
#1
Dec1808, 04:04 PM

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 zerofactor algebraic property. 



#2
Dec1808, 04:08 PM

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. 



#3
Dec1808, 04:16 PM

HW Helper
P: 1,344

Remember that [tex] 0 + a = a [/tex], [tex] 0 + 0 = 0 [/tex], and [tex] a  a = 0 [/tex], no matter what number you use for [tex] a [/tex]. Now the rules of arithmetic give this
[tex] \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*} [/tex] 



#4
Dec1808, 04:26 PM

P: 420

Why does multiplying by 0 equal 0?
You actually prove this at the beginning of an undergrad analysis/theoretical calculus class.




#5
Dec1808, 04:53 PM

P: 362





#6
Dec1808, 05:01 PM

Sci Advisor
HW Helper
Thanks
P: 26,167

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" 



#7
Dec1808, 05:35 PM

HW Helper
P: 2,693

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.




#8
Dec1808, 06:21 PM

P: 362





#9
Dec1808, 07:07 PM

Emeritus
Sci Advisor
PF Gold
P: 16,101

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) 



#10
Dec1808, 08:39 PM

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 



#11
Dec1908, 12:08 PM

HW Helper
P: 2,693





#12
Dec2208, 08:40 AM

P: 810

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). 



#13
Dec2208, 08:49 AM

P: 3,177

There isn't really any reason why, that's the way it's defined.




#14
Dec2208, 03:19 PM

P: 175

Mathematics is a human convention. That's how zero is defined in it. 



#15
Dec2208, 04:38 PM

P: 1,133





#16
Dec2208, 05:31 PM

P: 15,325

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 



#17
Dec2308, 12:32 AM

P: 1,133




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