Real Number System
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This document defines and provides examples of different types of real numbers: - Rational numbers can be expressed as ratios of integers and include integers, natural numbers, whole numbers, and numbers with terminating or repeating decimals. - Irrational numbers cannot be expressed as ratios and are non-terminating, non-repeating decimals like the square root of 2. - A Venn diagram is used to classify examples like 117, 0, -12.64039..., -1/2, 6.36, π/2, and √9/4 as rational or irrational numbers based on these definitions.
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This document describes different types of numbers in the real number system. It explains that real numbers include both rational and irrational numbers. Rational numbers can be written as fractions a/b, and include integers, terminating decimals, and repeating decimals. Irrational numbers cannot be written as fractions and their decimal representations do not terminate or repeat, such as pi and the square root of 2. The real number line provides a way to visualize real numbers as points along a line.
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Number System
A number system represents numbers using digits or symbols in a consistent manner. It allows unique representation of numbers and performing arithmetic operations. Common number systems include decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Each position in a number represents a power of the base. For example, in the decimal number 1457, the 1 is in the thousands place with a value of 1000.
Lesson 1.2 the set of real numbers
1. The document discusses subsets of real numbers including natural numbers, whole numbers, integers, and rational numbers.
2. Natural numbers are used for counting and start at 1. Whole numbers are formed by adding 0 to the natural numbers. Integers are formed by adding the negatives of natural numbers to whole numbers.
3. Rational numbers can be expressed as fractions a/b where a and b are integers and b is not equal to 0. Their decimal representations either terminate or repeat.
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Real Number System
- 1. 1.8 The Real Number System Created by ? Edited by Eddie Judd, Crestwood Middle School Standard: NS4
- 2. Real Numbers • Real numbers consist of all the rational and irrational numbers. • The rational numbers have many subsets: – Natural Numbers – Whole Numbers – Integers
- 3. Natural Numbers • Natural numbers are the set of counting numbers. {1, 2, 3,…}
- 4. Whole Numbers • Whole numbers are the set of numbers that include 0 plus the set of natural numbers. {0, 1, 2, 3, 4, 5,…}
- 5. Integers • Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…}
- 6. Rational Numbers • Rational numbers are any numbers that can be expressed as a ratio (in the form of , where a and b are integers, and b ≠ 0. • They can always be expressed by using terminating decimals or repeating decimals. b a
- 7. Terminating Decimals • Terminating decimals are decimals that contain a finite number of digits. • Examples: 36.8 0.125 4.5
- 8. Repeating Decimals • Repeating decimals are decimals that contain a infinite number of digits. • Examples: 0.333… 7.689689… FYI…The line above the decimals indicate that number repeats. 9 . 1
- 9. Irrational Numbers • Irrational numbers are any numbers that cannot be expressed as a ratio, • They are expressed as non-terminating, non- repeating decimals; decimals that go on forever without repeating a pattern. • Examples of irrational numbers: – 0.34334333433334… – 45.86745893… – (pi) – b a 2
- 10. Other Vocabulary Associated with the Real Number System • …(ellipsis)—continues without end • { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. • Finite—having bounds; limited • Infinite—having no boundaries or limits • Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.
- 11. Venn Diagram of the Real Number System Irrational Numbers Rational Numbers
- 12. Example • Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply. a. 117 b. 0 c. -12.64039… d. -½ e. 6.36 f. g. -3 h. 2
- 13. To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram. 9 4 2 1 Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers -12.64039… 117 0 6.36 9 4 -3 2
- 14. Solution • Now that all the numbers are placed where they belong in the Venn diagram, you can classify each number: – 117 is a natural number, a whole number, an integer, and a rational number. – is a rational number. – 0 is a whole number, an integer, and a rational number. – -12.64039… is an irrational number. – -3 is an integer and a rational number. – 6.36 is a rational number. – is an irrational number. – is a rational number. 9 4 2 1
- 15. FYI…For Your Information • When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.