The Math Mojo Chronicles » exponents

The Power of Zero

Brian — Mon, 07 Jun 2010 00:27:23 +0000

This entry might be better titled, “The Power of Zero to Confuse Us.” (no, not Confucius!)

A curious reader (the best kind!) asked:

I have a math question:

8/ -m⁰ + n⁰ I cannot figure if it is 8 / 0 = 0 or 8/ -2 = -4.

Please help me figure this problem out.

Professor Homunculus replies:

Hi. That’s a great question, because it involves so many interesting principles. Are you aware of these facts?:

Any non-zero number (even a negative number) to the zero power is considered to equal one. (One quick explanation why: x5/x5 = x⁽⁵⁻⁵⁾ = x0. But anything (besides zero) divided by itself is 1, so x5/x5 = 1.) Check out: http://mathmojo.com/interestinglessons/exponentsntothe0power/exponentsntothe0power.html

Zero to the zero power is “indeterminate” (therefore not always useful to solve equations). Check out: http://mathmojo.com/interestinglessons/n_to_the_0_power/n_to_the_0_power.html

8/0 does not equal 0. Any non-zero number divided by zero is undefined (therefore not useful to solve equations). That is important to know.Check out: http://www.mathmojo.com/interestinglessons/division_by_zero/division_by_zero_1.html

Scenario 1: Assuming that n is zero

Right off the bat, we can see that if n were zero, then n⁰ would be “indeterminate”, thus making the equation not ultimately solvable. Some people say the answer “should be” 1. I think that in order to show that someone understands the ambiguity of 0⁰, 0⁰ should be generally expressed as “indeterminate, or 1, depending…”

Here is a very useful explanation of zero to the zero power from the very useful website, “Ask Dr. Math” at: http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

“Other than the times when we want it to be indeterminate, 0⁰ = 1 seems to be the most useful choice for 0⁰ . This convention allows us to extend definitions in different areas of mathematics that would otherwise require treating 0 as a special case. Notice that 0⁰ is a discontinuity of the function f(x,y) = x^y, because no matter what number you assign to 0⁰, you can’t make x^y continuous at (0,0), since the limit along the line x=0 is 0, and the limit along the line y=0 is 1.

“This means that depending on the context where 0⁰ occurs, you might wish to substitute it with 1, indeterminate or undefined/nonexistent.

“Some people feel that giving a value to a function with an essential discontinuity at a point, such as x^y at (0,0), is an inelegant patch and should not be done. Others point out correctly that in mathematics, usefulness and consistency are very important, and that under these parameters 0⁰ = 1 is the natural choice.”

By the way, one of the many reasons to avoid calculators for such things is that many of them (including the google calculator) return the result ) 0⁰=1. That is because calculators do not have actual brains, just like the people who insist that elementary school math be done on calculators.

Although 0⁰=1 may be the “most useful” answer in many situations, it is not the answer. Being dependent on calculators would prevent someone from understanding the deeper meaning of 0⁰. This is a terrible problem with calculator-dependency.

Blindly accepting that 0⁰=1 makes us no better than performing chimps. It means we can “do” something, but we don’t know what it is we are doing.

So, if we assume that n is zero, then the answer is either “unsolvable because a term is indeterminate,” or , if we assume that n⁰=1, then -m⁰ + n⁰ = 2, thus making 8/ -m⁰ + n⁰ = 8/2 = 4 Neither answer should get you “points off” on a test, but I’m sure most testers don’t understand the implications of the question anyway, so who knows what they would accept as correct!

The best answer is the most complete, and the one that demonstrates that you know what you are talking about.

Scenario 2: Assuming that n is not zero

Assuming that n is not zero, here’s my answer and my reasoning:

-m⁰ is equal to 1, as is n⁰. That makes -m⁰ + n⁰ = 2, thus making 8/ -m⁰ + n⁰ = 8/2 = 4

So to sum up:

Since the question does not specify if n can or cannot equal zero, we can’t assume that it cannot equal zero. Therefore, it might, or it might not. So if n = 0, our answer would be as explained above: either “unsolvable because a term is indeterminate,” or “4.” If n does not equal zero, then the answer would be 4.

I hope that helped,

Brian (a.k.a. Professor Homunculus)

Exponents of the Zero Power

Brian — Tue, 07 Aug 2007 06:35:28 +0000

Someone wrote in to ask:

4⁰ * 5³ is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The answer actually should not be zero, and here’s why:

Because 4 to the 0 power is 1, not 0.

So 4⁰ * 5³ would be 1 x 5³ which is 125.

Any integer raised to the zero power equals 1.

That is hard for most people to believe, so I wrote a little piece to explain why it makes sense. Here it is:

In modern mathematics, we usually use a base system to represent our numbers.

You know that we have units, tens, hundreds, etc. in our base-ten system.
Well, we also represent those columns in terms of 10 to the nth power. In other words, the thousands column is represented by 10³. So 8,000, for example is 8 *10³.

8,300 would be (8 * 10³) + (3 * 10²).
8,320 would be (8 * 10³) + (3 * 10²) + (2 * 10¹).

Now comes the problem. You see how the base stays the same, and the exponent gets smaller? To represent the units column, mathematicians have accepted the convention that 10⁰ will always equal 1.

That keeps the rule going. So 8,325 would be:
(8 * 10³) + (3 * 10²) + (2 * 10¹) +(5 * 10⁰).

Because other bases systems, (base 2, base 3, etc.) work the same way, we have further accepted the convention to be n⁰ always equals 1, no matter what the base.

Take the binary (base 2) number 1011 for example. What that means is
(1 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰).

That is the same as 8 + 0 + 2 + 1,

which is the number 11 in our normal base 10 system.

As long as we keep n⁰= 1, then the units column of any base will always mean how many ones there are in it.

Math is a network of "conventions" mankind has accepted to make it work. It is based on rules and axioms that are the most "convenient". They may seem hard to figure out at first, but when you get down to it, the rules that we use are basically the best we can come up with to get the things done that we want to accomplish.

Every once in awhile some genius comes up with a rule that makes something even simpler than how we have been doing it up until now, and that becomes the new convention. But that rule has to be solidly based on what has come before, and may not break any of the other rules.

Maybe someday you will be one of the people who comes up with something that explains, or enables something that until now was done by a convention that needed improving.

Have fun!

Professor Homunculus

Multiplication, Addition of Exponents

Brian — Wed, 23 May 2007 12:06:23 +0000

Which is greater, x²*y² or x²+y²?

I am thinking about an example from a GRE (graduate record exam) book
that was shown to me.
I think it was "Which is greater, x²+y² or (x+y)²?

Here is the poop on how to think about examples like that. When in doubt – substitute
(if you can) for whole numbers. (In the original post, I had written real numbers instead of whole numbers. See the comment below about this by astute reader Randall Jones for important information about the difference that makes in this equation.)

So, try, say, "Which is greater, 5²+3² or
(5+3)²?"
In the first case, 5² = 25 and 3² = 9, so it would be
25+9, which equals 34.
In the second case, you would first do the 5+3 (because parenthesis come first
in the order of operations) and get 8. Then you would square that, and get
64, which is clearly greater than 34.
Therefore (5+3)² is greater than 5²+3².

For an easy substitution you can do in your head in seconds, substitute 1s for x and for y:
= x²+y² or (x+y)²
= 1+1 or 2 ²= 2 or 4

What if the example had been a bit different, though? What if it had been:
"Which is greater, x²*y² or (x*y)² (using multiplication instead
of addition)?

This article is continued at Mathmojo.com.