# If we say A is weaker than B by a factor of x, then should x be smaller or bigger than 1?

In math, we always need to derive different algorithms to get a tighter error bound. It may be correct almost surely that: the error bound of B is better than that of A by a factor x (x is bigger than 1). However, I am not sure if it is right to say: the error bound of A is weaker than that of B by a factor x (x is samller than 1)?

• There is no defined relationship between strength and size. Commented May 3, 2016 at 11:29
• Based on the answer you selected, it looks like this was a math problem and not an English language problem. Am I correct? Commented May 4, 2016 at 2:15

My intuition is that you use the number bigger than 1, and Googling bears me out.

### If B is better than A by a factor of x, then A is weaker than B by a factor of x.

Not only is it correct, but it is equivalent to this statement:

### If B == A * x, then A == B / x

There should be no restriction on `x` though. No need to say (x bigger than 1) or (x smaller than 1). Whatever `x` is in one statement, it should be the same in the other.

Take the starting statement: `B is better than A by a factor of x`. What "better" means can be confusing since it is unique to your exact problem, so I'll just use the value (rather than the algorithm) of B and compare that with the value of A.

Now we have: `B > A by a factor of X`.

The phrase "by a factor of, means when either increasing or decreasing a value by multiplying it by some factor. Since we are measuring how "good" an algorithm is, we aren't just comparing two numbers on the normal scale.
If we have a logarithmic scale (base 10) where 5 is the worst an algorithm can perform, and 8 is the best an algorithm can perform, then the values 6 is better than the value 5 by a factor of 10. Similarly, 5 is worse than 6 by a factor of 10.

Below I provide links and show details about how to find this equation, but in this section I'll only solve it.

If we have some initial value, `i`, and we increase it by a factor of some unknown, `x`, and we end at some final value, `f`, we can solve for the factor:

``````(f / i) == f
``````

If we had decreased `i` by a factor of `x` and ended at `f`, we could solve for the factor like this:

``````(f / i) == (1 / f)
``````

See the inverse relationship?

We can use this to show that A is weaker than B by a factor of X.

``````Initial value = A
Final value   = B
Factor        = X

(A / B) = X

(B / A) = 1 / X
``````

So to get the factor, when we increase A by a factor of X to get to B, we find the factor is X. And if we decrease by a factor, i.e. go from B to A, we take the inverse and see the factor is 1/X.

If we give X a number, let's say 5, we can see the property in action:

``````(A / B) = 5
=> A = 5B

(B / A) = (1 / 5)
=> B = 1/5A
``````

You can plug in either value of A or B into the other equation to see they are both true.

The usage of saying that some value is decreased by a factor of some value isn't too common in pure mathematics (excluding grammar school math), but you do see it in branches like economics and physics.

See the definition of factor according to the Cambridge Dictionary's listing:

a ​number or ​variable (= ​letter or ​symbol) that is being ​multiplied in a ​product (= ​result of ​multiplying):

According to this definition, a factor is simply some number or variable being multiplied to another.

We can use the definition of "factor" to see how the following examples make sense:

Suppose you invested \$100, and after a time, your investment was worth \$300. The final value (\$300) would be three (3) times the initial value. We would say that your investment had increased by a factor of 3.

On the other hand, if you made a poor investment, and the value decreased from \$100 to \$25, then the final value would be a quarter (1/4) of the initial value. We would say the investment had decreased by a factor of 4.

This short article goes on to show how the final value divided by the initial value is equal to the factor when increasing a value by a factor. It also shows the factor is the reciprocal of that value when decreasing by a factor.

Whatever the common usage may be, "decreased by a factor of" is confusing, and should be deprecated. Rather say decreased to (state the fraction here) of instead: decreased to a quarter of, for example.
One of the reasons the decreased by usage is bad, is that it's used to mean what it shouldn't. If you say decreased by a factor of 4, then what factor of 4 are you talking about - 1, 2, 4? B is not decreased by a factor of the decrease, the factor you're talking about has to be a factor of B!

Do you mean you end up with what you started with, less 1/4, or 1/4 of what you started with? In the first case, starting with 100, you end up with 75; in the second, with 25.
This is not all. It leads to something like rainfall has decreased by 200%, which means we must somehow be giving rain back. Starting with 100mm, for example, we now have -100mm.
Journalists tend to use this style a lot, which just proves that innumeracy is part of their job description.

• +1 Frank, I think you had some really notable points that needed to be said. I completely agree that the usage of "decreased by a factor of" should be deprecated, especially outside of the colloquial of mathematics. However, I did feel that from reading your post it seemed as though there was absolutely no common practice or understanding of the phrase, even within mathematics. I disagree with that particular point. I think that in mathematics (and only mathematics), there is an agreed practice for the usage of the phrase. See my answer for what that exact usage is. Commented May 4, 2016 at 2:14
• Thank you, Matthew. All through my answer, I'm referring to common usage. Mathematicians know what they're talking about: if they didn't, it wouldn't be mathematics! Commented May 4, 2016 at 7:24

if B = A*x and A weaker than B then x should be greater than 1.

Example :

First one:

`A = 2; x = 5;// (x>1)`

`B = 2*5 = 10; //A weaker than B`

Second one:

`A = 2; x = 0.8;// (x<1)`

`B = 2*0.8 = 1.6; //B weaker than A`