Arithmetic

I

Introduction

Arithmetic, branch of mathematics that arises from counting, the most basic mathematical operation. Arithmetic encompasses various ways of counting, or manipulating numbers: addition, subtraction, multiplication, and division. The ancient Greek word arithmētikē combined the words arithmos, meaning “number,” and technē, referring to an art or skill. Thus, arithmetic means the art of numbers. The numbers used in arithmetic may be positive integers (whole numbers), negative integers, fractions, or decimals. For the history of arithmetic and mathematics, see Mathematics.

Different civilizations throughout history have developed different kinds of number systems. Although the ancient Babylonians used a system based on the number 60, all modern cultures employ a system in which objects are counted in groups of ten, probably because humans have ten fingers and tend to use them in counting. This system is called the decimal, or base 10, system.

II

Operations With Positive Integers

We generally count with positive integers. This unending sequence of whole numbers starts with 1. Each subsequent number in the sequence is one more than the number before: 1, 2, 3, 4 …. Alternating numbers starting with 1 (1, 3, 5 …) are called odd numbers, while every other number starting with 2 (2, 4, 6 …) is called an even number.

In numbers with two or more digits, every digit has a so-called place value. In the decimal system, the place value increases from units, or ones, to tens, hundreds, thousands, and higher as the number of digits increases from right to left. We can also say that each place increases by a power of 10. A power of a number is the number of times the number is multiplied by itself. In the number 1,111, for example, the place value on the far right is a unit or ones value; the place value just to its left is 10 (1 × 10); the next one to the left is 10 × 10, or 100; and the place value on the far left is 10 × 10 × 10, or 1,000. The number 2,534, then, is equivalent to (2 × 1,000) + (5 × 100) + (3 × 10) + (4 × 1).

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A

Adding Positive Integers

The arithmetic operation of addition is basically a means of counting quickly and is indicated by the plus sign (+). We could place 4 apples and 5 more apples in a row, then count them individually from 1 to 9. Addition, however, makes it possible to count all of the apples in a single step (4 + 5 = 9).

We call the end result of addition the sum. The simplest sums are usually memorized. This table shows the sums of any two numbers between zero and nine:

To find the sum of any two numbers from 0 to 9, locate one of the numbers in the vertical column on the left side 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 that lies at the intersection of the column and row that have been selected. For example, 6 + 7 = 13.

We can easily add long lists of numbers with more than one digit by repeatedly adding one digit at a time. For example, if the numbers 27, 32, and 49 are listed in a column so that all the units are in a line, all the tens are in a line, and so on, finding their sum is relatively simple:

First add the units (7 + 2 + 9); they total 18. Then add the digits in the tens place (2 + 3 + 4); they total 9, but this means 9 tens, or 90. In the last step, add the total of the units to the total of the tens:

We can skip the second step, adding the sum of the units to the sum of the tens, by using a shortcut called carrying. Carry the 1 in 18, which stands for 1 ten, over to the tens column and add it directly to the digits there:

Add the digits in the tens column, including the carried 1, and place the sum, 10, just to left of the units sum. The result is 108. Similarly, when adding numbers with three or more places, we can carry digits to the hundreds place, thousands place, or beyond.

B

Subtracting Positive Integers

The arithmetic operation of subtraction is the opposite of addition and is indicated by the minus sign (-). If we take 5 apples away from 9 apples, subtraction tells how many apples remain without our actually counting them. The simple sums memorized for addition are used in reverse for subtraction. For example, the result of 9 minus 5 is 4 because 4 is the number we would have to add to 5 for a sum of 9. The end result of subtraction is called the difference.

It is possible to subtract 23 from 66 by counting backward 23 integers from 66, one number at a time, or by taking away 23 items from a collection of 66 and counting the remainder. Either way we would reach 43. The rules of arithmetic for subtraction, however, provide a much quicker method for obtaining the answer. We can subtract large numbers by repeatedly subtracting one digit at a time. First align the numbers under one another, units under units, tens under tens, as in addition:

Subtract the units: 6 - 3 = 3. Then subtract the tens column: 6 – 2 = 4. The results of these two single-digit subtractions, written side by side, provide the answer:

Subtraction is a bit more complicated if we need to subtract a larger digit from a smaller one. For example, when subtracting 47 from 92, the units value (7) of 47 is greater than the units value (2) of 92. We can handle this situation using a procedure called borrowing, which is like carrying in reverse. Ten units can be borrowed from the tens column—that is, from the 9 of 92—leaving 8 in the tens column. Bring the 10 over to the units column and add it to the 2 already there, giving 12 in that column from which 7 can then be subtracted:

Complete the subtraction by taking 4 away from 8 in the tens column, which gives 4. The answer, or difference, is 45.

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