Arithmetic

Arithmetic is the study of mathematics related to the manipulation of real numbers. The two fundamental properties of arithmetic are addition and multiplication. When two numbers are added together, the resulting number is called a sum. For example, 6 is the sum of 4 + 2. Similarly, when two numbers are multiplied, the resulting number is called the product. Both of these operations have a related inverse operation which reverses or "undoes" its action. The inverse operation of addition is subtraction. The result obtained by subtracting two numbers is known as the difference. Division is the inverse operation of multiplication and results in a quotient when two numbers are divided. The operations of arithmetic on real numbers are subject to a number of basic rules, called axioms. These include axioms of addition, multiplication, distributivity, and order. For simplicity, the letters a, b, and c, denote real numbers in all of the following axioms.

There are three axioms related to the operation of addition. The first, called the commutative law, is denoted by the equation a + b = b + a. This means that the order in which you add two numbers does not change the end result. For example, 2 + 4 and 4 + 2 both mean the same thing. The next is the associative law which is written a + (b + c) = (a + b) + c. This axiom suggests that grouping numbers also does not effect the sum. The third axiom of addition is the closure property which states that the equation a + b is a real number.

From the axioms of addition, two other properties can be derived. One is the additive identity property which says that for any real number a + 0 = a. The other is the additive inverse property which suggests that for every number a, there is a number −a such that −a + a = 0.

Like addition, the operation of multiplication has three axioms related to it. There is the commutative law of multiplication stated by the equation a × b = b × a. There is also an associative law of multiplication denoted by a × (b × c) = (a × b) × c. And finally, there is the closure property of multiplication which states that a × b is a real number. Another axiom related to both addition and multiplication is the axiom of distributivity represented by the equation (a + b) × c = (a × c) + (b × c).

The axioms of multiplication also suggest two more properties. These include the multiplicative identity property which says for any real number a, 1 × a = a, and the multiplicative inverse property that states for every real number there exists a unique number (1/a) such that (1/a) × a = 1.

The axioms related to the operations of addition and multiplication indicate that real numbers form an algebraic field. Four additional axioms assert that within the set of real numbers there is an order. One states that for any two real numbers, one and only one of the following relations is true: either a < b, a > b or a = b. Another suggests that if a < b, and b < c, then a < c. The monotonic property of addition states that if a < b, then a + c < b + c. Finally, the monotonic property of multiplication states that if a < b and c > 0, then a × c < b × c.