13. [Exploring Number]

Transcription

1 1. [Exploring Number] Skill 1.1 Using order of operations involving a mix of ( ),,, + or MM MM Simplify inside the brackets. Multiply ( ) and/or divide ( ) in order from left to right. Add ( + ) and/or subtract ( ) in order from left to right. Q. 18 (9 ) + A. 18 (9 ) subtract inside the brackets division before addition a) c) b) d) ( ) 8... e) (1 + 8) (7 + 6)... f) g) 8 (11 4)... h) 0 (1 9) i) 6 (14 + 7)... j) 8 (7 )... k) 4 + (9 4) 7... l) 6 (1 + )... m) n) 40 (1 + ) page 1 Maths Mate./6.1 Skill Builder 1

2 Skill 1. Using order of operations involving powers and ( ),,, + or MM MM Simplify inside the brackets. Simplify the power. Always multiply ( ) and/or divide ( ) in order from left to right. Always add ( + ) and/or subtract ( ) in order from left to right. Q. (6 + ) A. (6 + ) (6 + 10) 16 6 multiply within brackets first add inside the brackets a) ( ) 1 c) ( + ) b) ( 7) d) ( 4 + 6) e) ( + 8) 4 f) (7 + ) 8... g) + (1 6)... h) 8 + (1 8)... i) (4 + )... j) ( 4 + 8)... k) + (1 + 8)... l) 6 + (7 + 1)... m) (10 1) (0 )... n) (10 ) (1 ) page 16 Maths Mate./6.1 Skill Builder 1

3 Skill 1. Rounding decimal numbers to a given place. MM MM Rounding terminating decimals to a given place Circle the digit to the right of the requested place. If this digit is: 0, 1,, or 4 (< ) - round down - keep the digit in the requested place the same., 6, 7, 8 or 9 ( ) - round up - add 1 to the digit in the requested place. Rounding recurring decimals to a given place Write the first 4 digits after the decimal point. (see skill 7.14, page 84) Apply the procedure described above for terminating decimals. Q. Round 0.46 to decimal places. A a) Round 0.1 to 1 decimal place b) Round 7.89 to 1 decimal place. c) Round 1.4 to 1 decimal place. d) Round to decimal places e) Round 4.79 to decimal places. f) Round 4.1 to decimal places. g) Round.89 to 1 decimal place. h) Round 0.96 to decimal places. i) Round 4.7 to decimal places. j) Round.4 to decimal places k) Round 0.6 to decimal places.... l) Round 1.7 to decimal places.... m) Round 4.8 to decimal places.... n) Round 0.16 to decimal places.... page 17 Maths Mate./6.1 Skill Builder 1

4 Skill 1.4 Writing rational approximations of simple irrational numbers. MM MM Circle the digit to the right of the requested place. If this digit is: 0, 1,, or 4 (< ) - round down - keep the digit in the requested place the same., 6, 7, 8 or 9 ( ) - round up - add 1 to the digit in the requested place. Hint: To write a decimal number correct to two decimal places is the same thing as rounding off to the the nearest hundredth. Irrational number Rational approximation Q. cos Write the rational approximation of cos 4 correct to two decimal places. A circle the third digit 7 round up by adding 1 to 0 a) b) Write the rational approximation of correct to two decimal places. 1 Write the rational approximation of correct to two decimal places. 0 4 < round down by keeping 6.46 c) d) Write the rational approximation of correct to two decimal places. 4 Write the rational approximation of correct to two decimal places. 0 e) π Write the rational approximation of π correct to three decimal places. f) ϕ (the golden ratio) Write the rational approximation of ϕ correct to three decimal places. g) sine Write the rational approximation of sine 1 correct to three decimal places. h) tan Write the rational approximation of tan 60 correct to three decimal places. i) e (Euler s number) Write the rational approximation of e correct to two decimal places. j) Write the rational approximation of correct to three decimal places. 10 page 18 Maths Mate./6.1 Skill Builder 1

5 Skill 1. Writing very large and very small numbers in scientific notation. MM MM basic numeral Very large.4 10 scientific notaion Product of: Number 1 and < 10 Power of 10 with positive index basic numeral Very small scientific notaion Product of: Number 1 and < 10 Power of 10 with negative index To write a number greater than or equal to 1 and less than 10 in scientific notation Move the decimal point after the first non-zero digit and count the places moved. Write this number ignoring the zeros at either end. To find the power of 10 Write base 10 with an index equal to the number of places moved. Check the sign of the index: Decimal point moves left positive index. Decimal point moves right negative index. Q. Write m, the distance from earth to moon, in scientific notation. number 1 and < 10 A places 8 zeros Index +8 because of the 8 places and large original number (>.84) a) 7000 written in scientific notation is: 4 A) B) C) places 7000 Index zeros b) written in scientific notation is: 6 A) B).1 10 C) c) Some people have as many as hair follicles on their head. Write this in scientific notation.... d) By 010 the world s population is estimated to reach 6.84 billion people. Write this in scientific notation.... e) written in scientific notation is: 4 A).4 10 B) C) places index zeros f) written in scientific notation is: A).1 10 B).1 10 C) g) Write m, the diameter of a human hair strand, in scientific notation.... h) In diamonds the distance between two Carbon atoms is m. Write this in scientific notation.... page 19 Maths Mate./6.1 Skill Builder 1

6 Skill 1.6 Writing a number in scientific notation as a basic numeral. MM MM basic numeral scientific notation Product of: Number 1 and < 10 Power of 10 with positive index Very large scientific notation Product of: Number 1 and < 10 Power of 10 with negative index basic numeral Very small If the power of 10 is positive: Move the decimal point to the right as many places as the power of 10. Add zeros as place holders if necessary. Example: Hint: By convention If the power of 10 is negative: Move the decimal point to the left as many places as the power of 10. Add zeros as place holders if necessary. Example: If the result is less than 1, write a zero in the units place. Hint: By convention 0.7 not.7 Q. Write m, the diameter of optical fibre, as a basic numeral. add zeros as place holders A index 4 move decimal point 4 places left a) is the scientific notation for: A) 600 B) C) index places right c).1 10 is the scientific notation for: A).1000 B) C) 10 B b) is the scientific notation for: A) B) C) d) is the scientific notation for: A) B) C) e) Earth s atmosphere extends upward for m. Write this as a basic numeral. f) Write , China s population in 010, as a basic numeral.... g) The size of a red blood cell, mm, is scientific notation for: A) B) 8000 C) h) The size of a virus,. 10 mm, is scientific notation for: A) B) C) i) Write m, the radius of a hydrogen atom, as a decimal number. j) Write 10 7 m, the size of a speck of dust, as a decimal number page Maths Mate./6.1 Skill Builder 1

7 Skill 1.7 Using order of operations involving negative numbers. MM MM Simplify within the brackets. Simplify the power. Always multiply ( ) and/or divide ( ) in order from left to right. Always add ( + ) and/or subtract ( ) in order from left to right. Use the addition, subtraction, multiplication and division rules of negative numbers. Q. 48 ( 8 4) + A. 48 ( 8 4) + 48 ( 1) subtract inside the brackets division before addition a) 6 (4 + 6) b) 4 (7 + 9) c) (10 ) ( 7) d) (11 ) (1 9) e) (6 10) ( 4 8) f) ( 4) ( ) g) ( 9 ) (1 7) i) 8 1 ( 6) h) ( 8 + ) (6 1) j) 14 ( 10)... k) 4 (11 9)... l) 7 + (1 4)... m) 4 ( ) n) o) (7 + 8) p) (6 + ) page Maths Mate./6.1 Skill Builder 1

8 Skill 1.8 Recognising whole numbers and integers. INTEGERS Negative integers Zero Positive integers WHOLE NUMBERS MM MM Whole Numbers Negative integers, fractions, terminating decimals, recurring decimals and infinite non-recurring decimals are not whole numbers apart from these situations: 1) Any positive fraction whose numerator is divisible by the denominator. 1 4 ) Any positive decimal with only zeros after the decimal point ) Any square root of a perfect square. Q. Choose the whole numbers from this list: 8 1 7,,, 0,.6, 0 a) Choose the whole numbers from this list: 7.4, 89,, 1, 14, Integers Fractions, terminating decimals, recurring decimals and infinite non-recurring decimals are not integers, apart from these situations: 1) Any fraction whose numerator is divisible by the denominator. ) Any decimal with only zeros after the decimal point. ) Any square root of a perfect square. A. 7 is negative, so not a whole number is a whole number is a fraction, so not a whole number.6 is a decimal, so not a whole number 8 So, 0, 0 are whole numbers. b) Choose the whole numbers from this list: 67, 0.7, 4,, 1, c) Choose the whole numbers from this list: , 18,.9,,, d) Choose the whole numbers from this list:, , 4,, 1, 8.967, 7 e) Choose the integers from this list:., 11,.14, 1,, 7 f) Choose the integers from this list: 16.14,,, 0.7, 4 g) Choose the integers from this list: 8 7,.607,, 90, h) Choose the integers from this list: π 4,, 0., 18, 0 4 page 14 Maths Mate./6.1 Skill Builder 1

9 Skill 1.9 Recognising rational and irrational numbers. MM MM A number is rational if: It can be written as a fraction of integers. Hint: All integers are rational numbers., 700, 16,, 1 All terminating decimals are rational..16,.7469 All recurring decimals are rational Q. Which numbers are rational? A ) B ) C ) π 11 D ) A number is irrational (not rational) if: It can be written as a decimal, but not as a fraction. It has infinite non-recurring digits after the decimal point. Example: Hint: Square roots of prime numbers and rational numbers that are not perfect squares are irrational numbers., 18 A. is irrational, because is not a perfect square is rational, because it is a recurring decimal. π is irrational, because it has infinite non-recurring digits after the decimal point (π ) 11 is rational, because it is a fraction. So B and D are rational. a) Is 7 a rational or an irrational number? b) Is 4.16 a rational or an irrational number? 48 c) Is a rational or irrational number? irrational d) Is a rational or irrational number? e) Is 60 a whole number, an integer or an irrational number? 7 g) Is a whole number, an integer or a rational number? f) Is a whole number, an integer or a rational number? h) Is a whole number, an integer or an irrational number? i) Which is not a rational number? A ) 6 B ) C ) π D ) 9 k) Which is an irrational number? A ) B ). C ) 4 D ) m) Which numbers are rational? A ) 8 B ) 6.9 C ) D ) j) Which is not a rational number? A ) 0.1 C ) B ) D ) l) Which is an irrational number? A ).6 B ) 6.1 C ) 7 D ) 10 n) Choose the rational numbers from this list: 8 01, π, 0,, 6 11 page 14 Maths Mate./6.1 Skill Builder 1

10 Skill Recognising classes of numbers. REAL NUMBERS IRRATIONAL π, ϕ, e,,,, cos 0 RATIONAL, , 7 4 4, 0.6, 10 9 Integers...,,, 1, 0, 1,,,... MM MM Natural (Whole Numbers) 0, 1,,, 4,, 6,... Hint: Rational numbers include integers, terminating decimals and recurring decimals. Irrational numbers include infinite non-recurring decimals. included in, included in, included in Irrational number included in Q. Which classification describes 81? A integer A ) integer and irrational rational B ) rational and real real number C ) irrational and rational natural D ) real and natural irrational So B is the correct description. a) Use true and false to complete this table: b) Use true and false to complete this table: 4.7 Integer Rational Irrational Real false true false true 00 Integer Rational Irrational Real c) Use true and false to complete this table: d) Use true and false to complete this table: π Integer Rational Irrational Real Integer Rational Irrational Real 14 e) Use true and false to complete this table: f) Use true and false to complete this table: 6 Integer Rational Irrational Real 6 9 Integer Rational Irrational Real g) Which classification describes ? A ) integer and rational B ) rational and real C ) integer and irrational D ) irrational and real i) Which classification describes ? A ) integer and irrational B ) irrational and real C ) integer and rational D ) rational and real h) Which classification describes 49? A ) integer and rational B ) irrational and real C ) integer and irrational D ) rational and irrational 7 j) Which classification describes? A ) integer and rational 4 B ) irrational and real C ) rational and real D ) rational and irrational page Maths Mate./6.1 Skill Builder 1

11 Skill Comparing and ordering rational and irrational numbers. MM MM Express the numbers as decimals. Round the decimal numbers to one or two decimal places, as needed in the question. Examples: π correct to decimal places correct to decimal places correct to decimal places Find the perfect squares greater than (>) and less than (<) the number under the square root. Compare and order the terminating decimals. Q. Place in ascending order: 10 10,,.1, π, 1 express numbers as decimals A π <.16 <.1 <. < OR π < 10 <. 1 < < 1 The answer is π, 10,. 1, 10, 1 round to decimal places a) Which number is greater? 1 or 16 < 1 < 4 < 1 < b) Which number is greater? 7 or 8 c) Which number is smaller? 1.41 or d) Which number is greater? 8 or e) Which number is greater? π or 9 f) Which number is smaller? or 6 g) Which number is greater? 18 or 4 h) Which number is smaller? or.1 i) Place in ascending order: j) Place in ascending order: ,,.,,.4 8,,,.76, page 14 Maths Mate./6.1 Skill Builder 1

12 page Maths Mate./6.1 Skill Builder 1