Zero Power

[sologuitar]

Let me begin by saying that I am doing a self-study of college algebra. I love mathematics. I regret not majoring in math back in my student days. With that said, I hope this forum will allow me to post questions that I get stuck with in my review of college algebra. 

Let a = any integer.

Why does a^(0) = 1? 

[Eise]

One simple way to see it:

• a^3 = (a^4)/a
• a^2 = (a^3)/a
• a^1 = (a^2)/a = a
• a^0 = (a^1)/a = a/a = 1

Does that make sense?

 

[sologuitar]

14 hours ago, Eise said:

One simple way to see it:

• a^3 = (a^4)/a
• a^2 = (a^3)/a
• a^1 = (a^2)/a = a
• a^0 = (a^1)/a = a/a = 1

Does that make sense?

 

Let me see.

You are saying that, for example, a^0 is the same as (a^1)/a = a/a = 1.

How do you go from (a^1)/a to a/a?

Here is my logic considering the fact that the numerator and denominator have the same base a. 

I understand that any variable has a power of 1. Yes?

For example: x = x^1, y = y^1, z = z^1, etc. Back to my example. 

(a^1)/a = a^(1 - 1) = a^0 = 1. The jump from a^0 = 1 is not clear for me. 

 

[swansont]

2 minutes ago, sologuitar said:

How do you go from (a^1)/a to a/a?

a^1 = a

[Sensei]

a^1, a^2, a^3 are easy - anybody can do such calc even in memory.

But mathematicians want to have universal functions, in this case pow(a,x).

So, take piece of paper and make graph of f(x)=a^x, with a couple well-known x (a is constant and can be used any integer >= 2).

Then use x=0.5, 0.25, 0.2, 0.125, 0.1, 0.01 etc. and you should see curve goes closer and closer to 1.

Then draw line between them.

With a=2, and default x in range -2 to +2, you will get such curve:

https://www.wolframalpha.com/input?i=f(x)%3D2^x

If you want to change range, use f.e.

https://www.wolframalpha.com/input?i=f(x)%3D2^x%2C+x+range+-10+to+10

(Wolfram Alpha, and Excel/Spreadsheet are must-have for mathematicians these days)

(you can draw such graph in Excel/Spreadsheet too)

 

[sologuitar]

5 hours ago, swansont said:

a^1 = a

Ok. I get it now. 

4 hours ago, Sensei said:

a^1, a^2, a^3 are easy - anybody can do such calc even in memory.

But mathematicians want to have universal functions, in this case pow(a,x).

So, take piece of paper and make graph of f(x)=a^x, with a couple well-known x (a is constant and can be used any integer >= 2).

Then use x=0.5, 0.25, 0.2, 0.125, 0.1, 0.01 etc. and you should see curve goes closer and closer to 1.

Then draw line between them.

With a=2, and default x in range -2 to +2, you will get such curve:

https://www.wolframalpha.com/input?i=f(x)%3D2^x

If you want to change range, use f.e.

https://www.wolframalpha.com/input?i=f(x)%3D2^x%2C+x+range+-10+to+10

(Wolfram Alpha, and Excel/Spreadsheet are must-have for mathematicians these days)

(you can draw such graph in Excel/Spreadsheet too)

 

Thanks but I am not trying to get technical here. I am simply reviewing material learned long ago. In fact, I think there are video clips on YouTube that explain this concept without using too much mathematical jargon.