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### Division Explained in 5 Minutes

Division Explained in 5 Minutes

Division is a very important concept in mathematics and especially in the study of the positive integers 1,2,3,...

When considering integers mathematicians say one divides the other if there's no remainder. For example, 3 divides 12, but 5 does not because 12/5 gives a remainder.

What's happening during division is that the group of objects is being split into pieces of equal size, for example..

12 objects can be split into 2 equal size pieces, so 12 is divisible by 2
12 objects can be split into 3 equal size pieces, so 12 is divisible by 3
12 objects can be split into 4 equal size pieces, so 12 is divisible by 4
12 objects cannot be split into 5 equal size pieces, so 12 is not divisible by 5
12 objects can be split into 6 equal size pieces, so 12 is divisible by 6

Of course, when we split a group of objects into equal size pieces we don't include pieces with one element, or one piece that's the whole group itself.

So, can any group of objects be split into equal size pieces? It seems like the answer is yes. But it's not. There are some groups of objects that can never be split into equal size pieces. When this occurs we say the number of objects in the group is a prime number.

Consider a group of 17 objects. No matter how hard you try you'll never split this group into equal size pieces. That's because 17 is a prime number. And 17 is not alone, there are an infinite amount of prime numbers!

Division can be defined in terms of multiplication. Here's how it works..

For any number b we find a number b', called the inverse of b, such that b*b'=1. Then to divide any number by b we simply multiply the number by the inverse of b, c*b'=c/b, and this can be used as a definition of division in terms of multiplication. But notice that we cannot define division in terms of multiplication until we introduce the concept of an inverse. This concept appears in many branches of mathematics and applies to a wide range of mathematical objects, not just numbers.

The inverse b' is of course equal to 1/b or b^(-1) because b*(1/b)=1

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Content written and posted by Ken Abbott abbottsystems@gmail.com