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To clarify my confusion:

$88 \cdot 0.732$ is the same as $0.732 \cdot 88$

The latter being: 0.732 eighty-eight times: $0.732 + 0.732 + 0.732 + \cdots + 0.732$

But how to think about $88 \cdot 0.732$? that's $88\quad0.732$ times?

Can you see what's bothering me? I apologize if these are weird questions, but it's really bothering me.

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I'd re-express the decimal as a fraction $\frac{p}{q}$. Then you can apply your usual interpretations of multiplication and division... –  J. M. Sep 10 '11 at 1:21
    
Yeah, I'm aware of that, the fact that's bothering me is that $0 < x < 1$ numbers are parts of a whole, it sometimes hard to wrap my head around that... 88 times 0.732 is basically saying that 88 is the whole and you want 73.2% of that whole. or 732/1000... Which is actually a method of solving it, multiply by a thousand, do the integer-like multiplication, divide by 1000 (usually referred to as "moving the zeroes to the left from right") –  Curiosity Sep 10 '11 at 1:30
    
The "multiply by a thousand" bit looks sorta kinda redundant... –  J. M. Sep 10 '11 at 1:35
    
Think of a piece of elastic 1 unit long, on which the point 0.732 is marked. Stretch it to be 88 units long. Where the point originally at 0.732 ends up is 88 × 0.732. –  Rahul Sep 10 '11 at 3:33
    
People have a similar confusion about derivative of $x^2$ –  gaurav Sep 10 '11 at 13:02

4 Answers 4

up vote 4 down vote accepted

One way to think about real numbers is to see them as lengthes. In that setting, the product $a \times b$ represents the area of a rectangle whose sides are $a$ and $b$ (I'm not defining that precisely, but just relying on your intuition here). This explain easily why $a \times b = b \times a$.

Now in the special case where $n$ is an integer (and only in that case), we can check that $n \times x$ also has the meaning

$$n \times x = \underbrace{x + \ldots + x}_{n \textrm{ times}}$$

Indeed, it's easy to see why $(a + b) \times c = a \times c + b \times c$ (split a rectangle in two along the $a+b$ side) and $1 \times a = a$. So using this properties, we get

$$2 \times x = (1+1) \times x = x + x$$ $$3 \times x = (2+1) \times x = x + x + x$$ $$4 \times x = (3+1) \times x = x + x + x + x$$ $$5 \times x = (4+1) \times x = x + x + x + x + x$$

And so on.

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Draw a line $0.7$ units long. String $6$ of those together. That's $6\times 0.7$.

Now draw a line $6$ units long. Think about how to string $0.7$ of those together. Just $1$ of them is $6$ units long. $0.7$ is less than $1$, so $0.7$ of them would be shorter. It would be seven-tenths of that length.

Either way, you get the same length.

$0.7\times 6 = 6\times 0.7 = 4.2$

First look at this: $$ \begin{array} & 0 & & & & & & & & 0.7 & & & 1\\ | & & & & & & & & \downarrow & & & | \\ \hline | & & & & & & & & \uparrow & & & | \end{array} $$ and then at this: $$ \begin{array} & 0 & & & & & & & & 4.2 & & & 6\\ | & & & & & & & & \downarrow & & & | \\ \hline | & & & & & & & & \uparrow & & & | \end{array} $$

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It sounds like there is nothing magic about $0 \lt x \lt 1$. The same problem would exist with $1.732\times 88$ or $1.732 \times 2.438$. The real problem is extending the multiplication operation from the naturals, first to the rationals, then to the reals. As J. M. suggested, if you believe in the rationals it is natural to define $\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}$ (and if you don't, why worry about multiplication?). With $b=d=1$ this is multiplication on the naturals, then with $b=c$ and $d=1$ it feels right and go on. For the reals it is harder-as best I know you need to take convergent sequences or Dedekind cuts. It depends on how you define the reals or what sequence you develop their properties.

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The way to think about multiplication by a number between 0 and 1 is as multiplication.

The idea of viewing multiplication as repeated addition is an aide to help you start forming a mental concept of multiplication (and is a particular application of multiplication) -- but it is not meant to be the way you think about multiplication for the rest of your life!

But that aside, the phrasing of your question makes it sound like you are considering an interpretation of multiplication that covers products like $88 \times 4.732$ and $88 \times 1.732$ -- can you extend it to cover $88 \times 0.732$ by something like the following?

$88 \times 0.732$ is just like $88 \times 1.732$, but with one less copy of $88$.

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