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Estimation, Approximation and Standard form

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This document provides information about approximation and estimation in mathematics. It defines estimation as an intelligent guess made based on available information, while approximation is a nearly exact guess. Estimates help scientists make predictions before formal investigations. Examples are provided to demonstrate approximating values and rounding numbers to certain places. The concept of significant figures in numbers is also explained, with examples of writing numbers using a specified number of significant figures. The document concludes by introducing standard form as a concise way to write very large or small numbers, along with examples and activities working with numbers in standard form.

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息 Nsom Promise. All rights reserved. IB Mathematics Approximation and Estimation
Approximation and Estimation Estimation: Estimation is an Intelligent guess made about something base on some information. Approximation is guess that is nearly exact. Estimate in science help scientist to guess before doing actual investigation. Estimate gives an idea about some quantities off hand before proper findings or measurement. For example, if you were asked to pay electricity bill given that you consumed 12units and the cost of a unit being 64frs. The cost of 12 units is 768frs 770. Why would SONEL Company prefer to approximate this value t0 770frs?. What if the value was 764? Will they still approximate it to 770 or 775. Its better to be roughly right than precisely wrong(Allan Greenspan, U.S. Federal Reserve Chairman -retired) 息 Nsom Promise. All rights reserved. Introduction Activity: 1. Guess intelligently the length and width of the classroom and calculate its area 2. Estimate the amount of water you drink on daily basis. (1.5litres per day) 3. Guess the weight of your exercise book (250g) 4. Approximate the height and weight (mass) of your mate (1.7m, 60kg or 500N) 5. Estimate the weight of a football (396 to 463g) 6. Estimate the angle between the wall the floor
Post Mathematics for Mathematics HL and SL 息 Nsom Promise. All rights reserved.
Review: Operations with Fractions and Decimals 息 Nsom Promise. All rights reserved. Activity 1: Answers ) 3 4 ) Remark: To divide a decimal by a power of 10n, move n-places to the left. To multiply a decimal number by a power of 10n, move n-places to the right. Examples: a) 450歎 1000 b) 780 1000 Activity 1: Simplify as much as possible (Hint: Use BODMAS) ) 1 2 2 1 2 1 1 6 歎 2 1 3 ) 7 1 2 2 1 7 1 2 5 + 7 歎 3 3) 4 3 (2 1 2 ) 1 1 6 歎 2 1 3 4) 1.2 0.012 5) 9.02 100 6) 2 歎 100 7) 9.8+1.07+69 8) 5-4.667 9) 44.36 x 32 10) a) 0.3 0.2 b) 0.3 1000 b) 2 100 11) 0.2 x 1000
息 Nsom Promise. All rights reserved. Approximation and Estimation NB: Some numbers are too big to be rounded up or too small to be rounded down. To round up or down a decimal number (number with decimal point), We study the number of decimal places and we approximate accordingly. Examples: 1) Round 2475 to the nearest 10frs. (Ans: 2480frs ) 2) Put 2.963 to 1decimal place (Ans: 3.1) 3)Round 344frs to the nearest 5frs. Rounding Up and Down: To round up and down a number or decimal, Draw a vertical line (stroke) in front of the desired digit or unit If the digit after the line is 5 , round it up to one and add to the digit immediately before the line; the rest of the digits are considered zeros If the digit after the line is <5, round it down so that the digits before the line remain unchanged and those after the line are assumed zeros
息 Nsom Promise. All rights reserved. Approximation and Estimation Remark: The value of any digit depends on its position. The value of 2 in the number 1267 is 100 and in 0.324 is 100th or 2dp. Given 2768.043, what is the value of each digit? The value of 2=thousand, 3=hundred, 6=ten, 8=unit, .0=tenth (1dp), 4=hundredth (2dp), 3=thousandth (3dp) 1000 100 10 Unit 10th (thousand) (hundred) (ten) (tenth or 1dp) 100th (hundredth or 2dp) 1000th (thousandth or 3dp) (1000) Or 103 100 or 102 10 or 101 1 or 100 0.1 or 10- 1 0.01 or 10-2 0.001 or 10-3 Example: Copy and complete the table below Number 1dp 2dp Whole number or unit Neare st 10frs Nearest 100th Nearest Degree 444.525 5059.996 267.537
息 Nsom Promise. All rights reserved. Approximation and Estimation Answer: Check for correctness Number 1dp 2dp Whole number or unit Neare st 10frs Nearest 100th Nearest Degree 444.525 444.5 444.53 445 450 444.53 4450 5059.996 5060.0 5060.00 5060 5060 5060.00 50600 267.537 267.5 267.54 268 270 268.54 2680 Significant Figures In the number 4380, 4=first significant figure or digit, 3=second significant figure, 8=third significant figure. and 0 is not significant although 0 has a value. However, in 4308, 0 is significant. Remark: 0 between two whole numbers or digits is significant 0 at the beginning or end of a number is not significant If the digit after the required significant figure >=5, round up and add to the previous digit
息 Nsom Promise. All rights reserved. Approximation and Estimation Examples: Write the following numbers to 2 significant a) 0403670= b) 0.052407= c) 34.08945= d) 040567 (to 3 s.f.) Number 1dp 2dp 1sf 2sf 3sf Nearest whole number 42.546 0.9974 1.9995 295.6891
息 Nsom Promise. All rights reserved. Approximation and Estimation Examples: Write the following numbers to 2 significant a) 0403670=40 b) 0.052407=0.052 c) 34.08945=34 d) 040567=34.1 to 3 sf Number 1dp 2dp 1sf 2sf 3sf Nearest whole number 42.546 42.5 42.55 4 40 43.547 43 0.9974 1.0 1.00 1.0 1.00 0.997 1 1.9995 2.0 2.00 2 2.0 2.00 2 295.6891 295.7 295.69 3 30 296 296
息 Nsom Promise. All rights reserved. Introduction Activity: Write the biggest amount of money you dream to ever have in a life time Write a number that starts with 1 and then followed by 30 zeros. Do not write it in any other form? Write another number that starts with a decimal point, continues with 20 zeros and 2 at the end. Comment on this number What is the most convenient way of writing such a number with many figures? Write it down. What is the name given to this format of writing this number
息 Nsom Promise. All rights reserved. Standard Form Mathematicians or scientist use a convenient and economical manner or way to express very small numbers or very large numbers called the Standard Form. This form was adopted and made standard to be used all over the world. a 壱″ < , a 10+ . ｍ ″ ≠ a 10 . ｍ ″ ≠ Examples: Express the following numbers in standard form 1. 4500000 2. 0.0000045 3. 20 4. 5 5. 405 000 000 Activity 1: Express the following numbers in standard form 1. 10 2. 0.000104 3. 450000000000.0 4. 0.01 5. 9
Multiplication and Division of Numbers in Standard Form To multiply (or divide) numbers in standard form, multiply (or divide) the numbers (as) separately as well as the powers of 10 (102) ( 10)( b 10) = (a b)(10 10) ﾃ10 10 = ﾃ10 10 Examples: Evaluate 1. (2 105 )(4 103) 2. 3.6105 1.2101 3. (2 103)5 4. Convert to decimals: a) 2.5 x 10-2 b) 3.40 x 10-3 c) 0.4 x 104 息 Nsom Promise. All rights reserved. Activity 1: Simplify: 1. (4 102)( 20 108) 2. (3 103) 歎 ( 30 1010) 3. 4103 2.5104 4. (7 105) 歎 ( 1.4 108) 5. 5104 2100 6. (3 100) ( 4.004 103) 7. 4 ( 25 1012) 8. 4104 2 2.5102 3
息 Nsom Promise. All rights reserved. Review Exercise 1. Write the number 1 400 in a) Standard form b) two significant figures 2. Simplify 3 1 3 2 1 4 歎4 1 2 + 1 1 6 3. Evaluate 12.78103 9 101 Express your answer in a) in standard form b) correct to 2 significant figures c) correct to 3 decimal places
息 Nsom Promise. All rights reserved. END

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Estimation, Approximation and Standard form

  • 1. 息 Nsom Promise. All rights reserved. IB Mathematics Approximation and Estimation
  • 2. Approximation and Estimation Estimation: Estimation is an Intelligent guess made about something base on some information. Approximation is guess that is nearly exact. Estimate in science help scientist to guess before doing actual investigation. Estimate gives an idea about some quantities off hand before proper findings or measurement. For example, if you were asked to pay electricity bill given that you consumed 12units and the cost of a unit being 64frs. The cost of 12 units is 768frs 770. Why would SONEL Company prefer to approximate this value t0 770frs?. What if the value was 764? Will they still approximate it to 770 or 775. Its better to be roughly right than precisely wrong(Allan Greenspan, U.S. Federal Reserve Chairman -retired) 息 Nsom Promise. All rights reserved. Introduction Activity: 1. Guess intelligently the length and width of the classroom and calculate its area 2. Estimate the amount of water you drink on daily basis. (1.5litres per day) 3. Guess the weight of your exercise book (250g) 4. Approximate the height and weight (mass) of your mate (1.7m, 60kg or 500N) 5. Estimate the weight of a football (396 to 463g) 6. Estimate the angle between the wall the floor
  • 3. Post Mathematics for Mathematics HL and SL 息 Nsom Promise. All rights reserved.
  • 4. Review: Operations with Fractions and Decimals 息 Nsom Promise. All rights reserved. Activity 1: Answers ) 3 4 ) Remark: To divide a decimal by a power of 10n, move n-places to the left. To multiply a decimal number by a power of 10n, move n-places to the right. Examples: a) 450歎 1000 b) 780 1000 Activity 1: Simplify as much as possible (Hint: Use BODMAS) ) 1 2 2 1 2 1 1 6 歎 2 1 3 ) 7 1 2 2 1 7 1 2 5 + 7 歎 3 3) 4 3 (2 1 2 ) 1 1 6 歎 2 1 3 4) 1.2 0.012 5) 9.02 100 6) 2 歎 100 7) 9.8+1.07+69 8) 5-4.667 9) 44.36 x 32 10) a) 0.3 0.2 b) 0.3 1000 b) 2 100 11) 0.2 x 1000
  • 5. 息 Nsom Promise. All rights reserved. Approximation and Estimation NB: Some numbers are too big to be rounded up or too small to be rounded down. To round up or down a decimal number (number with decimal point), We study the number of decimal places and we approximate accordingly. Examples: 1) Round 2475 to the nearest 10frs. (Ans: 2480frs ) 2) Put 2.963 to 1decimal place (Ans: 3.1) 3)Round 344frs to the nearest 5frs. Rounding Up and Down: To round up and down a number or decimal, Draw a vertical line (stroke) in front of the desired digit or unit If the digit after the line is 5 , round it up to one and add to the digit immediately before the line; the rest of the digits are considered zeros If the digit after the line is <5, round it down so that the digits before the line remain unchanged and those after the line are assumed zeros
  • 6. 息 Nsom Promise. All rights reserved. Approximation and Estimation Remark: The value of any digit depends on its position. The value of 2 in the number 1267 is 100 and in 0.324 is 100th or 2dp. Given 2768.043, what is the value of each digit? The value of 2=thousand, 3=hundred, 6=ten, 8=unit, .0=tenth (1dp), 4=hundredth (2dp), 3=thousandth (3dp) 1000 100 10 Unit 10th (thousand) (hundred) (ten) (tenth or 1dp) 100th (hundredth or 2dp) 1000th (thousandth or 3dp) (1000) Or 103 100 or 102 10 or 101 1 or 100 0.1 or 10- 1 0.01 or 10-2 0.001 or 10-3 Example: Copy and complete the table below Number 1dp 2dp Whole number or unit Neare st 10frs Nearest 100th Nearest Degree 444.525 5059.996 267.537
  • 7. 息 Nsom Promise. All rights reserved. Approximation and Estimation Answer: Check for correctness Number 1dp 2dp Whole number or unit Neare st 10frs Nearest 100th Nearest Degree 444.525 444.5 444.53 445 450 444.53 4450 5059.996 5060.0 5060.00 5060 5060 5060.00 50600 267.537 267.5 267.54 268 270 268.54 2680 Significant Figures In the number 4380, 4=first significant figure or digit, 3=second significant figure, 8=third significant figure. and 0 is not significant although 0 has a value. However, in 4308, 0 is significant. Remark: 0 between two whole numbers or digits is significant 0 at the beginning or end of a number is not significant If the digit after the required significant figure >=5, round up and add to the previous digit
  • 8. 息 Nsom Promise. All rights reserved. Approximation and Estimation Examples: Write the following numbers to 2 significant a) 0403670= b) 0.052407= c) 34.08945= d) 040567 (to 3 s.f.) Number 1dp 2dp 1sf 2sf 3sf Nearest whole number 42.546 0.9974 1.9995 295.6891
  • 9. 息 Nsom Promise. All rights reserved. Approximation and Estimation Examples: Write the following numbers to 2 significant a) 0403670=40 b) 0.052407=0.052 c) 34.08945=34 d) 040567=34.1 to 3 sf Number 1dp 2dp 1sf 2sf 3sf Nearest whole number 42.546 42.5 42.55 4 40 43.547 43 0.9974 1.0 1.00 1.0 1.00 0.997 1 1.9995 2.0 2.00 2 2.0 2.00 2 295.6891 295.7 295.69 3 30 296 296
  • 10. 息 Nsom Promise. All rights reserved. Introduction Activity: Write the biggest amount of money you dream to ever have in a life time Write a number that starts with 1 and then followed by 30 zeros. Do not write it in any other form? Write another number that starts with a decimal point, continues with 20 zeros and 2 at the end. Comment on this number What is the most convenient way of writing such a number with many figures? Write it down. What is the name given to this format of writing this number
  • 11. 息 Nsom Promise. All rights reserved. Standard Form Mathematicians or scientist use a convenient and economical manner or way to express very small numbers or very large numbers called the Standard Form. This form was adopted and made standard to be used all over the world. a 壱″ < , a 10+ . ｍ ″ ≠ a 10 . ｍ ″ ≠ Examples: Express the following numbers in standard form 1. 4500000 2. 0.0000045 3. 20 4. 5 5. 405 000 000 Activity 1: Express the following numbers in standard form 1. 10 2. 0.000104 3. 450000000000.0 4. 0.01 5. 9
  • 12. Multiplication and Division of Numbers in Standard Form To multiply (or divide) numbers in standard form, multiply (or divide) the numbers (as) separately as well as the powers of 10 (102) ( 10)( b 10) = (a b)(10 10) ﾃ10 10 = ﾃ10 10 Examples: Evaluate 1. (2 105 )(4 103) 2. 3.6105 1.2101 3. (2 103)5 4. Convert to decimals: a) 2.5 x 10-2 b) 3.40 x 10-3 c) 0.4 x 104 息 Nsom Promise. All rights reserved. Activity 1: Simplify: 1. (4 102)( 20 108) 2. (3 103) 歎 ( 30 1010) 3. 4103 2.5104 4. (7 105) 歎 ( 1.4 108) 5. 5104 2100 6. (3 100) ( 4.004 103) 7. 4 ( 25 1012) 8. 4104 2 2.5102 3
  • 13. 息 Nsom Promise. All rights reserved. Review Exercise 1. Write the number 1 400 in a) Standard form b) two significant figures 2. Simplify 3 1 3 2 1 4 歎4 1 2 + 1 1 6 3. Evaluate 12.78103 9 101 Express your answer in a) in standard form b) correct to 2 significant figures c) correct to 3 decimal places
  • 14. 息 Nsom Promise. All rights reserved. END

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