Why is the 0 exponent of a non-zero number equal to 1?
People say that English is a funny language. I bet it is, but how else can you explain that we have noses that run and feet that smell?
Or tell the man that what his car transports is a shipment while if a ship does the same, it is a cargo.
Pretty crazy? I know. But we have not seen the end of the absurd when we take a walk down the mathematical lane.
A zero in English means nought if you are from the UK or thereabouts or naught if you happen to be an American. It means zilch or nothing.
But in mathematics, exponentiation, it does something magnificent.
If any number is raised to the power of zero, it automatically becomes 1. That is x0 = 1, with the exception of where x= 0 which may lead to leaving the expression undefined.
This expression is usually taught in the high school mathematics class. We often seem to take it as a law or convention. But nothing in maths is ever that. There is always a proof that something is what it was.
Not all the things we are taught in high school is redundant. I know you may ask what use is knowing that mitochondria are the powerhouse of a cell got to do with paying taxes to the government. That may be the one that may be a little bit redundant, but it may add to make you that awesome doctor that treats the Kim Kardashians and charge $5000 per hour. That is something that would help with the taxes.
Going back to the issue, sometimes we assume that a lot of things in mathematics are just that way. Some of the things we accept as such have proofs which are long lost and buried deep somewhere in a textbook in the dusty shelf of a library.
But when a number is raised to the power of zero, the answer is usually one when the number is not zero. The same is also true on the factorial of zero, i.e. 0! = 1.
Going back to the basics
We will see that the exponential function is a mathematical function which is of the form.
f (a) = xa
Here, x is the constant also known as the base. The a is the variable number which is the power of the function-any number can be here.
In mathematics, we will often run into the exponential-function base known as the transcendental number denoted by e which has the value of 2.71828.
We have in the situation f(x) = ex
When x (exponent) increases by 1, the value of e (the function base) will also follow suit and multiply by 1. The same happens if it decreases or gets divided.
In the beginning, we can only raise a constant by positive numbers. With time there was an expansion of the function's domain such that we can about raise a constant to any number starting off from a fraction, complex number, negative number including negative fraction, etc.
The exponential function is a mathematical representation of the number of times a number will have to undergo self-multiplication.
For example, 23 represents that the base function 2 undergo multiplication three times, i.e. 2x2x2 which of course is 8.
But when you try 64 1/3 we get an answer that is the outcome of the cube root of 64 which is 4.
i.e. 3√64 = 4
This conjecture invariably shows that the when a number is raised to a fractional exponent, that it expresses the root of the base number.
More example, a 5121/3 = 3√512= 8
But when an exponent is a negative number, the base number will have one divide it, and the negative exponent becomes positive.
For instance, 2-3 can be expressed as (1/2)3
Let's prove that x0 = 1 Assuming we have two variables xa and xb. If we divide the first variable by the second, xa/xb = xa+(-b) xa+(-b) = xa-b If a=b, then the right side of the equation gives x0. But xa/xb = xa+(-b)= x0 And we have a=b Then the left side of the equation is given as xa/xb = xa/xa = 1 x0 = 1 We have the proof REFERENCES
Let's prove that x0 = 1
Assuming we have two variables xa and xb.
If we divide the first variable by the second,
xa/xb = xa+(-b)
xa+(-b) = xa-b
If a=b, then the right side of the equation gives x0.
But xa/xb = xa+(-b)= x0
And we have a=b
Then the left side of the equation is given as
xa/xb = xa/xa = 1
x0 = 1
We have the proof