# Thread: Visualizing something-Integer multi, divide.

1. ## Visualizing something-Integer multi, divide.

Hi, I have kind of an off the wall question I guess:

Adding or subtracting an integer is easily visualized from just thinking of the number being across a line/ or bar. But for some reason I don’t visually see how multiplying or dividing an integer looks.
What I mean is, I know the rule and I don’t question it. But if you think about a fraction you can actually visualize a pie or something concrete. But how do you visualize a multiplication or division of an integer?

By the way just to be clear: I know that the rule is:
• If BOTH integers are positive, or BOTH integers are negative, the answer is positive.
• If one integer is negative, the answer is negative.

But why can’t I see a visual picture in my head with multiplying and dividing an integer as I can adding or subtracting one? Again I know the rule I'll stick with it but I remember rules visually and I don't see the picture.

2. Originally Posted by Waveform
But how do you visualize a multiplication or division of an integer?
Your question can be understood as a basic question or as a complex question. I will begin by understanding it as a basic question.
For example you have 2X3=6 but why that means adding 2 by itself 3 times OR adding 3 by itself 2 times,
$\underbrace{2+2+2}_3=6$ or $\underbrace{3+3}_2=6$
Note that is makes no difference how you mutiply integers thus $a\times b=b\times a$
Where,
$a\times b=\underbrace{a+a+...+a}_b$
That is how we can on a basic level visualize mutiplication.
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Since I do not know if you are asking a deep question, I will answer it in another way, by beginning to ask a question. What is an integer? This question is exceptionally difficult. Because you can provide a basic non-mathematical meaning to it but how do you on a mathematical level define such a concept? The answer are sets. Through this basic concept the foundations or mathematics lay. The person who succeded defining natural numbers was Guisseppe Peano.
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Now assuming you taken some set theory, here is where it gets fun.
One way to construct integers is like this,
define sets,
$<0>=\{\}$
$<1>=\{<0>\}\cup <0>$
$<2>=\{<1>\}\cup <1>$
$<3>=\{<2>\}\cup <2>$
....
Then define the integers as,
$0=|<0>|,1=|<1>|,2=|<2>|,3=|<3>|,...$

You can define addition/multiplication on these as,
$x+y=|\cup |$
$x\cdot y=|\times |$

You can define $x\leq y$ is there exists
$\phi: \to $ which is injective.

3. Amm, wow,
Well First off thanks for replying and taking the time to type all this out.
I’ll admit, this is the first time I’ve ever heard of sets, I’ve basically had one math class in high school and thought my self fractions on my own after HS.
Anyway, that link you gave on sets is a good insight but I have to be honest, I’m totally lost on what they are talking about. Some of the definitions are over my head. I ended up clicking on the little links on the pages which led me to deeper links just to try and understand what they were talking about.

I was pretty much following you at the top of your post till you got to the a x b= a + a + ….+ a stuff.

I have to read though this I guess or maybe I’m just tired.
sorry, as it is I’m abstract maybe making this deeper then it is.

4. Originally Posted by Waveform
I was pretty much following you at the top of your post till you got to the a x b= a + a + ….+ a stuff.
Sorry, bout that then ignore my what I wrote in the end. But you understand what I am basically trying to define, that $3\times 2$ is adding 3 to itself 2 times? If so then good, that is what mutiplication is. So given any integer $a$ and mutiply it by another integer $b$ then what you do is add $a$ to itself $b$ times that is what I mean by,
$\underbrace{a+a+...+a}_b$

5. This is what I’m talking about under the section titled: Multiplication and Division:
http://www.leeric.lsu.edu/bgbb/7/ecep/math/e/e.htm

Where is says Quote:
Multiply or divide any problems involving negative integers as usual. To decide if the final answer is negative or positive:
If BOTH integers are positive, or BOTH integers are negative, the answer is positive.
If one integer is negative, the answer is negative.

I guess what I’m not getting there is, How can (-7) X (-2) = Positive 14?

Am I jumping ahead? That page is titled AlgebraII, I haven't even had Algebra one yet.

6. Originally Posted by Waveform
This is what I’m talking about under the section titled: Multiplication and Division:
http://www.leeric.lsu.edu/bgbb/7/ecep/math/e/e.htm

Where is says Quote:
Multiply or divide any problems involving negative integers as usual. To decide if the final answer is negative or positive:
If BOTH integers are positive, or BOTH integers are negative, the answer is positive.
If one integer is negative, the answer is negative.

I guess what I’m not getting there is, How can (-7) X (-2) = Positive 14?

Am I jumping ahead? That page is titled Algebra II, I haven't even had Algebra one yet.
Keep going with what you doing. Asking questions and trying to get a mental picture are always good strategies.

Try this where it discusses addition, multiplication and division. Ignore the other parts. The link expands on PerfectHacker's repeated additions and explains why mutliplying two negatives results in a positive.

7. Cool, thanks for that link JackD

Printing it out now!