Basic Algebraic Properties of Real Numbers The numbers used to measure real-world quantities such as length, area, volume, speed, electrical charges, probability of rain, room temperature, gross national products, growth rates, and so forth, are called real numbers. They include such number as , , , , , , , and . The basic algebraic properties of the real numbers can be expressed in terms of the two fundamental operations of addition and multiplication. Basic Algebraic Properties: Let and denotes real numbers. (1) The Commutative Properties (a) (b)
The commutative properties says that the order in which we either add or multiplication real number doesn’t matter. (2) The Associative Properties (a) (b) The associative properties tells us that the way real numbers are grouped when they are either added or multiplied doesn’t matter. Because of the associative properties, expressions such as and makes sense without parentheses. (3) The Distributive Properties (a) (b) The distributive properties can be used to expand a product into a sum, such as or the other way around, to rewrite a sum as product: (4) The Identity Properties (a) (b)
We call the additive identity and the multiplicative identity for the real numbers. (5) The Inverse Properties (a) For each real number , there is real number , called the additive inverse of , such that (b) For each real number , there is a real number , called the multiplicative inverse of , such that Although the additive inverse of , namely , is usually called the negative of , you must be careful because isn’t necessarily a negative number. For instance, if , then . Notice that the multiplicative inverse is assumed to exist if . The real number is also called the reciprocal of and is often written as .
Example: State one basic algebraic property of the real numbers to justify each statement: (a) (b) (c) (d) (e) (f) (g) If , then Solution: (a) Commutative Property for addition (b) Associative Property for addition (c) Commutative Property for multiplication (d) Distributive Property (e) Additive Inverse Property (f) Multiplicative Identity Property (g) Multiplicative Inverse Property Many of the important properties of the real numbers can be derived as results of the basic properties, although we shall not do so here. Among the more important derived properties are the following. (6) The Cancellation Properties: a) If then, (b) If and , then (7) The Zero-Factor Properties: (a) (b) If , then or (or both) (8) Properties of Negation: (a) (b) (c) (d) Subtraction and Division: Let and be real numbers, (a) The difference is defined by (b) The quotient or ratio or is defined only if . If , then by definition It may be noted that Division by zero is not allowed. When is written in the form , it is called a fraction with numerator and denominator . Although the denominator can’t be zero, there’s nothing wrong with having a zero in the numerator. In fact, if , (9) The Negative of a Fraction: If , then
Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.
You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.Read more
Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.Read more
Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.Read more
Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.Read more
By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.Read more