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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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    What is the grammaticality issue here?
    – Kris
    Commented Feb 16, 2012 at 7:40
  • @Kris: It's about the why in grammar.
    – Noah
    Commented Feb 16, 2012 at 9:08
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    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
    Commented Feb 16, 2012 at 19:12

5 Answers 5

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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
    Commented Apr 4, 2012 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! :) Commented Apr 4, 2012 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
    Commented Jul 10, 2013 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. Commented Jul 11, 2013 at 1:27
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    ...but mostly I think it comes from the early schooling context of learning long division, where the operation is invariably described as answering the question "How many times does X go into Y?". In those early stages, no-one cares about awkward cases where the divisor is greater than the dividend. Commented Oct 21, 2014 at 12:40
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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
    Commented Feb 16, 2012 at 7:50
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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" Commented Feb 17, 2012 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
    Commented Jul 10, 2013 at 21:07
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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. Commented Feb 16, 2012 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. Commented Feb 16, 2012 at 18:00

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