# Why does multiplying by 0 equal 0?

by LightbulbSun
Tags: equal, multiplying
 P: 362 I know why it would equal to 0 if it was (0*0). But what about an actual number? Why does (100*0) equal to 0? You're not multiplying anything, but shouldn't it still equal to 100? If I have 100 cookies on the table, and I don't multiply it by anything, why do I suddenly have zero cookies on the table? I'm just trying to gain a conceptual understanding behind the zero-factor algebraic property.
 P: 740 One possible conceptual way to think about it is to say, multiplying 2 * 100 is like having 2 groups of 100 cookies on the table. Multiplying 1 * 100 is like having 1 group of 100 cookies on the table. Multiplying 0 * 100 is like having no groups of 100 cookies on the table, hence no cookies at all. However, I agree that this is may not be very intuitive. When dealing with 0, it is sometimes more difficult to match mathematical situations to real life situations. It is probably better to understand 0 * any number = 0 just as a consequence of several properties of numbers that we take for granted.
 HW Helper P: 1,319 Remember that $$0 + a = a$$, $$0 + 0 = 0$$, and $$a - a = 0$$, no matter what number you use for $$a$$. Now the rules of arithmetic give this \begin{align*} 0 \cdot a & = (0 + 0)a\\ & = 0 \cdot a + 0 \cdot a\\ \left(0 \cdot a - 0 \cdot a\right) & = 0 \cdot a\\ 0 & = 0 \cdot a \end{align*}
P: 420

## Why does multiplying by 0 equal 0?

You actually prove this at the beginning of an undergrad analysis/theoretical calculus class.
P: 362
 Quote by Tedjn One possible conceptual way to think about it is to say, multiplying 2 * 100 is like having 2 groups of 100 cookies on the table. Multiplying 1 * 100 is like having 1 group of 100 cookies on the table. Multiplying 0 * 100 is like having no groups of 100 cookies on the table, hence no cookies at all. However, I agree that this is may not be very intuitive. When dealing with 0, it is sometimes more difficult to match mathematical situations to real life situations. It is probably better to understand 0 * any number = 0 just as a consequence of several properties of numbers that we take for granted.
That's actually a very good way of thinking about it.

 Quote by statdad Remember that $$0 + a = a$$, $$0 + 0 = 0$$, and $$a - a = 0$$, no matter what number you use for $$a$$. Now the rules of arithmetic give this \begin{align*} 0 \cdot a & = (0 + 0)a\\ & = 0 \cdot a + 0 \cdot a\\ \left(0 \cdot a - 0 \cdot a\right) & = 0 \cdot a\\ 0 & = 0 \cdot a \end{align*}
Ah, so it's sort of a cancelation?
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P: 25,486
 Quote by LightbulbSun If I have 100 cookies on the table, and I don't multiply it by anything, why do I suddenly have zero cookies on the table?
Hi LightbulbSun!

No, if you don't multiply it by anything, you still have 100 cookies on the table.

This is a language thing

"not multiplying by anything" is not the same as "multiplying by nothing" …

"not multiplying by anything" means leaving it the same.

EDIT: I think the French don't have this problem …
they distinguish between (pardon my French! ) …
"multiplier par rien" and "ne multiplier par rien"
 HW Helper P: 2,692 This is far too natural to be confusing. Remember that multiplication is repeated addition. If you add 100, ZERO times, you have ZERO. If you want 100 as result, then you must add 100 ONE time.
P: 362
 Quote by symbolipoint This is far too natural to be confusing. Remember that multiplication is repeated addition. If you add 100, ZERO times, you have ZERO. If you want 100 as result, then you must add 100 ONE time.
Now it makes more sense to me. Thanks for the explanation.
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P: 16,094
 Quote by LightbulbSun I know why it would equal to 0 if it was (0*0). But what about an actual number? Why does (100*0) equal to 0? You're not multiplying anything, but shouldn't it still equal to 100? If I have 100 cookies on the table, and I don't multiply it by anything, why do I suddenly have zero cookies on the table? I'm just trying to gain a conceptual understanding behind the zero-factor algebraic property.
The problem is that natural language sort of pigeon-holes us into thinking "none" and "at least one" are conceptually different. As soon as you break through this barrier and become comfortable working with degenerate cases, stuff like this becomes easy.

For example, how many pennies do you have, if you have zero rows of N pennies each? (Or, as one would generally say in natural language, if you don't have any rows of N pennies each)
 P: 101 proof ox=o 0x = 0x + 0 = 0x + [ x + (-x)] = (0x + x) + (-x) = x( 0 +1) + (-x) = x +(-x) = 0 or 0x = 0 <===> 0x + x = 0+x <=====> x( 0 + 1) = 0 + x <===> x = 0 + x <===> x=x correct so 0x=0
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P: 2,692
 Quote by poutsos.A proof ox=o 0x = 0x + 0 = 0x + [ x + (-x)] = (0x + x) + (-x) = x( 0 +1) + (-x) = x +(-x) = 0 or 0x = 0 <===> 0x + x = 0+x <=====> x( 0 + 1) = 0 + x <===> x = 0 + x <===> x=x correct so 0x=0
How does a person learn to work with such simple, low-level ideas like that to prove what would otherwise seem so natural? Fantastic!
P: 810
 Quote by LightbulbSun I know why it would equal to 0 if it was (0*0). But what about an actual number? Why does (100*0) equal to 0?
Because it turns out simple and useful.

Multiplication, and any operation, is something defined by the mathematician. We could imagine a world where 0 * n = n. But it would break a lot of useful theorems. For instance, 0 * 1 + 1= 1 + 1 = 2. However, 0 * 1 + 1 = (0 + 1)*1 (by distribution), so 0 * 1 + 1 = 1, and so 1 = 2. We must conclude that addition no longer distributes over multiplication (disastrous!!).

In many definitions, when you get to the lowest possible value, the definition loses its literal intuitive meaning. One example is the factorial function, where 0! = 1. Factorial is often defined as the product: 1 * 2 * ... * n, but when n = 0, this definition doesn't make sense.

(Though there are other definitions that do, this is just one example).
 P: 3,128 There isn't really any reason why, that's the way it's defined.
P: 175
 Quote by loop quantum gravity There isn't really any reason why, that's the way it's defined.
We have the winner.

Mathematics is a human convention. That's how zero is defined in it.
P: 1,133
 Quote by symbolipoint This is far too natural to be confusing. Remember that multiplication is repeated addition. If you add 100, ZERO times, you have ZERO. If you want 100 as result, then you must add 100 ONE time.
P: 15,294
 Quote by Gear300 add it to what...anything?
Yes, anything.

126,
12,378,726,387,
0,
or simply n.

i.e.:

126 + 100*0 = 126
126 + 100*1 = 126 + 100

12,378,726,387 + 100*0 = 12,378,726,387
12,378,726,387 + 100*1 = 12,378,726,387 + 100

0 + 100*0 = 0
0 + 100*1 = 0 + 100

or simply

n + 100*0 = n
n + 100*1 = n + 100
P: 1,133
 Quote by DaveC426913 Yes, anything. 126, 12,378,726,387, 0, or simply n. i.e.: 126 + 100*0 = 126 126 + 100*1 = 126 + 100 12,378,726,387 + 100*0 = 12,378,726,387 12,378,726,387 + 100*1 = 12,378,726,387 + 100 0 + 100*0 = 0 0 + 100*1 = 0 + 100 or simply n + 100*0 = n n + 100*1 = n + 100
So...that would mean such is true for all systems?
P: 15,294
 Quote by Gear300 So...that would mean such is true for all systems?
I'm not prepared to say that. I am simply clearing up the possible confusion of the question "if you 'add' 100 times 0, what are you adding it to?".

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