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Thank you for a question and answers at Divide two into four and Divide two by four

However, can anybody explain why "dividing a pizza into 4" is different from "dividing 1 into 4"? 1 pizza and number 1 would be same one piece of object. As soon as if we add "pizza" after a or "1", It would be completely backward or flipped over. So it really does not make any sense to me. I would like to know the logic behind it. So for instance, when a teacher is explaining "1/4 as dividing 4 into 1" in math class, as soon as he add a word "pizza" as an example, it would become "dividing a pizza into 4". It would be very confusing. I wonder why such a confusing expression / grammar was created. Who started to use such a confusing mathematical notation, assuming " mathematical notation" was created after regular expression. I want to know the history behind it. Or maybe I should ask you this way, which expression is wrong? Both cannot be correct when sequence and positions are the same. From point of view of "into", mathematical expression seems to be wrong.But for instance, one day a mathematician started to use the expression "divide A into B" which is B/A even though general world expression was A/B when he invented the expression. Why did he do it? "Dividing A into B" made more sense to him? Most of you are brainwashed so I am pretty sure what I am saying may not make any sense to you but there must be a reason to flip over somehow. So I want to know why he flipped it over. For an example again, in C language world "=" is not equal. Equal is "==". "=" is casting a value on the right to a value on the left. So "divide A into B" had to be B/A somehow. Simply"divide A by B" was A/B but when he wanted to say "divide A ??? B" for B/A but the ??? was "into" somehow.

I still need an answer. Nobody has not answered my question yet. I am asking you about an issue of coexistence of conflicted definition.

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    This is more of an English Language Learners question topic. (A problem is that none of your uses quite fit practice. We'd say "...divide a pizza into fourths." or "...quarter a pizza" or "...divide a pizza into four pieces.". Saying "divide 4 into 1" is meaningless, really, but is sort of an example of "transforming this into that". "Divide 4 by 1" would be a more proper phrase, but either would logically mean the reciprocal of what you think (4÷1) and be similar to the second pattern you mention. The difference between raw numbers and real objects doesn't make that weird change.)
    – The Nate
    Commented Feb 18, 2019 at 19:12
  • I think you are getting confused on the phrase "How many times does denominator go into numerator" (numerator/denominator=numerator÷denominator) (e.g. "4 goes into 1 only a quarter of a time." or "4 goes into 1 no times leaving 4 remaining", depending on who's trying to explain what. Other examples: "3 goes into 10 3 times with 1 remaining." (=3 1/3=3.333 repeating) or "four goes into eight two times." (=8/4 =8÷4 =2))
    – The Nate
    Commented Feb 18, 2019 at 19:30
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    Because English, not mathematics.
    – Hot Licks
    Commented Feb 18, 2019 at 19:43
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    @TheNate I don't think "divide 4 into 1" is meaningless. It definitely has a meaning for me. Though that way of expressing 1/4 is rare as far as I know.
    – Zebrafish
    Commented Feb 19, 2019 at 1:11
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    When you divide 1 into 4, you get 4 equal pieces. When I divided the pizza I made into four an hour ago, I cut it into a half, a quarter, a sixth, and a twelfth, give or take. English isn't math, much to the disconcertion of many math teachers, programmers, and logical thinkers everywhere.
    – Ed Grimm
    Commented Feb 19, 2019 at 3:25

3 Answers 3

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In some mathematics contexts the construction divide X into Y means Y/X. Here you're speaking of the numbers X and Y.

  • Frankly, I would avoid this use altogether. It's fairly rare—I don't recall ever seeing it any context except elementary-school textbooks— and may be confusing. Say divide Y by X instead.

But when you divide one pizza into four you are not employing the construction mathematically but in its ordinary, everyday sense, where X and Y are not numbers but quantifiers: divide X objects into Y parts, meaning X/Y.

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  • Thank you for your answer. It is just like in grocery store, 2/$3 means 2 for $3, whereas in mathematical expression, it would be $3/2=$3 per 2, isn't it? But I wonder why such a confusing expressions exist. People just don't care about it? Therefore "Frankly, I would avoid this use altogether." would be one of right answers for me.
    – Shunsuke Akagi
    Commented Feb 18, 2019 at 20:13
  • @ShunsukeAkagi Where have you seen the construction 2/$3 in actual use? Or an example of anything similar?
    – The Nate
    Commented Feb 19, 2019 at 4:35
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You're missing context.

One is describing how to literally divide a real world object (a pizza into 4 pieces), and the other is describing how to perform a mathematical expression with words.

When describing mathematical notation, "Dividing 1 into 4" means 4/1, not 1/4.

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  • Thanks for your answer but I wonder why such a confusing expression / grammar was created. Who started to use such a confusing mathematical notation, assuming " mathematical notation" was created after regular expression. I want to know the history behind it.
    – Shunsuke Akagi
    Commented Feb 18, 2019 at 20:01
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In the pizza example, into reflects time’s arrow: pizza

In the second example, into represents reading left to right: division

Personally my experience supports this. When I’m tutoring kids who use long division, I use into; when I’m in class and we’re dealing with large numerators and denominators, I say over and use the reversed order.

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  • @ShunsukeAkagi You can express appreciation for an answer by upvoting or accepting it with the green checkmark
    – gen-ℤ ready to perish
    Commented Feb 18, 2019 at 21:30
  • You haven't answer my question yet. I hope you can find it.
    – Shunsuke Akagi
    Commented Feb 18, 2019 at 23:31
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    +1 for a nicely illustrated answer. The handwritten long division made a lot of sense.
    – user205876
    Commented Feb 19, 2019 at 1:39
  • Ironically, your use of "reflects" is similarly ambiguous to "into". I can see that you intend it to mean "represents", but in mathematics, we also use "reflects" in the sense of a mirror image, which would mean the arrow goes in the opposite direction!
    – Brendan W. Sullivan
    Commented Feb 20, 2019 at 17:37
  • @BrendanW.Sullivan If you like mathematics, you should go check out my page on that network
    – gen-ℤ ready to perish
    Commented Feb 20, 2019 at 20:01

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