My friend has recently been challenging me to solve some maths problems, the latest challange is to find a method of finding the answer of $2.5 \cdot 2.5$ without ever using multiplication. Now with whole numbers this is easy, you would just use repeated addition. Example: $3 \cdot 3= 3 + 3 + 3$. However this isn't possible with decimal numbers.(As far as I know) Is there a way to do this and the method also working for all other decimal numbers?
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If you have a finite decimal representation, you can do the same trick, just move the decimal (e.g. to perform $2.5\times2.5$ add $25$ to itself $25$ times and then shift the decimal 2 places to the left). This wouldn't work for irrational numbers though. – Peter Woolfitt May 06 '16 at 08:46
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Alternatively, for $2.5\times 2.5$ it isn't that hard to add $2.5$ twice and then add half again (in a sense using a combination of addition and division instead of multiplication). – Peter Woolfitt May 06 '16 at 08:49
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3@PeterWoolfitt : but moving the decimal is multiplying by 10. A tongue in cheek answer : $2.5*2.5 = \exp(\ln(2.5)+\ln(2.5))$ : I don't use multiplication, just addition, logarithm and exponentiation ^^ – Tryss May 06 '16 at 08:49
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@Tryss Well, any operation is going to end up being equivalent to multiplication... I guess the idea (as it seems to me) is that you never have to perform the standard multiplication algorithm. Heh, and that's a nice solution. – Peter Woolfitt May 06 '16 at 08:50
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A nice exercise is to prove the following that my dad taught me when I was a kid: to mulitply by itself any number ending in five, say $;X5;$ , just multiply $;X;$ by $;X+1;$ and then just attach $;25;$ at the end right to the product. For example, $;25\cdot25;=(2\cdot3)25=625;,;;145\cdot145=(14\cdot15)25=21025;$ , etc. With decimal numbers the similar trick is obvious: $;2.5\cdot2.5=6.25;$ . – DonAntonio May 06 '16 at 20:49