Real-Number-System

1. Real Number System

3. Rational Numbers • The set of numbers that can be expressed as the ratio of two integers. • Can be written in the form .

4. Irrational Numbers • The set of numbers that cannot be expressed as the ratio of two integers. • Irrational numbers have non-repeating decimals that are non-terminating. … … …

5. Integers • It comprises the negative integers, positive and zero. Negative Integers Zero Positive Integers

6. Non-Integers Fractions and Decimals (Terminating and Non-Terminating)

7. Natural Numbers (Counting Numbers) Whole Numbers

8. Properties of Addition

9. The Commutative Property of Addition states that the two Real numbers can be added in any order to get the same result. If a and b represent Real Numbers , then a + b = b + a. Examples:

10. The Associative Property of Addition.Illustrates that it doesn’t matter how we group or associate numbers in addition. If a, b, c represent Real Numbers , then (a + b)+c = a + (b + c). Example:

11. The Closure Property of Additionstates that the sum of any two Real numbers is also a real number . If a and b represent Real Numbers , then a + b = c where c .

12. The Identity/Zero Property of Addition.states that when we add zero to a number, the number remains the same. Hence, zero is referred as the additive identity. If a represents a number, then . Example:

13. The Inverse Property of Addition.illustrates those two numbers that are the same distance away from the origin, but on opposite directions are called opposites or additive inverses of each other. If a represents a number, then (-a) is its opposite or negative or its additive inverse so that . Example:

14. Properties of Multiplication

15. The Commutative Property of Multiplication states that the two Real numbers can be multiplied in any order to get the same result. If a and b represent Real Numbers , then (a) (b) = (b) (a). Example:

16. The Associative Property of Multiplication.Illustrates that it doesn’t matter how we group or associate numbers in multiplication. If a, b, c represent Real Numbers , then Example:

17. The Distributive Property of Multiplication Over Addition.demands multiplying a number to every number inside a parenthesis, then combine the results by addition If a, b, c represent Real Numbers , then Example:

18. The Closure Property of Multiplicationstates that the product of any two Real numbers is also a real number . If a and b represent Real Numbers , then a b = c where c .

19. The Identity Property of Multiplication.illustrates that whenever we multiply a number by 1, the product is the same number . If a represents a number, then . Example:

20. The Zero Property of Multiplication.tells that whenever we multiply a number by zero, its product is zero . If a represents a number, then . Example:

21. The Inverse Property of Addition.Explains that any number except zero has its reciprocal, and whenever this number is multiplied to its reciprocal, the product is equal to 1. We also call the two numbers as reciprocal of each other. If a represents a non-zero number, then Example:

22. Fundamental Operations with Real Numbers

23. Fundamental Operations with Real Numbers The concept of the absolute value of a real number is important to signed numbers. Signed numbers are numbers which are preceded by plus or minus sign. However, a number that has no sign is understood to be positive. The absolute value of a real number x denoted by |x| is defined as x, if x > 0 (i.e. is x positive) -x, if x < 0 (i.e. is x negative) 0, if x = 0 According to the definition, the absolute value of any nonzero number is always positive. For example, |4| = 4 ; |-4| = - (-4) = 4 ; |0| = 0

24. Rules Governing the Operations on Signed Numbers The operation on the set of real numbers is governed by the following rules: Rule 1: To add two real numbers with like signs, add their absolute values and prefix the common sign. For example, a. 2 + 5 = 7 b. – 2 + (-5) = -7 c. -23 + (-8) = -31 Rule 2: To add two real numbers with unlike signs, subtract the smaller absolute value from the bigger absolute value, and prefix the sign as that of the bigger absolute value. For example, a. 8 + (-11) = -3 b. -12 + 17 = 5 c. -25 + 43 = 18d. -8 + 11 = 3 e. -25 – (-17) = -8 f. 60 + (-80) = -20

25. Rule 3: To subtract two real numbers with like signs, change the sign of the subtrahend and proceed to algebraic addition (Rules 1 or 2). For example,a. 8 – 15 = 8 + (-15) = -7 d. 40 – 58 = 40 + (-58) = -18b. -23 – 15 = -23 + (-15) = - 38 e. -8 – 15 = -8 + (-15) = -23 c. -8 – (-15) = -8 + 15 = 7 f. -95 – (-80) = -95 + 80 = -15 Rule 4: To multiply (or divide) two numbers having like signs, multiply (or divide) their absolute values and prefix a plus sign. For example,a. 8 (2) = 16 b. (-8) (-2) = 16 c. (-5) (-3) = 15 d. 8 ÷ 2 = 4 e. -20 ÷ -4 = 5

26. Rule 5: To multiply (or divide) two numbers having unlike signs, multiply (or divide) their absolute value and prefix a minus sign. For example,a. (-10) (2) = -20 b. (-5) (3) = -15 c. -10 ÷ 2 = -5 d. -24 ÷ 6 = -4

27. Performing Operations on Series of Numbers In a series of numbers involving the basic operations in Arithmetic, the following give the order of performing the operations:

28. First, perform powers and extract roots.Illustration: Second, perform multiplication or division in order of occurrence.Illustration: ;

29. Third, perform addition or subtraction in order.Illustration: In the presence of parentheses, quantities within these symbols are to be performed first.Illustration: a.

30. b. c.

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