Multiplication as repeated addition

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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 $t*x$).

Answer 4

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.