{"id":82,"date":"2020-12-14T20:04:12","date_gmt":"2020-12-15T01:04:12","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/?post_type=chapter&p=82"},"modified":"2022-01-09T19:34:55","modified_gmt":"2022-01-10T00:34:55","slug":"numerical-calculations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/chapter\/numerical-calculations\/","title":{"raw":"Numerical Calculations","rendered":"Numerical Calculations"},"content":{"raw":"Engineering means that you will be calculating a lot of numbers and having to report those numbers.\u00a0 Your answers to any problem must be recorded with an accuracy that you can justify.\u00a0 You will use significant figures to show that accuracy.\r\n

Dimensional Homogeneity<\/h1>\r\nWhen you are writing an equation, all the terms in the equation must be expressed in the same units. We say that the terms of any equations must be \"dimensionally homogeneous\".\u00a0 The units must be consistent.\u00a0 This is often a great way to check if you are using the correct equation and the final answer to your solved problem.\u00a0 \u00a0Let us start of with the standard equation for straight line motion on a particle that is being accelerated.\u00a0 d = vo<\/sub>t + 1<\/sup>\/2<\/sub> at2<\/sup> where d is the displacement in metres,\u00a0 vo <\/sub>is the initial velocity in metres per second, t is the time in seconds and a is the acceleration in metres per second squared.\u00a0 Note that vo<\/sub>t units are (m\/s)(s) which is m.\u00a0 Note that the units for 1<\/sup>\/2<\/sub> at2\u00a0<\/sup> are (m\/s2<\/sup>)(s2<\/sup>) which is m.\r\n

Significant Figures and Engineering Notation<\/h1>\r\nThe number of significant figures contained in a written number is the way that the engineer shows the reader how accurate the number is.\u00a0 The number 4321 contains four significant figures.\u00a0 It shows that the last digit is significant that it could have been measured.\u00a0 The number might have been 4322 but it was 4321 as the \"1\" was measured.\u00a0 The problem is when the number is written as 4320.\u00a0 Was that last zero measured or not?\u00a0 Does it have three or four significant figures?\u00a0 The rules state that engineers are to be pessimistic, so if written as 4320 we state that it has three significant figures.\u00a0 It is possible though that the last zero was measured.\u00a0 Using engineering notation makes it clear.\u00a0 4.320(103<\/sup>) has four significant figures while 4.32(103<\/sup>) has three significant figures. The last zero in 4320 is important to tell us how big the number is, but it not significant.\u00a0 Often in engineering, words have a very specific meaning.\u00a0 Sometimes called jargon, but it the way that we communicate with each other so it is important to be aware of these rules and to follow them.\r\n\r\nIn this text we will generally give values to three significant figures.\u00a0 This is conveying an accuracy of about 1% and most of the data in engineering can be reliably measured to plus or minus one percent.\r\n\r\nIf the number is small, the zeros are the start are not significant.\u00a0 0.007 89 has three significant figures and in engineering notation it is written as 7.89(10-3<\/sup>).\r\n

Rounding Off<\/h1>\r\nRounding off the final value of a calculation must be done so that the final accuracy of the results will be the same as the original data.\u00a0 In general, any number ending in a number greater than five is rounded up and a number less than five is not rounded up.\u00a0 When it is five, it depends.\u00a0 If the digit before the 5 is an even number, then this digit is not rounded up.\u00a0 If the digit preceding the 5 is an odd number, then it is rounded up. Here are some examples.\r\n
    \r\n \t
  • 3.4567 rounded to three significant figures is 3.46.<\/li>\r\n \t
  • 2.341 rounded to three significant figures is 2.34.<\/li>\r\n \t
  • 67.25 rounded to three significant figures is 67.3.<\/li>\r\n \t
  • 0.2375 rounded to three significant figures is 0.238.<\/li>\r\n \t
  • 0.4555 rounded to three significant figures is 0.456.<\/li>\r\n<\/ul>\r\n

    Calculations or \"only round off at the end\"<\/h1>\r\nWhen a long sequence of calculations is to be done, it is best to store the intermediate results in the calculator or computer and only round off at the end when expressing the final result.\u00a0 Maintaining precision throughout is important. In this text we will generally round off the answers to three significant figures, even though the first data might be in given to only one significant figure.\u00a0 This is conveying an accuracy of about 1% and most of the data in engineering can be reliably measured to plus or minus one percent.\r\n\r\n ","rendered":"

    Engineering means that you will be calculating a lot of numbers and having to report those numbers.\u00a0 Your answers to any problem must be recorded with an accuracy that you can justify.\u00a0 You will use significant figures to show that accuracy.<\/p>\n

    Dimensional Homogeneity<\/h1>\n

    When you are writing an equation, all the terms in the equation must be expressed in the same units. We say that the terms of any equations must be “dimensionally homogeneous”.\u00a0 The units must be consistent.\u00a0 This is often a great way to check if you are using the correct equation and the final answer to your solved problem.\u00a0 \u00a0Let us start of with the standard equation for straight line motion on a particle that is being accelerated.\u00a0 d = vo<\/sub>t + 1<\/sup>\/2<\/sub> at2<\/sup> where d is the displacement in metres,\u00a0 vo <\/sub>is the initial velocity in metres per second, t is the time in seconds and a is the acceleration in metres per second squared.\u00a0 Note that vo<\/sub>t units are (m\/s)(s) which is m.\u00a0 Note that the units for 1<\/sup>\/2<\/sub> at2\u00a0<\/sup> are (m\/s2<\/sup>)(s2<\/sup>) which is m.<\/p>\n

    Significant Figures and Engineering Notation<\/h1>\n

    The number of significant figures contained in a written number is the way that the engineer shows the reader how accurate the number is.\u00a0 The number 4321 contains four significant figures.\u00a0 It shows that the last digit is significant that it could have been measured.\u00a0 The number might have been 4322 but it was 4321 as the “1” was measured.\u00a0 The problem is when the number is written as 4320.\u00a0 Was that last zero measured or not?\u00a0 Does it have three or four significant figures?\u00a0 The rules state that engineers are to be pessimistic, so if written as 4320 we state that it has three significant figures.\u00a0 It is possible though that the last zero was measured.\u00a0 Using engineering notation makes it clear.\u00a0 4.320(103<\/sup>) has four significant figures while 4.32(103<\/sup>) has three significant figures. The last zero in 4320 is important to tell us how big the number is, but it not significant.\u00a0 Often in engineering, words have a very specific meaning.\u00a0 Sometimes called jargon, but it the way that we communicate with each other so it is important to be aware of these rules and to follow them.<\/p>\n

    In this text we will generally give values to three significant figures.\u00a0 This is conveying an accuracy of about 1% and most of the data in engineering can be reliably measured to plus or minus one percent.<\/p>\n

    If the number is small, the zeros are the start are not significant.\u00a0 0.007 89 has three significant figures and in engineering notation it is written as 7.89(10-3<\/sup>).<\/p>\n

    Rounding Off<\/h1>\n

    Rounding off the final value of a calculation must be done so that the final accuracy of the results will be the same as the original data.\u00a0 In general, any number ending in a number greater than five is rounded up and a number less than five is not rounded up.\u00a0 When it is five, it depends.\u00a0 If the digit before the 5 is an even number, then this digit is not rounded up.\u00a0 If the digit preceding the 5 is an odd number, then it is rounded up. Here are some examples.<\/p>\n

      \n
    • 3.4567 rounded to three significant figures is 3.46.<\/li>\n
    • 2.341 rounded to three significant figures is 2.34.<\/li>\n
    • 67.25 rounded to three significant figures is 67.3.<\/li>\n
    • 0.2375 rounded to three significant figures is 0.238.<\/li>\n
    • 0.4555 rounded to three significant figures is 0.456.<\/li>\n<\/ul>\n

      Calculations or “only round off at the end”<\/h1>\n

      When a long sequence of calculations is to be done, it is best to store the intermediate results in the calculator or computer and only round off at the end when expressing the final result.\u00a0 Maintaining precision throughout is important. In this text we will generally round off the answers to three significant figures, even though the first data might be in given to only one significant figure.\u00a0 This is conveying an accuracy of about 1% and most of the data in engineering can be reliably measured to plus or minus one percent.<\/p>\n

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