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[Blazing Donkey]

Greetings to all.

Let me preface this by saying that my mathematical education is sorely lacking.

This is something that I have never been able to wrap my mind around: "Any number to the zero power = 1."

My understanding is that a number to a power is: that number multiplied by itself the number of times in the power. So 5 to the 2nd power is 5 x 5 ie. 5 times itself two times. If this is true, then it seems to me that a number times itself zero times means the number is not multiplied times itself, so it should equal the number.

If I have 5 and I don't multiply it at all, I can't understand how it could possibly equal 1 because I am not doing anything to the 5.

I would greatly appreciate if someone could explain this to me. Thank you for your time.

[Teh]

There are lots of ways of showing this - here's one:

a^(0) = a^(b-b) = a^(b).a^(-b) =a^(b)/a^(b) =1/1 =1

Actually, in the terms of your question:

5^3 = 1 x 5 x 5 x 5

5^2 = 1 x 5 x 5

5^1 = 1 x 5

5^0 = 1

[A_Seagull]

Here's another way:

As a sort of definition:

a^x . a^y = a^(x+y)

or a^x / a^y = a^(x-y)

then a^5 / a^5 = a^(5-5) = a^0

but a^5 / a^5 = 1

Hence a^0 = 1

Q.E.D.

[Blazing Donkey]

5^3 = 1 x 5 x 5 x 5

Why is the "1" added to it?

Why isn't it: 5^3 = 5 x 5 x 5 ?

[Teh]

(Nested quote removed.)


Why is the "1" added to it?

Why isn't it: 5^3 = 5 x 5 x 5 ?

It's not added, it's multiplied - it makes no difference to the maths, but it shows you a pattern.

Yet another way to understand this issue is to realise that we are in fact dealing with a continuous function:

y = a^x

To calculate y for any value of x, without plugging in x, take the limit (x +/- delta(x)) as delta(x) tends to zero. Try it on a calculator for x=0.

[Blazing Donkey]

BD: "Why is the "1" added to it? Why isn't it: 5^3 = 5 x 5 x 5 ?"

It's not added, it's multiplied - it makes no difference to the maths, but it shows you a pattern.

No, no.. I mean, why is "5^3 = 1 x 5 x 5 x 5" valid and "5^3 = 5 x 5 x 5" not valid?

[Philo_ben]

Rules were made to be broken.

[MazerRackhem]

(Nested quote removed.)


Why is the "1" added to it?

Why isn't it: 5^3 = 5 x 5 x 5 ?

A person can add as many 1x as they like to the beginning of a mathematics problem, I think that is the logic in Teh's second argument. However I think his later lines may have confused you unnecessarily, his original work explains the question. For any x:

x^0=x^(a-a)=(x^a)*(x^(-a))=(x^a)/(x^a)=1

Or in terms of x=5 and an arbitrary 'a':

5^0=5^(4-4)=(5^4)*(5^(-4))=(5^4)/(5^4)=625/625=1

[Trajectory]

I think there's an even simpler way to look at it.

How do you get from x³ to x²? Divide by x. How do you get from x² to x¹? Divide by x.

Continue this pattern and we can see to get from x¹ to x⁰ we divide by x again, and this gives us 1. We can also see that x^-1 is 1/x and so on.

[MazerRackhem]

That's another nice way of looking at it Trajectory, perhaps not quite as rigorous but probably more intuitively satisfying. Thanks for that.