g TION AL N UMBE RS IRR TION AL N UMBE RS eg NEGATIVE INTEGERS eg 9 22 ZERO POSITIVE INTEGERS eg 5 93 Natural Numbers These are the counting numbers 1 2 3 4 00 000 2 Whole Numbers These are the natural numbers including zero 0 2 3 50 3 Integer ID: 27010 Download Pdf

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g TION AL N UMBE RS IRR TION AL N UMBE RS eg NEGATIVE INTEGERS eg 9 22 ZERO POSITIVE INTEGERS eg 5 93 Natural Numbers These are the counting numbers 1 2 3 4 00 000 2 Whole Numbers These are the natural numbers including zero 0 2 3 50 3 Integer

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2009 Page 1 THE REAL NUMBE SY ST EM Summary EAL N UMBE RS ( INTEGERS ( FRACTIONS e.g. TION AL N UMBE RS IRR TION AL N UMBE RS e.g. , NEGATIVE INTEGERS e.g. –9, –22 ZERO POSITIVE INTEGERS e.g. 5, 93 Natural Numbers ( These are the counting numbers. 1 , 2, 3, 4, …., 00, 000, 2. Whole Numbers ( These are the natural numbers including zero. 0, , 2, 3, …, 50, 3. Integers ( These are the set of natural numbers together with their negatives and the number zero . …, 0, –9, …, –3, –2, , 0, , 2, 3, …, 6, Note that integers must be whole numbers. Therefore, is not an integer. 4. Rational

Numbers ( These are numbers that can be expressed in the form , where and are integers and ≠ 0. Examples are: ; ; –3 ; 8 All terminating (finite) decimals are rational numbers. Examples are: 0, 25 = 25 1 000 = 3,28 = 3 28 00 = 3 25 = 82 25 All non-terminating (non-finite) but recurring decimals are rational numbers. A dot is placed on top of the digit that is recurring. If three or more digits recur, we place a dot on the first and last digits that are recurring. SSON SSON

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Page Page Page Lesson 1 | Algebra Page 1 Page 5. Irrational Numbers ( These are numbers that are

not rational numbers. These numbers cannot be written as fractions with integer numerators and denominators. In decimal form, they are non terminating ( non finite) as well as non recurring. Generally, for the sake of identifying these numbers, we can say they are roots that do not simplify to rational numbers. Also expressions involving the number are irrational xamples = ,4 42 3562 = ,709975947 = 3, 592654 = 3, 6227766 = 9,86960440 6. Real numbers( The rational and irrational numbers together form the set of real numbers. 7. Non-real numbers These numbers are generally even powered roots of

negative numbers. Division by zero is undefined. xamples: –4 –5 00 ; 83 Activity 1 . ) = 0 if is rational if is irrational Determine ( ) + + (2 2. Given: (2 + )( – 3)( – 3)( + 4) = 0. Solve for if: 2. 1 2.2 2.3 2.4 2.5 3. Which of the following do not have real solutions? 3. 1 = 0 3.2 = Example Example Irrational since these are all non-terminating, non-recurring decimals. Irrational since these are all non-terminating, non-recurring decimals. Example Example Activity Activity

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Lesson 1 | Algebra Page 1 Page 3 Lesson 1 | Algebra Page 1 2009 Page 3 3.3 = 6 3.4 – 2) = –9

Solutions to Activity . ( ) + + (2 ) = + 0 + = 2. Note that = 2, which is rational. Also, –5 2. If a product of numbers is equal to zero then one of the numbers must be zero. In this example, we have 2 + = 0 or – 3 = 0 or – 3 = 0 or + 4 = 0 Therefore, = or = 3 or = . The equation + 4 = 0 cannot be solved in the real number system as the sum of two positive numbers cannot be zero. Remember that is greater than or equal to zero for all real values of , and solving the equation results in = –4 (even powered root of a negative). 2. 1 = 3 2.2 = 3 2.3 = ; 3 2.4 = 2.5 All of the above are real

numbers. Therefore, = or = 3 or = 3. 1 This equation does not have a real solution. You cannot divide 2 by any number to get zero. If we multiply both sides by , we get 2 = 0 which is false!! 3.2 By inspection, = 3. OR by multiplying both sides by , we get 2 = . = 3. The answer is a positive integer, (natural or whole number as well) and therefore is a real number. 3.3 = 6. = . Therefore the answers are irrational and hence real numbers. Always remember to include when square rooting both sides of an equation. 3.4 This equation does not have a real solution. The expression on the left

hand side is always greater than or equal to zero because it is a number raised to the power of 4. Any number, positive or negative, raised to an even power will be greater than or equal to zero. Thus, ( – 2) cannot be equal to minus 9.

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