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Let $a,b\in\mathbb{Q}$

How can I do the operation $a*b$ without knowing multiplication tables and without the help of a calculator? (Well, is it posible?)

If only one of them were whole numbers, it would be really easy: If $a\in\mathbb{Z}$ just add/substract $b$, $a$ times.

But if $a\in\mathbb{Q}$ then I'm missing a part of $b$ to add, the decimal part of $a$. If we had division, then it would be trivial, calculate $c=\frac{1}{a}$, and then $result=\frac{b}{c}$ but we cannot use divisions either.

Especially Lime
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Victor Oliva
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  • Do you have access to the numerator and denominator of those two fractions as separate integers? (At worst it would be "add $a$ to itself until you get an integer", at which point you can see how many times you had to add it to itself, and which integer you got.) – Arthur Dec 01 '17 at 10:22
  • There appears to be confusion about what situation you are actually in. In mathematics, saying you are given a rational number is usually taken to mean you are given an actual fraction. You appear (from the comments to an answer) to think about numbers being given using decimals, but most rational numbers don't have finite decimal expressions at all. – Marc van Leeuwen Dec 01 '17 at 10:49

2 Answers2

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Arthur basically already answered in his comment:
Add both numbers to itself until you get an integer, take note of how many times you did add it and which integer you got. This will give you $$a = \frac{a_n}{a_d}, \,\,\, b = \frac{b_n}{b_d}$$ with $a_n,a_d,b_n,b_d$ integers. Now, the product $$a*b = \frac{a_nb_n}{a_db_d}$$ can be computed using the multiplication in the integers; and you already gave the algorithm for that in your question.

Dirk
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  • I get these... However, this is based on the premise that you actually know how to represent rational numbers by a fraction between two integers... And if you have a decimal representation you can't do that by using addition/substraction, as well as the result is in fraction representation and you can't represent it in decimal if you don't know division – Victor Oliva Dec 01 '17 at 10:39
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    Of course you can. You assume that you can add rational numbers. Add a rational number to itself often enough and you will always get an integer. If you say that you can't to exact addition (e.g. you only have access to the first few digits of your numbers), then there is of course also no way to do exact multiplication. All you need to use is that $0.\overline{9} = 1$, should you encounter such an infinite sequence at some place in your numbers. – Dirk Dec 01 '17 at 10:40
  • When I was in 3rd grade I could add decimal numbers without knowing about fractions. This is the operation that's allowed. If we have $1.25$ as one of the numbers, you and I know this is $\frac{5}{4}$ and then we can apply this method, but if we don't know about fractions and/or how to convert from decimal to fraction then we can't use this method, because I need to know what $a_{n}$ and $a_{d}$ are. – Victor Oliva Dec 01 '17 at 10:44
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    $1.25 + 1.25 = 2.5$. $2.5 + 1.25 = 3.75$. $3.75+1.25 = 5$. Therefore, we see that $4*1.25 = 5$ and thus $1.25 = 5/4$. If you are asking for an approach that is understandable for someone who does not know about fractions rather than an algorithm that will only use addition then you should say so; however, I only see such an algorithm when the decimal expansion is finite, otherwise it might get difficult. – Dirk Dec 01 '17 at 10:50
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Supposing for simplicity $a,b>0$, write $a=\frac pq$ and $b=\frac rs$ as reduced fractions (just to save work), then $a+b=\frac{ps+qr}{qs}$, and all those products are products of integers , which you can do by repeated addition.

For multiplication, it is even easier, $ab=\frac{pr}{qs}$. The hardest part is to reduce the resulting fraction to lowest terms, for which you need the to apply the Euclidean algorithm for finding the $\gcd$. This can however be done by repeated subtraction instead of using division, if you must.

Marc van Leeuwen
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