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Why does "divide two into four" equal two, and "divide two by four" equal one half?

Correct if I am wrong, but this what I have learned recently.

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One and a half ? Do you mean just one half? –  simchona Feb 16 '12 at 7:22
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What is the grammaticality issue here? –  Kris Feb 16 '12 at 7:40
    
@simchona: Yes, fixed it. –  Noah Feb 16 '12 at 9:06
    
@Kris: It's about the why in grammar. –  Noah Feb 16 '12 at 9:08
    
As a AmE speaker from NE US, I have to say that I have a hard time recalling EVER hearing someone say "divide X into Y." I would not have been able to answer this question from someone without clarification of what they were asking. –  horatio Feb 16 '12 at 19:12
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5 Answers

up vote 4 down vote accepted

OP's confusion arises because "divide 2 into 4" is an idiomatic usage meaning perform a division operation, using 2 as the divisor, and 4 as the dividend.

symbolically...
4/2 = 2

clearest verbal form...
divide 4 by 2, giving the answer 2

idiomatic alternative...
divide 2 into 4, giving the answer 2

It's not the same usage as dividing a pizza into 4 [pieces], where you'd probably get 1/4 (a quarter) of a pizza as your share!

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I have got conflicting answers for this question. Your answer seems to be in conflict with Jim's answer. Any thoughts? –  Noah Apr 4 '12 at 2:02
    
@Noah: The other answerers either don't know of or forgot about the idiomatic usage referenced by my first link. I'm sure some people are unfamiliar with this usage because they've upvoted other answers, not mine. But you've obviously come across it or you wouldn't have asked the question. And obviously if "divide two into four" = two, it can't be the same usage as "divide a pizza into four", since that will give you four quarters, not two pizzas! Put a bounty on the question if you want some more people to think this one through and confirm that I'm right (or just believe me! :) –  FumbleFingers Apr 4 '12 at 2:49
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For me (an American) divide 2 into 4 is not the same as divide 2 by 4. This one: divide 2 into 4 uses 2 as the divisor, as you say: 4/2 ... but divide 2 by 4 I take to mean use 4 as the divisor, 2/4 . –  GEdgar Jul 10 '13 at 20:59
    
@GEdgar: You're quite right! I'm not sure how I managed to buy into what Jez had already answered at the time. Existing votes notwithstanding, I'll edit it to make more sense. –  FumbleFingers Jul 11 '13 at 1:27
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In math the term divide may have two different senses.

1) To use (a number) as a divisor. E.g: divided 5 into 35 = 7

5 is the divisor.

2) To subject (a number) to the process of division [by a divisior]. E.g: If you divide 6 by 2, you get 3

2 is the divisor.

10 divided into 100 = 10
100 divided by 10 = 10

Please check divide.

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I suspect that if you actually asked on math.se they'd tell you they don't normally use the phrasing "divide X into Y" –  FumbleFingers Feb 17 '12 at 2:43
    
@Fumble: On the Wikipedia page for long division ... en.wikipedia.org/wiki/Long_division ... read down a ways and find this: "In this example, 37 is to be divided into 1260257." –  GEdgar Jul 10 '13 at 21:07
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The expression of division using the word "into" comes from the form: "Two goes into four how many times?" Answer: two.
Dividing 2 by 4 means breaking(dividing) 2 into 4 equal parts ( 0.5 each).

You might think about it this way: No LaTeX in ELU!

Have a look at this link for an example of "goes into": http://www.themathpage.com/arith/divide-whole-numbers.htm

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Is this a canonical rule? There are exceptions to this implication of into in the context of division. –  Kris Feb 16 '12 at 7:50
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My experience of "divide [number] into [number]" was solely as regional spoken idiom. In different math textbooks used ca. 1965 in public schools both in suburban Seattle and on the South Side of Chicago, "40/8" was written out as "forty divided by eight." Outside Seattle it was said the same way. In Chicago, though, it was said "eight divided into forty." The discrepancy offers a good lesson about relying on numerical rather than verbal expressions when math is involved.

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"Divide 2 into 4" and "divide 2 by 4" seem semantically identical (2 / 4) to me; I don't recognize the former's meanining as 4 / 2.

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I'm assuming "divide 2 into 4" has the meaning "remove 2 from 4", hence the answer is 2. Looks like divide has the meaning subtract here. –  Vidya Murthy Feb 16 '12 at 8:00
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@user543101: That would be a seriously non-standard interpretation. Regardless of whether the preposition is by or into, subtraction would never be involved. –  FumbleFingers Feb 16 '12 at 18:00
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