# A value has 'reduced by factor of 3'. Does this make mathematical sense?

Last year there were 3000 observations, this year there are only 1000. This is described as showing a "fall by a factor of 3".

This phrase doesn't ring true. If a factor of 3 is a 1/3, then a fall by a third would be down to 2000. So the phrase is meant to represent a fall to a third.

Am I right in thinking the phrase 'by a factor of' can only refer to an increase?

cheers, Ian

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It makes perfect sense to me. But this is a question about language, not mathematics. – TonyK Sep 14 '11 at 11:11
I second @TonyK's comment that this is a question about language. In fact, I think this could be a good fit for English.SE (as well). – Srivatsan Sep 14 '11 at 11:20
@TonyK : You said the phrase made perfect sense. Did you interpret it as a fall to a 1/3 of the previous value, or a fall by a 1/3? Personally I see this a question regarding mathematical language - though such a tag didn't exist. – ianmayo Sep 14 '11 at 12:27
I just saw on the news earlier tonight that over the past ten years, robberies on Sydney trains had fallen by 114%. I'm not sure how to interpret the negative number of robberies. Perhaps criminals are now threatening to hurt people who don't take money from them. – Gerry Myerson Sep 14 '11 at 13:31
reduced by a factor of 3 seems OK to me. But I do object to 3 times less as ambiguous. – GEdgar Sep 14 '11 at 13:37

Last year there were 3000 observations, this year there are only 1000. This is described as showing a "fall by a factor of 3".

What is intended to be communicated is that there was a reduction or a decrease in a statistic to 1/3rd of the original value. Is this correct? What is the definition of a factor? Wolfram defines it:

Factor

A factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity. The determination of factors is called factorization (or sometimes "factoring").

Typically, in the context of non-algebraic factorization, the intent of the usage of factor is an integer factorization, but this is not a requirement of the definition. If you multiply factors together to obtain a result, what is the meaning of the phrase, "to reduce by a factor?" If the meaning is that one should divide by the factor, such that the result was the original quantity, then the usage was correct. However, if the meaning is that the result is reduced by multiplying by a factor (perhaps an arbitrary one between one and zero) then the usage was incorrect.

The phrase "fall by a factor" is therefore easily misunderstood outside of a broader context. A factor is a number you multiply by to get a result, and it is unclear without context whether one means the result is the number reduced by division or found by multiplication. If the "result" is the new number, the use of the word "factor" is wrong (perhaps "divisor" would be correct). The "result", outside of context of the numbers, is not well defined.

If you must emphasize the reduction, I recommend a context-free unambiguous verbal formulation stating that the "new result fell to 1/3rd of the prior statistic."

This question is at the intersection of English and Math. I think this is apropos to Math Stackexchange because you cannot do math without communication, and as this is a question about communicating a mathematical relationship, it is well suited for this site.

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