6 Numerical Calculations
Engineering means that you will be calculating a lot of numbers and having to report those numbers. Your answers to any problem must be recorded with an accuracy that you can justify. You will use significant figures to show that accuracy.
Dimensional Homogeneity
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”. The units must be consistent. This is often a great way to check if you are using the correct equation and the final answer to your solved problem. Let us start of with the standard equation for straight line motion on a particle that is being accelerated. d = vot + 1/2 at2 where d is the displacement in metres, vo is the initial velocity in metres per second, t is the time in seconds and a is the acceleration in metres per second squared. Note that vot units are (m/s)(s) which is m. Note that the units for 1/2 at2 are (m/s2)(s2) which is m.
Significant Figures and Engineering Notation
The number of significant figures contained in a written number is the way that the engineer shows the reader how accurate the number is. The number 4321 contains four significant figures. It shows that the last digit is significant that it could have been measured. The number might have been 4322 but it was 4321 as the “1” was measured. The problem is when the number is written as 4320. Was that last zero measured or not? Does it have three or four significant figures? The rules state that engineers are to be pessimistic, so if written as 4320 we state that it has three significant figures. It is possible though that the last zero was measured. Using engineering notation makes it clear. 4.320(103) has four significant figures while 4.32(103) has three significant figures. The last zero in 4320 is important to tell us how big the number is, but it not significant. Often in engineering, words have a very specific meaning. 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.
In this text we will generally give values to three significant figures. 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.
If the number is small, the zeros are the start are not significant. 0.007 89 has three significant figures and in engineering notation it is written as 7.89(10-3).
Rounding Off
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. In general, any number ending in a number greater than five is rounded up and a number less than five is not rounded up. When it is five, it depends. If the digit before the 5 is an even number, then this digit is not rounded up. If the digit preceding the 5 is an odd number, then it is rounded up. Here are some examples.
- 3.4567 rounded to three significant figures is 3.46.
- 2.341 rounded to three significant figures is 2.34.
- 67.25 rounded to three significant figures is 67.3.
- 0.2375 rounded to three significant figures is 0.238.
- 0.4555 rounded to three significant figures is 0.456.
Calculations or “only round off at the end”
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. 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. 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.