0.5 times 0.5 equals 0.25, but how does this work with repeated addition?

Question

So I'm trying to brush up on my math as an old adult, and I'm currently working through the very basics of math again. I'm trying to truly understand and visualize the various operations I'm engaging in, such as how one "moves" along the number line when one is multiplying two negatives, for example. It's proven more difficult than I thought.

One problem I have is my inability to visualize how I'm moving back and forth on the number line when I'm multiplying $0.5 \times 0.5 = 0.25$. When one multiplies, one is simply doing continuous addition. For example, $3$ times $5$ is merely $3 + 3 + 3 + 3 + 3 = 15,$ or $5 + 5 + 5 = 15$. You're adding a number x amount of times with itself.

This is all fine and dandy with whole numbers, but when I'm multiplying fractions, like $0.5$, I can no longer see how I'm moving along the number lines in regards to continuous addition to explain how I end up with $0.25$!

Is there some kind soul out there who can explain this?

Thanks in advance!

Answer 1

Does it help to think of it as moving halfway to $0.5$ (starting at $0$, of course)? In other words, you're adding in only half of the number $0.5$.

Answer 2

First, let's do 3 times 0.5, which we'll treat as 0.5 + 0.5 + 0.5. But instead of thinking of it happening all at once, imagine we take 3 seconds, and in the first second we move from 0 to 0.5, in the second second we move from 0.5 to 1.0, and in the third second we move from 1.0 to 1.5, and then we're finished and see our answer is 1.5.

Well, to do 0.5 times 0.5, we start our first second, where we're moving from 0.0 to 0.5, but half way through that second (which is the "0.5 times ..." part), we yell "Stop!". Guess where we are when "Stop!" is yelled. Yup, at 0.25

(BTW, to be pedantic, this does assume that we move at a constant speed, and don't speed up and slow down while moving. Since we are the ones who are construction this example, we can require/assume that this is how the motion takes place.)

Answer 3

Note that for positive integer $b$, $\frac{1}{b}\times a$ is the quantity $c$ such that

$$\underbrace{c+...+c}_{b \text{ copies}}=a.$$

In your example, $0.5\times 0.5=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$ since

$$\underbrace{\frac{1}{4}+\frac{1}{4}}_{2 \text{ copies}}=\frac{2}{4}=\frac{1}{2}.$$

We have as a general result

$$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}.$$

Answer 4

The model of multiplication as 'continuous addition' breaks down when dealing with other than whole numbers for exactly the reasons that you say, so instead we treat multiplication as satisfying some axioms — rules — that generalize the case of repeated-addition. For instance, $(a+1)\times c$ is equal to $a\times c+c$ — this is what repeated addition amounts to — and in fact, we can show the distributive property that $(a+b)\times c = (a\times c)+(b\times c)$ when all of $a$, $b$, and $c$ are whole numbers.

But this property is so handy to have that it'd be nice to not require them to be whole numbers to use it, and so in fact that's what we do. So how does this help find $0.5\times 0.5$? Well, we know that $0.5+0.5=1$, so let's multiplay both sides of that by $0.5$. $(0.5+0.5)\times 0.5$ $= 1\times 0.5 = 0.5$. But now we can distribute on the left to get $(0.5\times 0.5)+(0.5\times 0.5)=0.5$; in other words, whatever $0.5\times 0.5$ is, you have to add it to itself to get $0.5$. But that's just what $0.25$ is! And if you're paying close attention, you might realize that what I just described is effectively the definition of division in disguise; we say that $0.5\times x$ is exactly the number $y$ such that $y+y=x$.

Incidentally, if you want a mental model of multiplication that holds up better to non-whole numbers, I recommend thinking of multiplication as scaling the number line. Zero stays the same, but multiplying by $m$ stretches (or shrinks, if $m$ is less than $1$) the whole line by a factor of $m$. Multiplying by three is the same as stretching the number line out by a factor of three — those steps from $0$ to $1$, $1$ to $2$, etc. become 'counting by threes' instead of 'counting by ones' and so they become $0$ to $3$, $3$ to $6$ and so on. But then, we can think of multiplying by $0.5$ as shrinking the line by a factor of two: $6$ goes to $3$, $2$ goes to $1$, and $0.5$ goes to... $0.25$.

Answer 5

Since you specifically mention "visualize", note that multiplication is used to calculate the area of a rectangle (or square).

If you take a square of size 1x1, and you split both sides in half, you end up with 4 quadrants. Each of these 4 quadrants has edges 0.5x0.5, and together the 4 quadrants have size 1.00 Hence, each quadrant must have size 1/4=0.25, so 0.5x0.5=0.25

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Answer 6

Why you think that it is natural to go "$x$ times forward"? why not go backward? You "go forward" naturally because natural numbers are somehow extracted from nature rules. i.e. if you add $1$ and $1$ you know that the result is next to $1$ and we go forward.

In multiplication, do you really know you have to move forward? I don't know!

I usually use geometric block to do multiplication. $4\times 5$ is the number of $1\times 1$ blocks inside a rectangle with sides $4$ and $5$ that is $20$ $1\times 1$ blocks. in the case of $0.5\times 0.5$ we have a square with side $0.5$ and we want to know the number of $1\times 1$ blocks inside that. So expand the square side to the natural number $1$ and compute new multiplication that is $1\times 1=1$ and by drawing you can see that we have $3$ extra similar pieces out of $4$ pieces and we get $1-3/4=0.25$.

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You can define for yourself that for decimal part of numbers ($0<$ decimal $<1$) we go backward and for natural numbers we go forward.

Now can you calculate $1.5\times 1.5$?

Answer 7

For me it's easiest to convert one of the decimal numbers to fractions of whole numbers for visualizing the mutliplication on the number line.

So in your example we have $0.5 \times 0.5 = \frac{\color{red}1}{\color{blue}2} \times 0.5$. That would mean visualize the multiplication $\color{red} 1 \times 0.5 = \color{green}{0.5}$ as usual and then split the segment between $0$ and $\color{green}{0.5}$ on the number line into $\color{blue} 2$ equal parts. The length of each part gives the result.

Let's look at the example $0.75 \times 0.8$. We have $0.75 \times 0.8 = \frac{\color{red}3}{\color{blue}4} \times 0.8$, so we can visualize as $\color{red} 3 \times 0.8 = 0.8 + 0.8 + 0.8 = \color{green}{2.4}$. Now we split the segment between $0$ and $\color{green}{2.4}$ on the number line into $\color{blue} 4$ equal parts - each part has length $0.6$ which is the result of the calculation.

Answer 8

As other people (who are probably real mathematicians) have implied, as you get further on in your mathematical career, the less useful is to think of mathematical constructs being real. Instead it's helpful to think of them being useful (or in some cases elegant but of no practical use - although that's largely what people thought of number theory, before the invention of public key cryptography).

A friend of mine at school was exasperated when his younger sister thought negative numbers were 'wrong'. In a way though, she was correct: I've never seen -3 sheep or a -£10 note but if I have -£10 in my bank account, I can work out that if I deposit £100, I'll have £90 left. Similarly for irrationals or complex numbers etc.

This view was expressed by the famous mathematician Kronecker who said “Natural numbers were created by God, everything else is the work of men.” Except he said it in German.

To return to your original question though: imagine you have a number line and want to double a number, x. You get an imaginary rope, cut it to length x then lay it out from 0 to x then from x to 2x. This is easily generalized to multiplying by any natural number, a.

To divide by two you measure out a length of rope, then grab both ends and you have a length of x/2. You can generalise to divide by any natural number, b.

So now you can multiply by a/b (or equivalently divide by b/a) so you are now sorted for the rationals. You have to get a bit more elaborate to get to irrational and transcendental numbers but as I said at the start, at some point, you have to let simple physical models go to achieve mathematical enlightenment.