Arithmetic

These axioms apply to all real numbers. It is important to note that real numbers is the general class of all numbers that includes whole numbers, integers, rational numbers, and irrational numbers. For each of these number types only certain axioms apply.

Whole numbers, also called natural numbers, include only numbers that are positive integers and zero. These numbers are typically the first ones to which a person is introduced, and they are used extensively for counting objects. Addition of whole numbers involves combining them to get a sum. Whole number multiplication is just a method of repeated addition. For example, 2 × 4 is the same as 2 + 2 + 2 + 2. Since whole numbers do not involve negative numbers or fractions, the two inverse properties do not apply. The smallest whole number is zero but there is no limit to the size of the largest.

Integers are whole numbers that include negative numbers. For these numbers the inverse property of addition does apply. For these numbers, zero is not the smallest number but it is the middle number with an infinite number of positive and negative integers existing before and after it. Integers are used to measure values which can increase or decrease such as the amount of money in a cash register. The standard rules for addition are followed when two positive or two negative numbers are added together and the sign stays the same. When a positive integer is added to a negative integer, the numbers are subtracted and the appropriate sign is applied. Using the axioms of multiplication it can be shown that when two negative integers are multiplied, the result is a positive number. Also, when a positive and negative are multiplied, a negative number is obtained.

Numbers to which both inverse properties apply are called rational numbers. Rational numbers are numbers that can be expressed as a ratio of two integers, for example, ¹⁄₂. In this example, the number 1 is called the numerator and the 2 is called the denominator. Though rational numbers represent more numbers than whole numbers or integers, they do not represent all numbers. Another type of number exists called an irrational number which cannot be represented as the ratio of two integers. Examples of these types of numbers include square roots of numbers which are not perfect squares and cube roots of numbers which are not perfect cubes. Also, numbers such as the universal constants π and e are irrational numbers.

The principles of arithmetic create the foundations for all other branches of mathematics. They also represent the most practical application of mathematics in everyday life. From determining the change received from a purchase to calculating the amount of sugar in a batch of cookies, learning arithmetic skills is extremely important.