What is Division? An Intuitive Explanation

[Mr Davis 97]

Although of course I have been using division since my elementary years, I still can't out my finger on exactly what division is from a computation standpoint. For example, addition is adding objects (very intuitive), subtraction is adding the additive inverse of an object, and multiplication is adding an object to itself some number of times (again, fairly intuitive); however, I can't really explain what division is. I know that it is the inverse operation of multiplication, but it doesn't seem to be related to it in any intuitive way. I have tried to look up definitions of arithmetic division, but that just seems to add to my lack of understanding. There are definitions which relate to partitioning, there is a quotative definition, and there is a definition which says, "multiplication is the operation of ascertaining how many times one quantity is contained in another (Dictionary.com). In addition, I have no idea how division is used in other applications, such as ratios, proportionality, percentages etc. I don't completely understand how division is used to compare relative quantities, i.e. ratios. All of this said, it would be nice if somebody could give me an intuitive definition for division, how to compute it, along with how this definition fits into the notions of fractions and ratios.

 

[symbolipoint]

You make a large effort to complain about not knowing what is division or what it means while at the same time explaining very clearly what it means, therefore you have shown in your complaint that you really do understand the meaning of division very well.

 

[Vahsek]

If you are happy with your "meaning" of multiplication but not with the one for division, why don't you view division from a multiplication perspective? For example:

x/y = k is the same as saying x = ky. That is x/y is that number by which you multiply y in order to get x.

 

[bhillyard]

How about the idea of sharing.
I have 24 sweets to share amongst 6 people. How many does each person get.
Move on to things that can be subdivided - 24 kilogrammes of sugar shared (divided) between 5 people.

 

[Vahsek]

How about the idea of sharing.
I have 24 sweets to share amongst 6 people. How many does each person get.
Move on to things that can be subdivided - 24 kilogrammes of sugar shared (divided) between 5 people.

This doesn't work quite smoothly for division by real numbers or fractions in general. It's still a good perspective for natural numbers nevertheless.

 

[NascentOxygen]

This doesn't work quite smoothly for division by real numbers or fractions in general. It's still a good perspective for natural numbers nevertheless.

Evenly spread 586.3 grams of icing over 32.25 cup cakes. How many grams would 1 whole cake receive?

Seems deliciously smooth to me! http://www.makeathumbnail.com/thumbnails/image281337.png

 

[romsofia]

Cx=B, therefore C divides B because there exists a number x, that makes the equation C=B/x true.

 

[verty]

Have you seen the procedure of dividing a line segment into N parts? Look here.

 

[jeff davis]

Hello,
I just wanted to first state that I am no mathematician by any means. When i think of division, i just think of how many times one number can go into another. Two number 2s can fit inside the space reserved for one number 4. This means that 4 divided by 2 = 2.
It is related to multiplication because in multiplication you are taking groups of things (numbers of the same value) and combining them together a certain amount of times, and in division you are seeing how many groups (of the same valued number) you have combined when you multiplied them.
As far as fractions go: A fraction is just a way to express the numerator(top) divided by the denominator(bottom). Example 1/2 is equal to one divided by two, which is .5. Sometimes it is easier to express decimal values by a fraction. Expressing some decimals as fractions also will allow you to represent an exact amount for a number that would otherwise have a decimal string that repeats forever. Example: 1/3 is equal to one divided by three which is equal to .3333333333333(repeating forever). It is best to use the fractional 1/3 in your equations until the end(if you need a decimal) so you can have as accurate answers as possible.
If you are still curious about ratios and such, just let me know and i will try to explain what i can.

 

[Hertz]

When you divide one number by another number, the result is "the amount of times the bottom number would fit into the top number".

For example, 10/2 is equal to 5 because if you divide 10 total objects into groups of 2, you will have 5 groups total. I.e. if you have groups of 2, you need 5 of them to make a total of 10 objects. I.e. 10/2 = 5, or 10/5 = 2. Another example is 3/2. How many sets of 2 do you need in order to make 3 total objects? Well, the answer is that you need 1.5 groups of 2 to get 3 total objects. 1 group of 2 = 2 objects, .5 group of 2 = 1 object, so 1.5 groups of 2 = 3 objects.

 

[D H]

and multiplication is adding an object to itself some number of times (again, fairly intuitive)

This is the root of your problem. That definition happens to work for the natural numbers. What about rationals? What about irrationals? It's certainly valid to multiply $\pi$ and $e$, but do you make $\pi$ copies of $e$ (or $e$ copies of $\pi$)? The same goes for things like $\sqrt 6 \cdot \sqrt{15} \cdot \sqrt{10}$.

There's been a big debate amongst the math education community during the last six years regarding whether teaching multiplication as repeated addition is the wrong approach. One thing is certain: You eventually have to unlearn that concept if you want to progress toward more advanced mathematics.

Some teachers are using the concept of stretching to teach multiplication in a way that doesn't use the repeated addition concept. Get a decimeter stick and a ribbon of nice stretchy material. Rule that stretchy material so it mimics the decimeter stick. Now stretch the ribbon so the zeros line up on both the ruler and ribbon and the 2.5 cm mark on the ribbon lines up with the 10 cm mark on the ruler. You've just multiplied 2.5 by 4! You can see that by looking at the number on the ruler that aligns with the 1 cm mark on the ribbon: It's the 4 cm mark. Division goes hand in hand with multiplication. Just as this stretching shows that 2.5*4=10, it also shows that 10/4=2.5. And of course, 1/4=0.25, as can easily by seen by looking at the number on the ribbon that aligns with the 1 cm mark on the decimeter stick.

Google search for "Multiplication is not repeated addition" and you'll get plenty of hits on this topic.