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Basics of Arithmetic

by Ron Kurtus (revised 26 February 2013)

Arithmetic is a fundamental branch of Mathematics that is concerned with numbers and their operations.

The subject usually starts with counting and the number (or numeric) system. Then arithmetic considers the major operations on numbers, such as addition, subtraction, multiplication and division. A third area concerns applying these operations to fractions and decimals.

Questions you may have include:

This lesson will answer those questions.



Counting and the numeric system

Often people start counting using their ten fingers. Then they progress to counting objects without using their fingers. Finally, people simply count numbers without referring to objects.

Numbers have names

Each number has a name, such as one, two, three and so on. Associated with each names of a number or quantity of some object is a symbol that you can write down: 1, 2, 3 and so on.

System based on 10

Our numeric system is based on the total number of fingers that an average person has. We write down the first ten numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and then we start over with 11, 12, 13 and so on.

The ancient Maya of Central America used a system based on 20. In other words, they counted both on their fingers and their toes.

Ancient Babylonians—for some reason—used a numeric system based on 60.

Before you can start with arithmetic operations, you need to know how to count.

Arithmetic operations

The major arithmetic operations are addition, subtraction, multiplication and division.

Addition

If you have a certain number of objects and want to include another group of objects, you add them together.

Counting the total amount is made easier by knowing how to add. For example, if you have three objects and want to include four more objects, you can add 3 plus 4, which equals 7.

Otherwise, you will count 1, 2, 3, and then continue with the next four numbers,
4, 5, 6, 7 to reach the total of 3 plus 4.

The sign for addition is the plus sign: +. The sign for equals is: =.

Adding "6 plus 7 equals 13" is written:

6 + 7 = 13

Subtraction

Subtraction is going in the opposite direction of addition.

If you hold up 5 of your fingers and then put down 3 of them, you are subtracting 3 from 5. The result is 2.

The subtraction or minus sign is: . Thus, "13 minus 5 equals 8" is written:

13 − 5 = 8

Other ways of indicating subtraction are: "13 less 5" and "subtract 5 from 13."

Multiplication

Multiplication is repeated addition.

If you add 5 and 5, you get 10. If you add a third 5, you get 15. In this case, you have added 5 with itself 3 times, such as adding 5 + 5 + 5. Designating that as multiplication, you would say "3 times 5" or 3 × 5.

Multiplication symbols

There are several symbols used for multiplication: ×, · ("dot"), and *. All of the following mean the same thing:

3 × 5

3 · 5

3 * 5

Division

Division is the opposite of multiplication.

If you have the number 15, what number would you add 3 times to equal 15? Or you could say, "What number times 3 equals 15?"

Dividing 15 by 3 results in 5. Or 15 ÷ 3 = 5. Usually, you divide a larger number by a smaller number.

Division symbols

There are two division signs: ÷ and /. The following mean the same thing:

15 ÷ 3

15 / 3

Fractions and decimals

Fractions and decimals are numbers less than 1 that represent an incomplete division. Fractions and decimals can both be added to whole numbers.

Fractions

Fractions are representations of incomplete division. The fraction 3/4 means
3
divided by 4. Since you are dividing a smaller number by a larger one, the division process stops.

A fraction can be added to a whole number, resulting in a complex fraction. For example 2 + ¾ can be rewritten as the complex fraction . It is called two and three fourths.

Fractions can be added, subtracted, multiplied and divided, but the method and rules are somewhat different than for whole numbers. This is covered in a later lesson.

Decimals

A decimal is a representation of a fraction in the base of 10. For example, the fraction 3/10 is written as a decimal as 0.3 and 27/100 is written as 0.27.

Adding a decimal to a whole number results in a complex decimal. For example:

2 + 0.3 = 2.3

The advantage of decimals is that the arithmetic operations are more straightforward as compared with using fractions.

Summary

Arithmetic is a branch of Mathematics concerned with numbers and their operations. The first area in Arithmetic is counting and the number or numeric system. Then it considers the major operations on numbers, such as addition, subtraction, multiplication and division. The third area is applying these operations on fractions and decimals.


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