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Secondary 1 Mathematics Chapter 1 Real Numbers Practice 1 History of Real Numbers In legends, the first use of numbers traces all the way back to 35 000 B.C. Numerical representation was needed for various purposes (e.g. to keep track of the goods bought and sold, determine the amount of property ownership etc). People in the past kept track of quantities using tally marks. Tally marks used in Europe, North America and some parts of Asia Pacific. Tally marks used in China, Japan and Korea The Egyptian number system favours the use of picture symbols to represent numerals. This number system uses bases of 10. 1 10 100 1000 10 000 100 000 1 000 000 Staff Arch Coiled rope Flower Bent finger Tadpole Astonished man The Roman number system used letter symbols for numerals. These symbols are still commonly used today in clocks faces, successive political leaders or children with identical names and the numbering of annual events. Symbol I V X L C D M Number 1 5 10 50 100 500 1000 The Addition Rule: The Roman numeration system is based on an additive system in which symbols do not have place values. This means that a numeral is the total sum of the numbers represented by each symbol. Hence, II = 2, VIII = 8, XVII = 17, XXXVIII = 38. The Subtraction Rule: Learning Objectives: The subtraction rule states that when a smaller numeral is placed on the left of a larger To provide an aim for numeral, the smaller numeral has to be subtracted from the larger numeral. For example, 45 is students to achieve at written as XLV (−10 + 50 + 5 = 45) instead of XXXXV. the end of each lesson. • • • Identifying integers, rational numbers, natural numbers, whole numbers and real numbers; Representing numbers on a number line; Perform the four operations on positive and negative integers correctly. -- 1 -- S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd Introduction to the world of real numbers Natural numbers are numbers we use for counting. {1, 2, 3, 4, 5, 6} Whole numbers include the number zero. {0, 1, 2, 3, 4, 5, 6} Negative integers are numbers smaller than zero. {−1, − 2, − 3, − 4, − 5, − 6} Positive integers are numbers greater than zero. {1, 2, 3, 4, 5, 6} Zero, 0, is an integer that is neither positive nor negative. Rational numbers are numbers which can be expressed in the form of 22 are integers, and b ≠ 0 . Eg: , 7 7 2 o All integers, Eg: , − , 1 1 a , where a and b b 2 4 , − 3 5 5 1 o All terminating or repeating decimals, Eg: 0.375 = • 3 1 , 0.3 = 8 3 a , where a Summary: b An overview of key concepts and b are integers, and b ≠ 0 . They are non-terminating and non-repeating decimals. Irrational numbers are numbers which cannot be expressed in the form of Eg: π, e, 2, is provided to strengthen student’s understanding. 5 etc. Real Numbers, ℝ • The set of rational and irrational numbers Rational Numbers, ℚ Irrational Numbers • Numbers which can be expres sed a in the form of , where a and b are b integers, and b ≠ 0 . • Numbers which cannot be expressed in a the form of , where a and b are b integers, and b ≠ 0 • Examples are: π,e, 2, 5, etc. Integers, • Consists of positive and negative integers, and zero • = {..., -3, -2, -1, 0, 1, 2, 3, ...} Negative Integers, − • − = {-1, -2, -3, ...} Zero Fractions • Examples are: 22 1 3 5 , , , etc. 7 2 4 12 Positive Integers, + • Also known as natural numbers, ℕ • + = {1, 2, 3, ...} Whole numbers -- 2 -- S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd Number line -5 -4 -3 -2 -1 0 1 Negative Integers 4 3 2 5 Positive Integers / Natural Numbers Numbers which are smaller in values will be to the left and numbers with greater values will be to the right of the number line. Representing the numbers −2, − 10, 0, 1, 3 on a number line will be as follows: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 As much as we would love to show you everything, we cannot be showing you the best. Do drop by any JustEdu centre to view the full set! -- 3 -- S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd Example 1 Evaluate the following without using a calculator. a) 13 + 22 b) 17 − 8 c) 4 − 9 Solution: a) 13 + 22 = 35 d) −13 − 6 b) 17 − 8 = 9 + 22 13 -8 35 9 c) 4 − 9 =−5 17 d) − 13 − 6 =−19 -6 -9 -5 -19 4 Example 2 Evaluate the following without using a calculator. a) −12 + ( −10 ) b) −6 + ( −14 ) Solution: a) − 12 + ( −10 ) =−12 − 10 =−22 c) −8 − ( −9 ) d) −11 − ( −10 ) b) − 6 + ( −14 ) =−6 − 14 =−20 -10 -22 -14 -12 -20 -6 d) − 11 − ( −10 ) =−11 + 10 =−1 c) − 8 − ( −9 ) =−8 + 9 =1 + 10 +9 -8 -13 Teachers will go through examples in details to bring across key concepts. -11 1 -- 4 -- -1 S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd Multiplication of Integers Rules for signs: (+)× (+) = (+) Examples 5× 3 = 15 ( −) × (−) = ( +) ( +) × ( −) = ( −) ( −) × (+) = (−) ( −5) × ( −3) =15 5 × ( −3) =−15 ( −5) × 3 =−15 Rules for signs: (+) ÷ (+) = (+) Examples 15 ÷ 3 = 5 ( −) ÷ ( −) = ( +) ( +) ÷ ( −) = ( −) ( −) ÷ ( +) = ( −) ( −15) ÷ ( −3) =5 15 ÷ ( −3) =−5 ( −15) ÷ 3 =−5 Division of Integers Example 3 Evaluate the following. a) 6 × ( −5) b) ( −2 ) × ( −7 ) c) 112 d) ( −6 ) e) 83 f) h) ( −1) ( −11 + 10 ) × ( −3) a) 6 × ( −5) =−30 b) ( −2 ) × ( −7 ) =14 2 c) 11= 11 × 11 d) ( −6 ) = ( −6 ) × ( −6 ) g) ( −7 ) × 3 − ( −2 ) × 4 2 3 Solution: = 121 = 36 e) 83 = 8 × 8 × 8 f) = 512 g) 2 ( −1) = ( −1) × ( −1) × ( −1) 3 = −1 ( −7 ) × 3 − ( −2 ) × 4 = ( −21) − ( −8 ) = −21 + 8 = −13 h) ( −11 + 10 ) × ( −3) = ( −1) × ( −3) =3 -- 5 -- S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd Example 4 Evaluate the following. a) c) ( −21) ÷ ( −3) 0 ÷ ( −267 ) b) 125 ÷ ( −5) d) 987 ÷ 0 Solution: a) ( −21) ÷ ( −3) =7 b) 125 ÷ ( −5) =−25 c) 0 ÷ ( −267 ) =0 d) 987 ÷ 0 =∞ Exercise 1) Arrange the following in ascending order. 3 2 7 3 2 a) , 0.008, − , , − 5 8 7 3 1 3 b) − 3 216 , − , ( −2 ) , 0, − 49 3 c) − 2) • • • 157 π , , − 0.785, − 0.78, 200 4 11 4 Write down all the rational numbers in descending order. π , − ( 7 ) , 0.7, 25 112 3 3) 7 , 81 ( ) 4 81 , 7 + 1, − 7, 7 2 , −1 7 9 Use a calculator to evaluate each of the following. a) 3 7 c) −7 − −7 5 10 e) 2 3 15 625 − −1 8 7 4 7 4 5 d) × − ÷ − × − 8 3 8 6 7 742 − 702 b) 3 3 3 3 4 7 1 × − × 5 9 9 3 2 1 8 f) + − ÷ 3 125 4 9 5 Do drop by our centre to view the full set of materials. -- 6 -- S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd No calculators allowed for Questions 6 and 7. Proper workings must be shown. 4) Evaluate each of the following. 3 1 1 5 a) + −1 × 4 + ( −49 ) ÷ 15 × − ÷ − 3 8 8 4 1 2 23 1 1 b) − − ÷ − ÷ 3 − −2 2 5 30 2 4 3 1 1 1 1 − 3 −− c) −3 × 2 × − 3 8 144 5 2 −27 1 3 1 169 3 125 d) + − × −1 ÷ − × 512 4 4 4 100 -- 7 -- S1 | Chapter 1 Real Numbers | Practice 1 © JustEdu Holdings Pte Ltd