Approximations – Decimal Places & Significant Figures Mathematics Approximations – Decimal Places & Significant Figures

The aim of this powerpoint is to teach you techniques for approximating decimal numbers as well as very large or very small numbers using significant figures. EITHER Take notes as you go along, include some examples and write down any questions and your answers (which you can mark as you go along) OR At the end of the powerpoint, printout the notes called Calc2b

Decimal Places To approximate means to give a rough estimate, not an exact value. The symbol ‘≈’ means approximately equal to. Decimal places are the number of digits you must have in your final answer after the decimal point. Truncate means to cut off at a particular point (everything on the right become zeros or is ignored/lost).

Approximating to Decimal Places Imagine the decimal number you have been given truncated (i.e. cut off) after the specified number of decimal places. Now look at the next number to this on a number line (i.e. the last digit of your truncated value would be 1 bigger). Which of these two values is your original value closest to? If the digit on the right (called the decider) of the required number of decimal places is 5 or more, round UP, otherwise your truncated value is your answer.

Example 1 3.67083 approximated to 2 d.p. is 3.67 Give 3.67083 to 2 d.p. Truncated: 3.67 Next value: 3.68 Which is 3.67083 closest to? This ‘0’ is less than 5 so stick with the truncated value 3.67083 approximated to 2 d.p. is 3.67

Example 2 24.7728 approximated to 1 d.p. is 24.8 Give 24.7728 to 1 d.p. Truncated: 24.7 Next value: 24.8 Which is 24.7728 closest to? This ‘7’ is more than 5 so round UP to the next value 24.7728 approximated to 1 d.p. is 24.8

Practice Round each value to the number of d.p. quoted in brackets: 1) 4.873 (1 d.p.) 2) 0.7942 (2 d.p.) 3) 2.3549 (1 d.p.) 4) 5.12741 (3 d.p.) 5) 7.0663 (1 d.p.) 6) 4.4971 (2 d.p.) Work out YOUR answers FIRST, then click on to the next slide to check them.

Answers 1) 4.873 (1 d.p.) 2) 0.7942 (2 d.p.) 3) 2.3549 (1 d.p.) 4) 5.12741 (3 d.p.) 5) 7.0663 (1 d.p.) 6) 4.4971 (2 d.p.)  4.9  0.79  2.4  5.127  7.1  4.50

What next? If you haven’t made any notes or copied any examples, questions and answers out during the first half of this presentation, print out the notes called Calc2b. Read through the first page and make sure you answer any questions. Work through the MyMaths lesson (and then the online homework) called Rounding Decimals found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=accuracy/roundingDecimal&taskID=1004 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=accuracy/roundingDecimalOH&taskID=1004 Work through the MyMaths lesson (and then the online homework) called Decimal Places found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=accuracy/decimalplaces&taskID=1001 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=accuracy/decimalplacesOH&taskID=1001 Save and complete the worksheet called: DP-S1.xlsx If you feel confident enough, continue with the rest of this powerpoint

Significant Figures Start counting significant figures from the first NON-ZERO digit. 360.54 has 5 significant figures… The 3 is the 1st significant figure The 6 is the 2nd significant figure The 0 is the 3rd significant figure The 5 is the 4th significant figure The 4 is the 5th significant figure 0.000401 has 3 significant figures… The 4 is the 1st significant figure The 0 (after the 4) is the 2nd significant figure The 1 is the 3rd significant figure

Significant Figures Significant figures tell you the maximum number of non-zero digits you should have in your final answer. However, please remember that when carrying out a calculation or solving a problem, you should ONLY round your FINAL answer to a set number of s.f. (or d.p.). Any workings in-between should NOT be rounded at all.

Approximating to Significant Figures To approximate to significant figures you may need to round to the nearest unit, ten or hundred etc. OR you may need to approximate to a number of decimal places. Find the column value the number of significant figures refers to and round to that value or approximate to that number of decimal places.

Examples Give 367.54 and 0.00401 each to 2 s.f. In 367.54, the 6 is the 2nd s.f. It is in the ‘tens’ column so round 367.54 to the nearest 10 367.54 is between 360 and 370 but closer to 370 In 0.000401, the 0 after the 4 is the 2nd s.f. It is in the 5 d.p. position, so round 0.000401 to 5 d.p. 0.000401  0.00040

Practice Round each value to the number of s.f. quoted in brackets: 1) 4873 (1 s.f.) 2) 0.7942 (2 s.f.) 3) 235.49 (1 s.f.) 4) 5.12741 (3 s.f.) 5) 75663 (1 s.f.) 6) 497.135 (2 s.f.) Work out YOUR answers FIRST, then click on to the next slide to check them.

Answers 1) 4873 (1 s.f.) 2) 0.7942 (2 s.f.) 3) 235.49 (1 s.f.) 4) 5.12741 (3 s.f.) 5) 75,663 (1 s.f.) 6) 497.135 (2 s.f.) to nearest 1000  5000 to 2 d.p.  0.79 to nearest 100  200 to 2 d.p.  5.13 to nearest 10,000  80,000 to nearest 10  500

What next? If you haven’t made any notes or copied any examples, questions and answers out during the second half of this presentation, print out the notes called Calc2b. Read through page 2 of them and make sure you answer any questions. Work through the MyMaths lesson (and then the online homework) called Significant Figures found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=accuracy/significantfigures&taskID=1005 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=accuracy/significantfiguresOH&taskID=1005 Save and complete the worksheet called: SF-S1.xlsx Now move on to the Calc3a-MentalA powerpoint