Arithmetic functions and integer products
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Contents:
More formally, subtracting 3 is the inverse of adding 3. It is similar with division and multiplication. Just as people sometimes want to form sets of the same size into one larger set, they sometimes want to break up a large set into equal-sized pieces. Thus division by 3 undoes implicit multiplication by 3 and leaves you with the original amount.
More formally, dividing by 3 is the inverse of multiplying by 3. Two interpretations of division deserve particular mention here. If I have 20 cookies, and want to sort them into 5 bags, how many go in each bag?
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This is the so-called sharing model of division because I know in how many ways the cookies are to be shared. I can find the answer by picturing the 20 cookies arranged in 5 groups of 4 cookies, which will be the contents of 1 bag. If the cookies originally came out of 5 bags of 4 each, when I put them back into those bags, I will again have 4 in each.
Thus, division by 5 undoes multiplication by 5, or division by 5 is the inverse of multiplication by 5. The picture below shows the sharing model for this situation. In the sharing model also called the partitioning model or partitive division , you know the number of groups and seek the number in a group.
In the measurement model also called quotative division , you know the size of the groups and seek the number of groups. The circled numbers in the figures above and below illustrate a crucial difference between the two models: the order in which the cookies are placed in bags. In the sharing. Note that because multiplication is commutative, 5 bags of 4 cookies each is the same total number of cookies as 4 bags of 5 cookies each. Eventually students come to see the two kinds of division as interchangeable and use whichever model helps them with a particular division problem.
We might summarize the story so far by saying that there are two pairs of operations—addition and subtraction, and multiplication and division—and these are inversely related in the sense described above. However, this summary would not quite be correct. In fact, subtraction is not actually an operation on whole numbers in the same sense that addition is.
You can add any pair of whole numbers together, and the result is again a whole number. Sometimes, however, you cannot subtract one whole number from another. This situation can be described by using negative numbers: I have negative-two apples, meaning that after I give Bart all the apples I have, I still owe him two. What is happening mathematically is that I have bumped up against a subtraction problem, 3—5, for which there is no solution in whole numbers.
Mathematicians respond by inventing a solution for it, and they call the solution —2. But your problems are not over. You soon find that you cannot be. You get into situations in which you want to do arithmetic with them also.
Besides enlarging their idea of number, people have had to extend the arithmetic operations to this new larger class of numbers. They have needed to create a new, enlarged number system. The new system, encompassing both positive and negative whole numbers, is called the integers. How do people decide what arithmetic in this extended system is or should be? How do they create recipes for adding and multiplying integers, and what are the properties of these extended operations?
They have two guides: a intuition and b the rules of arithmetic, as described above and in Box 3—1. Fortunately, the guides agree. Consider first the intuitive approach: Think hard about a lot of different cases and decide what is the right way to add and multiply in each one. To use intuition, you need to think in terms of some concrete interpretation of arithmetic. The yield of financial transactions is a good one for these purposes. Here negative amounts are money you owe, and positive amounts are money that you have or are owed by someone else.
Continuing in this way, you can puzzle out what the sum, difference, or product of any two integers should be. The trouble with this approach is that it is somewhat contrived and depends upon making decisions about how to interpret each case in the particular context.
Another approach 6 is to use an exploratory method to reason how the operations should extend from the whole numbers. By means of somewhat lengthy reasoning, you can find out how to do arithmetic with integers. But are the regularities observed about the whole number system the rules in Box 3—1 still valid? Going through the cases again will show that they are.
So not only has the number system been extended from the whole numbers to all integers, but the arithmetic in the larger system looks very similar to arithmetic in the original one in the sense that these laws are still valid. Moreover, there are some new notable regularities that describe how the new numbers are related to the original ones. These are summarized in Boxes 3—2 and 3—3.
Functions for Arithmetic Operations
The extension of whole numbers to integers is an example of the axiomatic method in mathematics: basing a mathematical system on a short list of key properties. Something much more dramatic is also true. One can show that, if the goal is to extend addition and multiplication from the whole numbers to the integers in such a way that the laws of arithmetic of Boxes 3—1 and 3—2 remain true, then there is only one way to do it.
And the rules in Box 3—3 describe how it has to work. Recipes laboriously constructed by means of some sort of concrete interpretation of negative numbers are all completely dictated by this short list of rules of arithmetic. Its most famous success is the Elements of Euclid for plane geometry. Another rather striking thing has happened during this extension from whole numbers to all integers.
The reason for making the extension was to. Box 3—2 Additional Properties of Addition. Additive identity. Adding zero to any number gives that number. Additive inverse. Every number has an additive inverse, also called an opposite.
Arithmetic derivative
The opposite is the unique number that, when added to that number, gives zero. Subtraction and negation. Subtracting a number is the same as adding its opposite. Multiplication and negation. Negation is the same as multiplication by —1. Opposite of opposite. The opposite of the opposite of a number is the number itself.
Multiplying and dividing with integers
Now, in the integers, subtraction is a true operation in the sense that you can subtract any integer from any other. As described in the rule on additive inverses in Box 3—2 , for every integer, there is another integer, called its opposite or additive inverse, that counter-balances it: the two sum to zero. Thus, at least on a conceptual level, subtraction is merged into addition, and you really only need to have the single operation of addition to capture all the arithmetic of addition and subtraction.
As soon as subtraction is made into a true operation by extending the whole numbers to the integers, you also get additive inverses, which allows you to subordinate subtraction to addition. This sort of simplification illustrates a kind of mathematical elegance: Two ideas that seemed different can be subsumed under one bigger idea. As we show below, the analogous thing happens to division when you construct rational numbers.
That subordination is the best justification for why mathematicians talk about only the two operations of addition and multiplication when discussing number systems, and not all four operations recognized in school arithmetic. Forgetting for a moment the triumph with integers, return to the whole numbers and the problem of division.