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1.1 The Real Number System Types of Numbers: The following diagram shows the types of numbers that form the set of real numbers. Definitions 1. The natural numbers are the numbers used for counting. 1, 2, 3, 4, 5, . . . A natural number is a prime number if it is greater than 1 and its only factors are 1 and itself.
A real number is either a rational number or an irrational number. An irrational number is a nonrepeating, nonterminating decimal. The whole numbers consist of the natural numbers and 0. 0, 1, 2, 3, 4, . . . The natural numbers are also referred to as the counting numbers. 1, 2, 3, 4, . . .
SECTION 1.1 deals with the axioms that define the real numbers, definitions based on them, and some basic properties that follow from them. SECTION 1.2 emphasizes the principle of mathematical induction. SECTION 1.3 introduces basic ideas of set theory in the context of sets of real num-bers.
Given the fundamental importance of the real numbers in mathematics, it is important for mathematicians to have a logically sound description of the real number system. In particular, the adoption of set theory as the basic language for mathematics means that the real numbers need to be described formally in terms of set theory.
Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers.
make up what we call the collection of real numbers, which is denoted by R. Therefore, a real number is either rational or irrational. So, we can say that every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number. This is why we call the number line, the real ...
The Number System Identify the sets to which each of the following numbers belongs by marking an “X” in the appropriate boxes. Number Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers 1. − 17 2. −2 3. 37 9 − 4. 0 5. −6.06 6. 4 .56 7. 3.050050005... 8. 18 9. 0 −43 10. π 11. .634 12. 225 13 ...