlibm (OS dependent; link 2)
__ieee754_pow(double x, double y) { ...
27: comment: x^0 returns 1
110: comment: /* y==zero: x**0 = 1 */
111: if((iy|ly)==0) return one;

*** IEEE754 (a specification, link 3):
"[many]...extremely competent mathematicians...[involved in IEEE754]"

*** Math: it pops out.
6 reasons it is not entirely arbitrary at link 4.

1. http://svn.python.org/view/python/trunk/Modules/mathmodule.c?revision=82221&view=markup
2. https://github.com/JuliaLang/openlibm/blob/master/src/e_pow.c
3. https://stackoverflow.com/questions/7409082/android-kernel-libm-powfloat-float-implementation
4. http://scienceline.ucsb.edu/getkey.php?key=2626

Answer 9

@Chuma Umenze Why English when you can speak using Mathematical theorems, rules and definitions? 😆 If you take your time reading it and not get hung up on the terseness of the symbology it is a very good explanation.

Answer 10

Anything raise to the power of zero is equal to zero. Here is a simple way I understand it.

Say, you have 2^4. We can assume that 2^4 is equal to 4^2 (which is correct since both will give 16).

i.e. 2^4 = 4^2

using Indices we change the bases to same base. we use base 2. Our equation now becomes

2^2(2) = 2^2(2)

Cross-multiplying (Dividing) both will give

2^(4-4)

which is same as

(2^4) / (2^4)

where we cancel both power of 4, giving us

2/2

which gives us 1.

Well, this is how I understand it and the way I can explain it. I'm not a science guy sha.

Answer 11

@Chuma @Pawanx The only problem with that explanation is when we talk about 0^a, since you cannot divide by 0. (0 / 0 != 1)

So you would need a bit more elaboration to explain that 0^0 cannot be 0, in order for it to be 1.
Great, easy way to put it though!

Answer 12

You guys are complicating things . This has nothing to do with programming . It is a basic mathematical rule that can be proved via various complicated methods . Remember the rule "Anything raised to the power zero is 1 "

Answer 13

If I explain the limits (lim) in laymens terms, this is basically what I was saying:

0^0.01 = 1.07
0^ 0.001 = 1.007
0^0.000001 = 1.000000...7

We are slowly getting closer to 1.
Same from the other side:

0^-0.1 = 1.07
0^-0.01 = 1.007
0^-0.00000001 = 1.0000....

We still get close 1.
When I say 'continuous' I imply there's not going to be any issues by saying 0^0 is the same as 0^0.00000..(keep repeating)..1 and 0^-0.00000000.(keep repeating)..1

and since those are equal to (or very close to) 1.
0^0 must be 1.

Hopefully that helps. 😝

Answer 14

@Rrestoring faith are you kidding me? Speak in English!

Answer 15

No confusion
simply anything power 0 is always 1

Answer 16

@Steven Schneider To me those symbologies looks like Egyptian manuscripts. I only have a high math knowledge. I don't even know what is lim.

Answer 17

@Rrestoring faith I will read up on that. 'll try find something that explains it better in layman's understand. Thank you.