Expert: Ahmed Salami - 6/22/2011

Question
QUESTION: I've gotten into a debate with someone over the nature of applying percentage labeling to numbers.

Am I correct when I assert that any number in existence (given that it is 100% of itself) is therefore 100% greater than zero?

This seems like a obvious mathematical certainty to me.

The debate comes into play when I asserted that increasing zero by any value is an increase of 100%, because any value relative to itself is 100% and zero has no value and percentage does not apply to it. There's no such thing as 100% of zero anymore than there is 50% of zero.

Ergo, if you go from zero to any non zero value, it's mathematically sound to claim this is a 100% increase. For the record, I was referring to a positive number which was one, although any positive number is yielding the exact same result (100% increase).

ANSWER: Hi Robert,
I'm sorry this is terribly late.
I'd be happy to talk more about this, but right now i'll just make a few comments.

First of all, it sounds incorrect to say that any number in existence is 100% greater than zero. The important thing is to realise that we're talking relative to zero here. So, to say that 'increasing zero by any value is an increase of 100%' would be wrong.
Any number x cannot be 100% greater than zero.
As an illustration, do you agree that 3 is a 200% increase in 1? If so, then 3 has to an infinite percentage increase in 0.
You should realise that the percentage value would always refer to the number we're operating on. So, if you go from zero to any non zero value, its a 100% increase regarding that number, not zero, which is why you cant phrase your assertion like that.
But of course zero would be 100% lesser than any number in existence.
Consider this, 3 is a 200% increase in 1 but 1 is not a 200% decrease in 3.

Regards

---------- FOLLOW-UP ----------

QUESTION: If I may pose a math problem then:

If 2 is increased to 4, this increase would be expressed, as a percentage, as 100%. Correct?
If 1 is increased to 3, this increase would be expressed, as a percentage, as 200%. Correct?
If 100 is increased to 110, this increase would be expressed, as a percantage, as 10%. Correct?

What about the following examples?

0 increased to 4
0 increased to 3
0 increased to 110

Expressed as a percentage, what is the answer for those three increases?

ANSWER: Hi Robert,
Usually, percentages are used to indicate changes in non zero numbers and the definition is sensible that way. Mathematically though, an increase in zero to any other number, whatever it may be, is simply an infinite increase.
If you increased 1 to 3, then the percentage increase is
(3 - 1)/1 x 100 = 200%
If you increased 0 to 3, then the percentage increase is
(3 - 0)/0 x 100 which would then be an infinite percentage increase.

Regards

---------- FOLLOW-UP ----------

QUESTION: Hello Ahmed,

I must admit, your initial formula had me quite confused! I was reading it like so:

(x-y)/1 x 100 = %

Which had me wondering where the 1 came from and noting it wasn't correctly calculating other ratios. I then concluded you actually meant this:

((x/y)-1) x 100 = %

until I realized your equation was actually this:

(x-y)/y x 100 = %

I was beginning to write a confused reply until I caught myself...that would've been embarassing. My bad!

You said you end up with an infinite percentage increase. I'm not following this logically, because the ratio is quite clearly a very finite increase.

You also said zero is indeed 100% lesser than any number in existence (I'm obviously not worrying about negative values here!).

If zero is indeed 100% lesser than any (positive) number in existence, then it logically follows that any number in existence is 100% greater than zero. This seems like a tautological assertion, whereas the only rule is no number in existence can be anything other than 100% greater than zero (when dealing with positive values).

Ergo, this the position I was operating from. Any number in existence qualifies as 100% greater than zero (or simply worded differently, zero is 100% lesser) by virtue of having an actual value relative to a non value.

I attribute this to the quirk of applying percentage value to a non value concept. So long as I'm asserting an increase of 100% relative to zero (no other percentage value has meaning relative to zero), any value in existence can be arbitrarily picked at random and be a 'correct' answer. The only other recourse would be to claim increasing zero by 100% yields zero, which is an increase of nothing and cancels out the very assertion claimed to be made (an increase!).

I sincerely hope I'm not boring or frusterating you with my mathematically limited knowledge, as I find this train of thought very interesting.

Answer
Hi Robert,
First of all, your concerns are not boring or frustrating. Curiosity stands you in good stead for good scientific thinking.
Now, i'll address some of the statements.
"I'm not following this logically, because the ratio is quite clearly a very finite increase". Well, the increase is finite but the ratio is not. You need to remember that a ratio is actually a relative term. Increasing different numbers by the same amount doesnt result in the same percentage increase. Going from 1 to 3 is a 200% increase while from 2 to 4 is just a 100% increase even though the numerical increase is 2 in both cases.

"If zero is indeed 100% lesser than any (positive) number in existence, then it logically follows that any number in existence is 100% greater than zero". Consider this, again, 3 is a 200% increase in 1 but 1 is not a 200% decrease in 3. You should know that when you make a statement involving two numbers and an associated percentage, the percentage change is referring to a particular one of them considered as the base value. So, the statement '3 is a 200% increase in 1' actually means '3 is a 200% (of 1) increase in 1. For the same reason, '1 is not a 200% decrease in 3' because '1 is not a 200% (of 3) decrease in 3. But 1 is a 66.7% decrease in 3.

"Any number in existence qualifies as 100% greater than zero". If we say that 4 is 100% greater than 0, then 4 would be how much percentage greater than 2? 50%?
By calculation, if we increased x to y, how do we find the percentage increase? I'm sure you agree that it is
(y - x)/x . 100
The question for you is why are we dividing by x and not y???

Regards