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Test Your Knowledge - and learn some interesting things along the way. Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!
And yes, 'gift' is a verb. It'll cost you nothing to read. We're intent on clearing it up 'Nip it in the butt' or 'Nip it in the bud'?
We're gonna stop you right there Literally How to use a word that literally drives some pe Is Singular 'They' a Better Choice?
We are thankful for obscure words. The act or process of dividing. Cell division. The operation of determining how many times one quantity is contained in another; the inverse of multiplication.
Published by Houghton Mifflin Company. Similarly, to support division of any integer by any other, the realm of numbers must expand to the rational numbers.
During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results.
Loosely speaking, since division by zero has no meaning is undefined in the whole number setting, this remains true as the setting expands to the real or even complex numbers.
As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers.
In keeping with this change of viewpoint, the question, "Why can't we divide by zero? Answering this revised question precisely requires close examination of the definition of rational numbers.
In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory.
First, the natural numbers including zero are established on an axiomatic basis such as Peano's axiom system and then this is expanded to the ring of integers.
The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers.
This relation is shown to be an equivalence relation and its equivalence classes are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed for verifying transitivity.
The above explanation may be too abstract and technical for many purposes, but if one assumes the existence and properties of the rational numbers, as is commonly done in elementary mathematics, the "reason" that division by zero is not allowed is hidden from view.
Nevertheless, a non-rigorous justification can be given in this setting. It follows from the properties of the number system we are using that is, integers, rationals, reals, etc.
The concept that explains division in algebra is that it is the inverse of multiplication. For example, . In general, a single value can't be assigned to a fraction where the denominator is 0 so the value remains undefined.
I'm going to divide it into 3 groups. So the best way to divide it into 3 groups is I can have 1 group right there, 2 groups, or the second group right there, and then, the third group.
And then each group will have exactly how many bell peppers? They'll have 1, 2. So 6 divided by 3 is equal to 2. So the best way or one way to think about it is that you divided the 6 into 3 groups.
Now you could view that a slightly different way, although it's not completely different, but it's a good way to think about it. You could also think of it as 6 divided by 3.
And once again, let's say I have raspberries now-- easier to draw. And here, instead of dividing it into 3 groups like we did here.
This was 1 group, 2 group, 3 groups. Instead of dividing into 3 groups, what I want to do is say well, if I'm dividing 6 divided by 3, I want to divide it into groups of 3.
Not into 3 groups. I want to divide it into groups of 3. So how many groups of 3 am I going to have?
Well, let me draw some groups of 3. So that is one group of 3. And that is two groups of 3. So if I take 6 things and I divide them into groups of 3, I will end up with 1, 2 groups.
So that's another way to think about division. And this is an interesting thing. When you think about these two relations, you'll see a relationship between 6 divided by 3 and 6 divided by 3.
Let me do that right here. What is 6 divided by 2 when you think of it in this context right here? When we think about 6 divided by 2 in terms of dividing it into 2 groups, what we can end up is we could have 1 group like this and then 1 group like this, and each group will have 3 elements.
It'll have 3 things in it. So 6 divided by 2 is 3. Or you could think of it the other way. You could say that 6 divided by 2 is-- you're taking 6 objects: 1, 2, 3, 4, 5, 6.
And your dividing it into groups of 2 where each group has 2 elements. And that on some level is an easier thing to do. If each group has 2 elements, well, that's the 1 right there.
They don't even have to be nicely ordered. This could be one group right there and that could be the other group right there.
I don't have to draw them all stacked up. These are just groups of 2. But how many groups do I have? I have 1, 2, 3. I have 3 groups.
Free Trial. Season 4: End of Watch Learn More. December 8 , Season 4 — End of Watch is Live! Read More. If you want your students to experience success in learning division, please make sure they know their multiplication facts to 81, how to multiply by 0 and how to multiply by If they don't know these things, this is going to take a lot longer.
On this page you will find many Division Worksheets including division facts and long division with and without remainders. We start off with some division facts which as you know are just the multiplication facts expressed in a different way.
The main difference is that you can't divide by 0 and get a real number. If you really want your students to impress, say at their dinner table when their parents ask them what they learned today, you can teach them that division by zero is undefined.
The rest of the page is devoted to long division which for some reason is disliked among some members of the population.
Long division is most difficult when students don't know their multiplication facts, so make sure they know them first. Oh, we already said that.
What about a long division algorithm We adamantly say, yes! The reason that you and your ancestors used it is because it is an efficient and beautiful algorithm that will allow you to solve some of the most difficult division problems that even base ten blocks couldn't touch.
It works equally well for decimals and whole numbers. Long division really isn't that hard.