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I was helping my brother with some multiplication and he ended up asking me "Why are we doing multiplication, the use and history of it?" I replied to him that multiplication is nothing but repeated addition, Suppose we have $3\times 4 =4+4+4 = 12 $. But later he asked me how is $1.678\times 3 $ interpreted, as we can't add $3$ "$1.678$" times, So How could I justify it? Is multiplication just a defined operation that was extended to a broader set of numbers? How could I explain it to him in simple terms, Is there anyhow multiplication that is "defined"

He doesn't know commutativity multiplication $1.678\times 3$ could also be interpreted as 1.678 added thrice (commutativity), but he resists it as it is read as $1.678$ times $3$ which means $3$ being added $1.678$ times and not the other way!

PS - commutativity doesn't help $1.678\times 3.14$ type questions too! Also, the number "1.678" taken is just an example taken to show the limitations in expressing multiplication as repeated addition

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    $\begingroup$ Assuming commutativity will not help much to answer what is 1.678 x 3.14 $\endgroup$
    – Surb
    Aug 31, 2022 at 14:49
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    $\begingroup$ I'd just explain it as an area of a rectangle. $3\times 4$ is $12$, becuase in a rectangle with $3$ times $4$ squares, there are overall $12$ squares. Now what if we cut a square in half? What if we cut a square into $0.678$ section? etc. $\endgroup$
    – Daniel P
    Aug 31, 2022 at 15:00
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    $\begingroup$ The quick version is that we started by defining multiplication of naturals as repeated addition. This extended to multiplication of integers. We then defined rational numbers and multiplication of rationals: $\frac{a}{b}\times\frac{c}{d}=\frac{a\times c}{b\times d}$. That is part of the definition, not just a result. We then define the real numbers and multiplication of reals in whatever way is preferred, either by dedekind cuts or by cauchy sequences, in either event having used multiplication of rationals for the inbetween steps. $\endgroup$
    – JMoravitz
    Aug 31, 2022 at 15:00
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    $\begingroup$ The full explanation would need to get into the gritty details of how real numbers are defined in the first place, and that is often requiring more mathematical maturity than one might have when first asking this question. Suffice to say, yes we are aware of the question and yes we were careful how things are rigorously defined. Until you are ready, just know that "it works" and know the properties. Actually going through the effort of defining the real numbers formally isn't usually done until late into undergrad college or early graduate school. $\endgroup$
    – JMoravitz
    Aug 31, 2022 at 15:04
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    $\begingroup$ Best way is through area, I think. A rectangular tub that's 3 units wide and 5 units deep holds 15 square units of water. (Ignore the third dimension for now.) What if it's 3 units wide and 1.618 deep? $\endgroup$ Sep 1, 2022 at 2:08

4 Answers 4

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Think of multiplication as finding an area. Say you have a rug that's $2 \times 3$ meters. You can divide it into $2 \times 3 = 6$ one-square-meter sections, like so.

┌──────┬──────┬──────┐
│      │      │      │
│      │      │      │
│      │      │      │
├──────┼──────┼──────┤
│      │      │      │
│      │      │      │
│      │      │      │
└──────┴──────┴──────┘

To show commutativity, just rotate the rug 90°. Now it's “$3 \times 2$” instead of “$2 \times 3$”, but the area is still 6 m².

┌──────┬──────┐
│      │      │
│      │      │
│      │      │
├──────┼──────┤
│      │      │
│      │      │
│      │      │
├──────┼──────┤
│      │      │
│      │      │
│      │      │
└──────┴──────┘

Now, let's consider $1.678 \times 3$. This is like my first example, except that one dimension isn't a whole number of units.

┌──────┬──────┬──────┐
│      │      │      │
│      │      │      │
│      │      │      │
├──────┼──────┼──────┤
│      │      │      │
│      │      │      │
└──────┴──────┴──────┘
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

This time, you can't just count the boxes in the drawing to find the area, because some of them aren't whole square meters.

But you can chop off two 0.322-meter strips from one of the incomplete squares, and rearrange them to make the other two incomplete squares a whole square meter.

┌──────┬──────┬──────┐
│      │      │      │
│      │      │      │
│      │      │      │
├──────┼──────┼══════╛
│      │      │▒▒▒▒▒▒▒
│      │      │▒▒▒▒▒▒▒
│      │      │▒▒▒▒▒▒▒
└──────┴──────┘▒▒▒▒▒▒▒

So now we've got 5 complete 1 m² squares, plus a tiny bit (0.034 m²) extra. And this, I hope, should visually demonstrate what “$1.678 \times 3 = 5.034$” means.

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One way to justify it would be using division: $1.678=\frac{839}{500}$ so $1.678\times4=\frac{839\times4}{500}$

Cutting a few pies into 500 pieces each, adding 4 slices 839 times and then thinking about what fraction the pieces make compared to the original singular pie.

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Your original definition of multiplication is correct. Multiplication is simply repeated addition of a number from an interval 0 to another number. To visualize this with decimals, imagine a massive hour glass with sand in it that dumps x amount of sand into the container at the bottom every second (at a constant rate). Multiplication is the amount of sand that got dumped into the bottom chamber when you have the hour-glass tipped over for t seconds, pouring out x sand per second (or $t*x$).

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  • $\begingroup$ So how do you compute $e\times\pi$? $\endgroup$
    – FShrike
    Sep 9, 2022 at 22:26
  • $\begingroup$ sand coming down at a rate of pi per second for a total of e seconds $\endgroup$
    – CR130
    Sep 11, 2022 at 1:21
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Here's another way of thinking about multiplication, at least for the "counting numbers" $0,1,2,3,4...$. The basic idea is that multiplication is a way of getting new true equations from ones we already know to be true. This doesn't answer your entire question about multiplication of more complicated numbers, but perhaps it can still offer an interesting perspective on multiplication as repeated addition.

Let's say we start with an equation in terms of addition that we know is true, like $2+3=5$. Sometimes, we might be interested in learning what other similar equations are also true, given that we know this one is true. How can we find such equations?

Let's say we have a process for transforming numbers that turns true additive equations into true additive equations. Call this process $T$, for "transformation", and let it act on individual numbers. Considering the above example, we want to find a process $T$ so that $T(2) + T(3) = T(5)$. But how can we figure out what $T$ does to each number?

Well, we can figure out what $T$ must do to zero. Consider the true additive equation $4+0 = 4$. We want $T(4)+ T(0) = T(4)$. Since $T(4)$ is just some number, we can subtract it from both sides of this equation, and find $T(0) = 0$. That's a start! But what does $T$ do to other numbers?

Let's now consider the equation $1+1=2$. We want $T(1) + T(1) = T(2)$. Once we figure out what $T(1)$ is, that means we can figure out what $T(2)$ is, just by adding $T(1)$ to itself! Similarly, we can figure out $T(3)$ in terms of $T(1)$ by considering $1+1+1=3$ and noticing this means that $T(3) = T(1) + T(1) + T(1)$. So, once $T(1)$ is set, we can figure out what $T$ must do to each of $0,1,2,3,...$ and so on.

Let's set $T(1)=3$ and see what happens. Then $T(2) = T(1) + T(1) = 3+3 = 6$. And $T(3) = T(1) + T(1) + T(1) = 3+3+3=9$. Similarly, $T(4) = 12$ and $T(5) = 15$. Now we can try this out on the equation $2+3=5$. We get $T(2) + T(3) = 6 + 9 = 15 = T(5)$. So, we have turned $2+3=5$ into the equation $6+9=15$! We have obtained a new true equation from an old one.

Notice that by setting $T(1) = 3$ we end up getting an operation that "multiplies each number by 3". You can get multiplication by other numbers in this way by setting $T(1)$ to other values.

To conclude: looking for transformations that transform true additive equations into true additive equations leads us to multiplication. The fact that multiplication is repeated addition corresponds to the fact that how our transformation acts on larger numbers is forced by how it acts on 1.

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  • $\begingroup$ This one is interesting :) $\endgroup$
    – Shashaank
    Sep 11, 2022 at 15:30

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