Numerical Computation 1 Numerical Computation Click on the computer image at the bottom right for a direct web link to an interesting Wikipedia Math Site. Prepared by: Richard Mitchell Humber College

1.1 - THE NUMBER TYPES 1.1-The Number Types

1.1-PLACE VALUE Whole Numbers Decimal Numbers 213 - Nearest Whole Number 51 700 - Nearest Hundreds 710 010 - Nearest Tens 52 - Nearest Whole Number 517 000 - Nearest Thousands Decimal Numbers 0.105 - 3 DP’s/Nearest Thousandths 0.21 - 2 DP’s/Nearest Hundredths 11.051 - 3 DP’s/Nearest Thousandths 20.005 - 3 DP’s/Nearest Thousandths 7.00 - 2 DP’s/Nearest Hundredths 1.1-The Number Types

1.1-EXACT NUMBERS Counted Quantities have no uncertainty. Whole Numbers and Fractions have no uncertainty. (Only when not measured and not in Decimal Form) Defined Numbers have no uncertainty. 4 wheels (exactly counted) 501 roses (exactly counted) 17 letters (exactly counted) 2 010 washers (exactly counted) 81 700 cars (exactly counted) 24 hours in a day (exactly counted) 0 1 2 3 10 516 21 019 20 000 1.1-The Number Types 1 inch = 25.4 mm (exactly measured by definition)

1.1-APPROXIMATE NUMBERS Measured Quantities have some degree of uncertainty. (Last SD and/or Decimal Place is often estimated visually on a scale or meter). Decimal Numbers have some degree of uncertainty. (Both measured and non-measured) Estimates have some degree of uncertainty. Decimal Form of Fractions and Irrational Numbers. 217 m 1.75 cm 13.0 inches 1 000 m/s 50 112 Hz 2.02 lbs 0.00051 km 0.0020 mph 0.217 2 157.0 cm 3.00 12.05 ft 1.1-The Number Types approximately 3 500 people about 2 710 cars were built 2/3 (exact form) equals 0.6667 (approximate decimal form) ∏ (exact form) equals 3.1428571 (approximate decimal form) (exact form) equals 1.732 (approximate decimal form)

1.1-SIGNIFICANT DIGITS 0.152 - 3 SD’s 0.0177 - 3 SD’s Whole Numbers 217 - 3 SD’s 274 Ō00 - 4 SD’s 210 015 - 6 SD’s 31 Ō00 - 3 SD’s 1 000 201 - 7 SD’s 8 003 - 4 SD’s 24 900 - 3 SD’s 5 101 110 - 6 SD’s 310 400 - 4 SD’s 4 Ō00 - 2 SD’s Decimal Numbers 0.152 - 3 SD’s 0.0177 - 3 SD’s 0.0005 - 1 SD 0.000583 - 3 SD’s 11.25 - 4 SD’s 2.008 - 4 SD’s 2.0500 - 5 SD’s 15.05 - 4 SD’s 1.1-The Number Types

1.1-SIGNIFICANT DIGITS SUMMARY 1.1-The Number Types

1.1-ACCURACY vs PRECISION Accuracy is Determined by Significant Digits 1.255 - 4 SD’s of Accuracy 23 800 - 3 SD’s of Accuracy 0.0050 - 2 SD’s of Accuracy 125 - 3 SD’s of Accuracy 310.03 - 5 SD’s of Accuracy 0.002 - 1 SD of Accuracy Precision is Determined by Place Value 1.255 - 3 DP’s of Precision (Nearest Thousandths) 23 800 - Nearest Hundred’s of Precision 0.0050 - 4 DP’s of Precision (Nearest Ten Thousandths) 125 - Nearest Whole Number of Precision 310.03 - 2 DP’s of Precision (Nearest Hundredths) 0.002 - 3 DP’s of Precision (Nearest Thousandths) 1.1-The Number Types

1.1-ROUNDING SUMMARY Example 8 Example 9 1.1-The Number Types

1.2 - NUMERICAL OPERATIONS

1.2-RULE of PRECISION 1 Decimal Place (Approximate) 3 Decimal Places (Approximate) USE: The Rule of PRECISION (1 Decimal Place) When adding or subtracting approximate numbers, keep as many decimal places in your answer as the number having the fewest decimal places in the question. Use the least precise place value if there are no decimal places in the question. 1.2-Numerical Operations

1.2-EXAMPLE 13 (Approximate) Nearest Hundreds (Exact) Nearest Ones USE: The Rule of PRECISION (Nearest Hundreds Place) When adding or subtracting approximate numbers, keep as many decimal places in your answer as the number having the fewest decimal places in the question. Use the least precise place value if there are no decimal places in the question. 1.2-Numerical Operations When using exact numbers, treat them as if they had more Significant Digits than any of the approximate numbers in the question.

1.2-RULE of ACCURACY 5 Significant Digits (Approximate) 3 Significant Digits (Approximate) USE: The Rule of ACCURACY (3 Significant Digits) When multiplying or dividing approximate numbers, keep as many significant digits in your answer as the number having the fewest significant digits in the question. 1.2-Numerical Operations

1.2-RULE of ACCURACY 3 Significant Digits (Approximate) Do not count these as Significant Digits. USE: The Rule of ACCURACY (3 Significant Digits) When multiplying or dividing approximate numbers, keep as many significant digits in your answer as the number having the fewest significant digits in the question. 1.2-Numerical Operations When using exact numbers, treat them as if they had more Significant Digits than any of the approximate numbers in the question.

1.2-RULE of ACCURACY 4 Significant Digits (Approximate) 3 Significant Digits (Approximate) USE: The Rule of ACCURACY (3 Significant Digits) When multiplying or dividing approximate numbers, keep as many significant digits in your answer as the number having the fewest significant digits in the question. 1.2-Numerical Operations

1.2-RULE of ACCURACY 4 Significant Digits (Approximate) Do not count these as Significant Digits. USE: The Rule of ACCURACY (3 Significant Digits) When multiplying or dividing approximate numbers, keep as many significant digits in your answer as the number having the fewest significant digits in the question. 1.2-Numerical Operations When using exact numbers, treat them as if they had more Significant Digits than any of the approximate numbers in the question.

1.2-POWERS Powers 1.2-Numerical Operations

1.2-ROOTS Roots 1.2-Numerical Operations

1.3 - ORDER of Operations 1.3-Order of Operations

1.3-EXAMPLE extra 1.3-Order of Operations USE: The Rule of EXACT NUMBERS. Exact numbers produce ‘exact answers’ and do not need to be rounded off.

1.3-EXAMPLE 37 1.3-Order of Operations USE: The Rule of ACCURACY (3 Significant Digits). Be sure to keep all of the digits used in each calculation and only round off at the end.

1.4 - Scientific and Engineering Notation

1.4-EXAMPLES Large Numbers 346 = 3.46 x 102 (3 SD’s) 2 700 = 2.7 x 103 (2 SD’s) 5 101 000 = 5.101 x 106 (4 SD’s) 31Ō 000 = 3.10 x 105 (3 SD’s) Small Numbers 0.0000931 = 9.31 x 10-5 (3 SD’s) 0.008300 = 8.300 x 10-3 (4 SD’s) 0.00000950 = 9.50 x 10-6 (3 SD’s) 1.4-Scientific and Engineering Notation

1.4-CALCULATOR SKILLS Normal Mode (30 000) x (215 000) = 6 450 000 000 (30 000) + (215 000) = 245 000 (500 000) + (300 000) = Exponential Mode (EXP or EE or key) (3.00 x 104) x (2.15 x 105) = 6 450 000 000 (3.00 x 104) + (2.15 x 105) = 245 000 Scientific Mode (Mode or FSE key) (3.00 x 104) x (2.15 x 105) = 6.45 x 109 (3.00 x 104) + (2.15 x 105) = 2.45 x 105 USE: The Rule of EXACT NUMBERS when using exact numbers. 1.4-Scientific and Engineering Notation USE: The Rule of ACCURACY when using approximate numbers (Significant Digits).

1.4-EXAMPLES RULE: Digits from the decimal part of the number are grouped into sets of three. RULE: For numbers less than 1, separate the digits following the decimal point into groups of three. 1.4-Scientific and Engineering Notation

1.4-SUMMARY Scientific Notation Engineering Notation 1.4-Scientific and Engineering Notation

1.5 - Units of Measurement 1.5-Units of Measurement

1.5-EXAMPLE 47 Conversion Factor: 1 ft. = 0.3048 m OPTIONAL FORMULA Multiply by the conversion factor that will cancel the units you wish to eliminate. 654.5 ft. x (0.3048 m / 1 ft.) Conversion Factor: 1 ft. = 0.3048 m 1.5-Units of Measurement

1.5-EXAMPLE 48 Conversion Factor: 1 ha = 2.471 acres 1.5-Units of Measurement

1.5-EXAMPLE 51 Conversion Factor: 1 km = 1000 m 1.5-Units of Measurement

1.5-EXAMPLE 52 Conversion Factor: 1 N = 1.0 x 105 dynes 1.5-Units of Measurement

1.5-EXAMPLE 53 Conversion Factor: 1 cu.ft. = 7.481 gal.(U.S.) 1.5-Units of Measurement

1.6 - Substituting into Equations and Formulas

1.6-EXAMPLE 57 1.6-Substituting into Equations and Formulas

1.6-EXAMPLE 58 Convert all units to pounds and inches. 1.6-Substituting into Equations and Formulas

1.7 - Percentage 1.7-Percentage

1.7-FRACTIONS-DECIMALS-PERCENTS 1.7-Percentage

1.7-EXAMPLES 1.7-Percentage

1.7-STRATEGY A 1.7-Percentage

1.7-STRATEGY B RATE (%) (Change) PART (Change) FORMULA BASE (Original) 1.7-Percentage

1.7-EXAMPLE 59 PART (Change) 1.7-Percentage

1.7-EXAMPLE 60 PART (Change) 1.7-Percentage

1.7-EXAMPLE 61 BASE (Original) 1.7-Percentage

1.7-EXAMPLE 62 BASE (Original) 1.7-Percentage

1.7-EXAMPLE 63 RATE (%) 1.7-Percentage

1.7-EXAMPLE 64 RATE (%) 1.7-Percentage

1.7-STRATEGY Type I: Percent Change Type II: Percent Efficiency Type III: Percent Error 1.7-Percentage Type IV: Percent Concentration

1.7-EXAMPLE 65 Type I: Percent Change 1.7-Percentage

1.7-EXAMPLE 66 Type I: Percent Change 1.7-Percentage

1.7-EXAMPLE 67 Type II: Percent Efficiency 1.7-Percentage

1.7-EXAMPLE 68 Type III: Percent Error 1.7-Percentage

1.7-EXAMPLE 69 Type IV: Percent Concentration 1.7-Percentage

Copyright