Multiplication and Division of Decimals

[jimgavagan]

What is division? 12 ones split up into 3 things is 4 ones (per thing.) 1 one split up into 100 things is 1/100 ones (per 1 thing.) Thus, "5 ones split up into 1/100 things is 500 ones (per 1 thing)" is probably better understood as "5 ones split up among hundredths of a thing is 500 hundredths, or 1 one per hundredth of a thing." On the other hand, for non-decimals, I would write that "5 ones split up among one-hundred things is 1/20 (0.05) ones per thing."

Fine. Logical.

Still, how is dividing something by a fraction (0 < x < 1) the same as multiplying something by the inverse of that fraction?

If X * Y = X / (1 / Y), and "X ones split up among 1/Y of a thing is X*Y "1/Y's," or 1 one per "1/Y" of a thing," then what would be the complement terminology in multiplication to the "split up among" terminology in division?

 

[cepheid]

You can "verbalize" all you want. I'm sure everyone has his or her favourite way of expressing something in English in such a way that it "makes the most sense" to him or her. I tend to think of 5/3 as "divide 5 into three parts". Similarly, I tend to think of 5/(1/3) as "divide 5 into 'one-third' parts" which I interpret to mean, "make it so that 5 is 1/3 of the whole" (the whole would therefore be 15). Another way to think about it: the result of the operation has to give you 5 back when it has the inverse operation applied to it (which is multiplication by 1/3).

But at the end of the day, NONE of this 'conceptualizing' matters. Mathematics is about definitions. Those definitions do not have to agree with your intuition nor do they have to fall within the framework of any sort of real-world example/application (the sorts of things on which your intuition is usually based). The only *requirement* is that these definitions be internally consistent (i.e. consistent amongst themselves). We define the multiplicative inverse (a.k.a. reciprocal) of some number A to be another number B such that:

A*B = 1

It follows from this definition and other basic mathematical rules that:

B = 1/A [1]

However, it also follows that

A = 1/B [2]

i.e. if B is the reciprocal of A, then A must be the reciprocal of B, otherwise we'd have to revise our definition of multiplication. Substituting the expression for B in [1] into [2]. you get:

A = 1/(1/A) [3]

So, the fact that dividing by the reciprocal of a number is the same as multiplying by that number is just something that is required for the self-consistency of mathematics ([3] must follow from [2] and [1]). That is really the end of the story.

 

[jimgavagan]

I follow, but I'm just wanting to know how to think about multiplying numbers.

To illustrate:

"Similarly, I tend to think of 5/(1/3) as "divide 5 into 'one-third' parts" which I interpret to mean, "make it so that 5 is 1/3 of the whole" (the whole would therefore be 15)."

I understand this, but this it's the "make it so that 5 is 1/3 of the whole" part that I do differently. Instead, I think of 5 different things split up three times each, so that the new number of parts is 15.

It's that type of "visualization" that I'm asking about, but as it applies to multiplication, especially decimals.

 

[cepheid]

I understand this, but this it's the "make it so that 5 is 1/3 of the whole" part that I do differently. Instead, I think of 5 different things split up three times each, so that the new number of parts is 15.

See, that's the thing. I don't know if I can help you with your conceptualization, since I "intuit" things in a different way from you. In particular, your description above is much more intuitive to me as a description of 5*3, since each of the 5 objects multiplies (becoming a triplet) and therefore [each of] these triplets occurs 5 times.

It's that type of "visualization" that I'm asking about, but as it applies to multiplication, especially decimals.

What do you mean by decimals? You haven't mentioned any decimals at all in your previous posts. Are you asking how you should "interpret" something like 5/0.45? Because I would first convert that into 5/(9/20) = 5*20/9 :tongue2:

I mean, if you wanted to at least stick closer to the numbers in the decimal, you could do something like multiply 5 by 4/10 and then multiply 5 by 5/100, and add the results of these two multiplications together. But I don't think there is any way to do the arithmetic mentally (or even by hand) without resorting to a conversion of the decimal into some sort of fractional form. At that point you could then try "interpreting" that in terms of some sort of real-world partitioning example if you really wanted to. However, as I've stated before, this would just be an analogy or an application of the arithmetic that would be of questionable usefulness for gaining mathematical understanding.

 

[jimgavagan]

Bingo, we just showed it with our different descriptions of the same thing. Your description of 5*3 is the same as my description of 5 / (1/3), showing both that they are the same operation and that they have the same conceptual visualization.

Well done!

P.S. - By decimals, I meant the 0 < x < 1 fractions. :P

 

[cepheid]

Bingo, we just showed it with our different descriptions of the same thing. Your description of 5*3 is the same as my description of 5 / (1/3), showing both that they are the same operation and that they have the same conceptual visualization.

Umm...okay. I think the point I was trying to make was that "conceptual visualizations" of arithmetic operations are arbitrary and vary from person to person. They can't really be used to "show" anything. But if my discussion of your example has help you to "understand intuitively" why it must be true that 5*3 = 5/(1/3), then I was glad to be of help.

P.S. - By decimals, I meant the 0 < x < 1 fractions. :P

Okay. Granted, anything written in our base 10 number system is a "decimal representation" of a number, but I think what people usually mean by "decimals" (vs. fractions) are numbers that explicitly have a decimal point in them and have digits to the right of that point, as an alternative to expressing those numbers in fractional form. So, calling 1/2 a "decimal" is kind of confusing -- especially when you are specifically talking about numbers less than 1. I would call 1/2 and 0.5 the "fractional" and "decimal" representations of the same number (respectively).

 

[jimgavagan]

Yes, "intuitively understand," thank you again.

This is something I got from a friend that might "show" why x*y = x / (1/y) is true.

8 / 16 = .5
8 / 8 = 1
8 / 4 = 2
8 / 2 = 4
8 / 1 = 8
8 / .5 = 16
8 * 2/1 = 8 / .5

Yes or no?

Why?