I'm struggling to explain to a child the rules of division, why we add 0's , decimal points, and when we do those things. I know how to do it by memory, but I can't justify those steps. ( I think it is called long division. ) I need to understand the rules. All the material that I found so far are for kids, I feel like I need a more mature view about this algorithm. Thank you
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1I assume you mean decimal point by comma. – Kenny Lau Aug 10 '17 at 17:53
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1It is very easy to find sites such as this. – Weather Vane Aug 10 '17 at 17:54
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1You might want to use the column model (where you specify the place values of each digit). – Kenny Lau Aug 10 '17 at 17:56
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1Do you just need to divide two rational numbers, represented by finite decimals? (That falls under the arithmetic tag.) Or do you really mean two real numbers, represented by Cauchy sequences or similar? (That falls under real-analysis.) – Chris Culter Aug 10 '17 at 18:01
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that's right, two rational numbers represent by finite decimals such as 23,52 divided by 6,4 – Victor Aug 10 '17 at 18:05
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long division is an example of a recursive algorithm. Going to extremely long lengths to try to hide this fact and not talking about other more intuitive types of recursion is a problem I think. – James Arathoon Aug 10 '17 at 20:21
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I would highly recommend Sybilla Beckmann's book "Mathematics for Elementary Teachers".
This is a college level book specifically addressing all of your questions: they "why's" of elementary school mathematics. Included are explanations of the meaning of the 4 basic operations, why the standard algorithms for computing these operations make sense, etc.
Steven Gubkin
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Not sure if this is what you're after, but:
To justify the $0$'s and placement of the decimal point after division, taking your example:
$$23.52\div6.4=(2352\times0.01)\div(64\times0.1)$$ $$=(2352\div64)\times(0.01\div0.1)$$ $$=(2352\div64)\times0.1$$
From applying that method in a similar way to the general case, you see that you get the rule:
Multiply the result by $10^n$ after applying the division ignoring the decimals
where $n$ isthe number of places after the decimal in the divisorminusthe number of places after the decimal in the dividend'.
Shuri2060
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