Arithmetic

I

Introduction

Arithmetic, literally, the art of counting. The word comes from the Greek arithmētikē, which combines the ideas of two words: arithmos, meaning “number”, and technē, referring to an art or skill.

The numbers used for counting are called the positive integers. These are generated by adding 1 to each number in an unending series. Different civilizations throughout history have developed different kinds of number systems. One of the most common is the one used in all modern cultures, in which objects are counted in groups of ten. Called the base 10, or decimal, system, it is the one that is used in this article.

In the base 10 system, integers are represented by digits expressing various powers of 10. For example, take the number 1534. Every digit in this number has its own place value, and these increase by another power of 10 as they move to the left. The first place value is a unit value (here, 4 × 1); the second is 10 (here, 3 × 10, or 30); the third place value is 10 × 10, or 100 (here, 5 × 100, or 500); and the fourth place value is 10 × 10 × 10, or 1,000 (here, 1 × 1,000, or 1,000).

II

Fundamental Definitions and Laws

Arithmetic is concerned with the ways in which numbers can be combined through addition, subtraction, multiplication, and division. The word number also includes negative numbers, fractions, and algebraic irrational numbers. The arithmetic laws of addition, multiplication, and distribution are as those for algebra.

A

Addition

The arithmetic operation of addition is indicated by the plus sign (+) and is a means of counting by increments greater than 1. For example, four apples and five apples could be added by putting them together and then counting them individually from 1 to 9. Addition, however, makes it possible to add, or compute, sums more readily. The simplest combinations of sums must be memorized. The following table represents the sums of any two digits from 0 to 9:

To find the sum of any two numbers between 0 and 9, locate the first of the numbers in the vertical column at the left of the table and the other number in the horizontal row at the top. The sum is the number in the body of the table at the intersection of the particular row and column that have been selected.

Arithmetic also makes it possible to add long lists of large numbers by applying simple rules that make the work quite easy. For example, it would be possible to add 27 + 32 + 49 by first counting to 27, then counting 32 more times after 27, and then another 49 times beyond that to get the total, or sum. But if the numbers are listed so that all the units are in one column, all the tens are in another column, and so on, the addition is relatively simple:

First the units (7 + 2 + 9) are added; they total 18. Then the digits in the tens place (2 + 3 + 4) are added; they total 9, but this means 9 tens, or 90. This can now be written as

In the last step, the total of the units is added to the total of the tens. This procedure may be shortened by carrying the 1 of 18, which stands for one ten, over to the tens column and adding it directly with the digits there.

The digits in the tens column are added, and the sum 108 is the result. Similarly, in adding numbers with three or more places, numbers may be carried to the hundreds, thousands, or higher places.

B

Subtraction

The arithmetic operation of subtraction is indicated by the minus sign (-) and is the opposite, or inverse, operation to addition. Again, it is possible to subtract 23 from 66 by counting backwards from 66 or by taking away 23 items from a collection of 66, until one reaches the remainder of 43. The rules of arithmetic for subtraction, however, provide a much simpler method for obtaining the answer. First the numbers are aligned under one another, units under units, tens under tens:

Subtraction is a bit more complicated if, in any column, the digits of the subtrahend are larger than those of the minuend. This can be handled in a manner analogous to carrying, but known as borrowing in subtraction. For example, to subtract 46 from 92, a 1 can be borrowed from the tens column—that is, from the 9 of 92—leaving 8 in the tens column.The 10 is brought over to the units column and added to the 2 already there, giving 12 (notated ¹2) in that column from which 6 can now be subtracted:

The subtraction is completed by carrying out the difference in the tens column, that is, by taking 4 away from 8 to give 4; the answer is, therefore, 46.