Multiplication by zero

[noravoningersleben]

Hi all,

Here is another general question from me:

I know that we can only divide both sides of an equation by a variable if we're certain that the variable doesn't equal zero because division by zero is illegal. What happens, though, if we're trying to multiply by a variable and we're not sure if that variable is equal to zero or not? In multiplication it doesn't matter, right? We can go ahead and multiply by the variable even if we don't know whether the variable equals zero or not, correct?

Thanks in advance for your help!

[shyamprasadrao]

Dividing by zero is same as multiplying by zero

Lets look at an example,
X and Y are integers

a * X = a * Y this can be rewritten as

X /a = Y /a I am just cross multiplying.

Unless you are sure that the variable is not zero do not multiply it with the equation.

When multiplyin by negative numbers in an inequality be sure to change signs
X > Y

-X < -Y

Hope this helps

[RonPurewal]

Dividing by zero is same as multiplying by zero

whoa, no. this is not true.

notice that your argument seems to work for any number "a". this is troubling, because, according to this "argument", dividing by ANY number "a" is the same as multiplying by "a".
this is definitely not true. for instance, dividing by 2 is clearly not the same thing as multiplying by 2.

what your argument DOES show is something that is already well known: namely, that, IF "A" IS NONZERO,
* you can multiply both sides of an equation by "a", OR
* you can divide both sides of an equation by "a".

these facts are well known, but it never hurts to reiterate them.

Lets look at an example,
X and Y are integers

a * X = a * Y this can be rewritten as

X /a = Y /a I am just cross multiplying.

"cross multiply" is not a real operation.
(also, technically, you aren't "cross multiplying" here anyway; you're actually UNDOING "cross multiplication".)

to get from the first to the second of these, you're actually DIVIDING by a^2.
if a = 0, then that's a problem, since division by 0 is not allowed, ever.

--

here's the original:

In multiplication it doesn't matter, right? We can go ahead and multiply by the variable even if we don't know whether the variable equals zero or not, correct?

correct.

this is only true for EQUATIONS, though.

if you have an INEQUALITY, you can't multiply by a variable unless you know whether it is positive or negative.
one side effect here is that you shouldn't multiply INEQUALITIES, especially those containing "<" or ">", by something that could be 0.

[noravoningersleben]

Thanks, got it! But - you write that I shouldn't multiply inequalities, especially those that contain "<" or ">" by a variable that could be zero. Are there any equalities, though, that don't contain "<" or ">"? I thought containing "<" or ">" is what makes an inequality and inequality...?

[Ben Ku]

All "equalities" (actually, equations) have equal signs, so you can multiply or divide by variables that are not 0 without worrying about changing the equation.

Because multiplying or dividing by negative numbers changes the direction of the inequality, you should NEVER multiply or divide an inequality by a variable, unless you know for sure whether it's positive or negative.

I think when Ron stated

if you have an INEQUALITY, you can't multiply by a variable unless you know whether it is positive or negative.
one side effect here is that you shouldn't multiply INEQUALITIES, especially those containing "<" or ">", by something that could be 0.

he meant that there are OTHER Inequalities (as opposed to equations), such as "≤" and "≥" where it's a combination of both an inequality and equation.