Addition and Subtraction of Integers

The origins of the symbols "+" and "-" to denote operations of addition and subtraction are mirky. It is very plausible that, like the ampesand "&", the plus symbol began as an abbreviation for the Latin et, and. But the Bakhshali Manuscript estimated to date at about third or fourth century A.D., used "+" for subtraction. Both symbols, with their now common meanings, first appeared in print in 1489, although the latin cross (a cross with one segment alongated) in a horizontal position was used by Fermat, Rolle, L'Hospital, and De Moivre as late as the eighteenth century.

Addition signifies joining two groups in counting or two measurements, say when two sticks are placed end to end. (In passing, the numbers added are both called addends, the result is the sum.) R. Aharoni observes that there are two different modes of addition: static and dynamic.

 Static situation  Dynamic situation
 I have 5 hardcover and 3 papeback books. How many books do I have?  I had 5 hardcover books and have bought 3 paperbacks. How many books do I now have?

The distinction between two modes of addition is important as it highlights the differences in meaning of the operation of subtraction. According to [Aharoni, p. 69] children find static subtraction difficult.

The static situation is best suitable for the introduction of the commutativity of addition, the dynamic for the introduction of the associativity.

Commutativity of Addition

If I have a certain number of books, of which 5 are hardcover and 3 paperbacks. There is a definite number of books in my possession. To determine this number, I may count first the hardcover books and then the paperbacks to obtain 5 + 3 = 8. Or I may count first the paperbacks and then the hardcoverbooks, getting 3 + 5 = 8. In fact, it does not matter how I go about counting the books: one group first and then the other, or disregarding the nature of book cover altogether. The result is always the same.

Associativity of Addition

I have 5 hardcover and 3 paperback books. I bought 2 paperbacks. How many books do I have now? Answer: counting different covers separately, 5 + (3 + 2). On the other hand, I had (5 + 3) before the purchase, therefore, presently, I have (5 + 3) + 2 books. The two amounts have to be the same, so that 5 + (3 + 2) = (5 + 3) + 2.

The previous example illustrates the situation in which the second addend has been changed. By the commutativity of addition, we could as well change the first addend. So, more generally,

When one of the addends changes by a certain amount, the sum changes by the same amount.

(Two applets help accustom a student with this rule of change. The Rule of Change of Addends let's one change one of the addends and observe the corresponding change of the sum. Less, Equal, More is a little different. The sum now does not change automatically so that when an addend changes an equality becomes inequality. One needs to change the sum manually to turn it back into equality.)

Three Meanings of Subtractions

 Removal  Whole-part  Comparison
 There were 5 apples. John removed 3. How many apples have been left?  There are 5 apples. 3 are red, the rest are green. How many green apples are there.  One plate contains 5 apples, another 3. How many more apples are there on the first plate?

Subtraction has a strong association with removal. [Aharoni, p. 73] reports that, when asked for a subtraction story, all pupils in his first grade class came up with a story of removal.

The result of subtraction is called the difference. The number being subtracted is the subtrahend. The number from which the subtrahend is subtracted is the minuend.

Repeated subtractions act as an aggregated subtrahend:

a - b - c = a - (b + c) and
a - b - c - d = a - (b + c + d), and so on.

Thus increasing the subtrahend decreases the result by same amount. With the minuend it is the opposite: increasing the minuend increases the result by the same amount.

References

  1. F. Cajori, A History of Mathematical Notations, Dover Publications, 1993
  2. R. Aharoni, Arithmetic for Parents, Sumizdat, 2007

Related material
Read more...

  • Less than, Equal to, Greater Than Symbols
  • The Rule of Change of Addends
  • Less, Equal, More
  • Demonstration of the Distributive Law
  • The Idea of Counting
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