if we have ab what does it mean exactly ? lets assume I have 1+1+1+1+1 .. a times, then 1a=1+1+1...(a times)

[Ryan$]

Hi guys, I just want to verify about something that I find it sometimes hard ..
if we have ab what does it mean exactly ? lets assume I have 1+1+1+1+1 .. a times, then 1a=1+1+1...(a times) but what's confusing me is it include the first element that we add? I mean in my example of 1+1+1.... , will 1*a is the result of the whole sum of numbers of "1" with including the first element there(the first element of "1+1+1+... " is 1 because it's first element)?!)


thanks alot

 

[JeffM]

Try this as a recursive definition of multiplication of non-negative integers.

[MATH]a, \ b \in \mathbb Z_{\ge 0} \implies[/MATH]
[MATH]a * b = 0 \text { if } a = 0, \text { but}[/MATH]
[MATH]a * b = (a - 1) * b + b \text { if } a > 0.[/MATH]
You are correct that defining multiplication as repeated addition is meaningless if we are talking about multiplying by an integer less than 3. Addition is a binary operation, meaning that you need two numbers to do one addition, and consequently you need at least 3 numbers to do repeated addition. But notice that the definition above does mean that when we are multiplying by an integer greater than 2, the result is exactly the same as for repeated addition. How things are explained to children aged 7 is not necessarily a useful way to define things for more advanced students.

 

[Dr.Peterson]

For clarity of language, I prefer to describe the multiplication ab (in the repeated addition sense, applicable only to positive integers) as starting with 0, and adding a, b times: 0+a+a+...+a. This way, the count (multiplier) b can be any positive integer, even 1 (or, in fact, it could even be zero and still make sense). And the multiplier is actually the number of additions, which is not true if you think of it as adding b a's, a+a+...+a. (My approach is also equivalent to JeffM's recursive definition.)

Now, when we move on from counting numbers and work with signed integers, then rational numbers, and beyond, we have to leave behind this "(optionally) repeated addition" concept, and generalize the meaning, extending the definition to new kinds of numbers in a way that retains consistency with the properties of multiplication. So by the time you learn algebra, this definition is obsolete.

Also, I have followed your understanding of the multiplier as the second number; multiplication can also be introduced taking the first number as the multiplier (as JeffM did), and either way works just as well.