The multiplication sign may be regarded as a form of "shorthand". In that sense, it could be described as "one step up" from addition. Instead of writing 7 + 7 + 7 + 7 + 7, we put 7 ´ 5.
The next step up from multiplication is "indexing" (or "raising to the power"). Thus, 7 ´ 7 ´ 7 ´ 7 ´ 7 may be abbreviated 75. That is also written as 7 ^ 5 (for example in computer programs).
Many people take it for granted that 5 ´ 7 gives the same answer as 7 ´ 5. However, we are simply lucky that it does (as indeed we are with x + y, which is always equal to y + x).
Although this "law of reversal" just happens to hold in what I have called the first and second steps of arithmetic notation (i.e. with addition and multiplication) - it does break down on the third step.
Compare for example 32 = 3 ^ 2 = 9 with 2 ^ 3 = 8. They are obviously not equal. Most of the time, in fact, x ^ y produces a completely different answer from y ^ x.
Are we entitled to ask for a reason why this reversal (or "commutative") law had to work in the first and second steps but then suddenly cease to be valid in the third step? Certainly, if that change-over had taken place lower down, then 5 ´ 7 and 7 ´ 5 would have produced different results (and we would then have had to learn twice as many multiplication tables!)
Everyone knows that ab = ba, but are we wrong to ask whether that can
actually be proved mathematically? (not just for integers, which is easy,
but for all values of a and b).
Another rule of arithmetic states that (a ´ b) ´ c = a ´ (b ´ c). This is known as the "associative law of multiplication".
For example, 2 ´ 3 ´ 4 produces the same result regardless of whether we perform the (2 ´ 3) first and then multiply by 4, or whether we start by multiplying (3 ´ 4) and then double that.
Once again, this is not taught formally to young children - it is simply taken for granted.
However, multiplication did not have to follow that rule; (once
again, we are just lucky that it does). When we climb up one step higher,
the associative law does not hold. In other words, (a ^ b) ^ c is not often
equal to a ^ (b ^ c).
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