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Introduction: Numerical calculations

It is important to consider the following aspects when doing calculations and reporting the answers to engineering problems.

Dimensional Homogeneity

The logic implies that only physical quantities with identical dimensions and units can be added to each other or subtracted from one another. For example it makes sense to add masses of objects, but it is nonsense to add the mass to velocity. Moreover, terms on both sides of an equation should have the same dimension and unit. Multiplication and division of physical quantities with different dimensions and units are logically allowed. It is obvious that all units in the calculations comply with a system of the units chosen in advance.

Rounding Numbers

Rounding a number means replacing the number with a different one being approximately equal to the original number. Rounding can have two main proposes:

  1. Simplifying the number to make it easier to report. For example: 400 is simpler to report than 398, or the number \pi is usually approximated as 3.14.
  2. Avoiding misleading accuracy when reporting computed or measured quantities. For example, it is not appropriate to report the mass of a person as 61.23453409\rm kg when measuring with a body weight scale (bathroom scale).

Rounding a number needs a rule or an instruction. It is usually told how to round a number, or it is understood from the context (e.g. the accuracy of the device of the measurements). There are two ways for declaring an instruction of rounding a number.

  1. Rounding to a named place. For example rounding a number to the nearest tenth, or rounding a number to a specific number of significant digits (see the next section).
  2. Rounding to a specific number of places. For example rounding a number to four decimal places.

Once the place (target place) that a number should be rounded to is known, the digit one place to the right of the target place is truncated. Then, if the place (right to the target place) is 5 or greater, the target digit is rounded up by 1. Otherwise, the target digit is left as it is. The next step is to replace any digits to the right of the target digits with zeros if they are to the left of the decimal point and/or to delete the digits if they are to the right of the decimal point. Calculation tip: use a vertical line, |, to separate the target place from the rest of the digits. The target place falls just before the vertical line.

Example 1: Consider the number 9647:

  • 9647 rounded to the nearest ten is 9650 (hint: 964\vert 7).
  • 9647 rounded to the nearest hundred is 9600 (hint: 96\vert 47).
  • 9647 rounded to the nearest thousand 10000 (hint: 9\vert 647).

Example 2: Consider the number 2.0623:

  • 2.0623 rounded to the nearest unit is 2 (hint: 2\vert .0623).
  • 2.0623 rounded to the nearest tenth is 2.1 (hint: 2.0\vert 623).
  • 2.0623 rounded to the nearest hundredth is 2.06 (hint: 2.06\vert 23).
  • 2.0623 rounded to the nearest thousandth 2.062 (hint: 2.062\vert 3).

Example 3: Consider the number \pi =3.14159265\dots :

  • \pi rounded to to five decimal places is 3.14159 (hint: 3.14159\vert 265\dots).
  • \pi rounded to to four decimal places is 3.1416 (hint: 3.1415\vert 9265\dots).
  • \pi rounded to to three decimal places is 3.142 (hint: 3.141\vert 59265\dots).

Example 4: Consider the numbers 4.296 and 5.98:

  • 4.296 rounded to the nearest hundredth (or two decimal place) is 4.30 (hint: 4.29\vert 6).
  • 5.98 rounded to the nearest tenth (or one decimal place) is 6.0 (hint: 5.9\vert 8).

In the last example, zeros should be included to indicate the decimal place the numbers are rounded to. It is therefore wrong to write 4.3 and 6 respectively for the above cases.

Significant figures

Significant figures or significant digits of a number are digits that show us the accuracy of a measurement or calculations based on the measurement. Each significant figure adds accuracy to the reported measurement. Since significant figures are to demonstrate the accuracy of measurements, the number of significant figures in a number directly depends on the precision of the measuring instrument. The significant figures of a reported measurement contains the digits that we are reasonable sure of. Usually, the last digit has some uncertainty, but it is reasonable to be reported.

For example, the length of an object measured by a ruler with the accuracy of tenth of centimetre i.e. millimetre can be reported as 2.34\rm cm. The numbers 2 and 3 are certain because we can observe the marks on the ruler, but, 4 has some uncertainty because we can see the edge of the object ends somewhere between two millimetre marks on the ruler. However, we estimate the last digit on our engineering judgment. Obviously, it does not make sense to report a thousandth place for this measurement (e.g. 2.347), because even the hundredth place is itself uncertain.

As noted in the example, the significant figures include the certain digits and the first uncertain digits out of a measurement. There are some rules to make a digit significant or non-significant. Non-significant digits are placeholder zeros. A placeholder zero is either a trailing zero in a non-decimal number, or a leading zero at the beginning of a decimal number less than 1. Here, are some examples before formally declaring the rules of the significant figures:

Example 1: These measurements have four significant figures (three certain digits and one uncertain): 2351, 2002, 21.26, 0.2314, 0.3006

Example 2: These measurements have only one significant figure (one certain digit): 1000, 0.000001. The zeros are just placeholders determining the order of the numbers. The measurements can be written as 1\times 10^3 and 1\times 10^{-6}.

Example 3: These measurements have three significant digits (two certain digits and one uncertain): 0.0000361, 0.0000540, 1.01

Example 4: These measurements have four significant digits (three certain digits and one uncertain): 100.0, 541.0, 6520. (note the decimal point after zero; zero is not just a place holder in this case).

Remark: a measurement reported as 100 has only one significant figure, but a measurement reported as 100.0 has four significant figures. The zero after the decimal point in 100.0 tells us that the measurement was accurate to the tenths place; it happened to be zero in this case.

Remark: the significant digits of a number with trailing zeros may sometimes be ambiguous. For example, if a measurement is reported as 1000, it might have been 999.9 or 1000.4 rounded up or down to 1000. Therefore, 1000 has four significant digits. Another example is when 1000 is accurate to the tens place and now the tens place happens to be zero. In this case 1000 has four significant digits. To remove this ambiguity, 1000 is written as 1000. with a decimal point added to indicate four significant figures and accuracy up to the tens place. Another notation for this purpose is to underline the trailing zero before which the zeros are significant (including the underlined zero). for example, 100\underline 0 has four significant digits and 10\underline 00 has three significant digits.

Example 5: These examples shows how to indicate the accuracy to a certain place:

7 is accurate to the units place (one significant figure).
1007 is accurate to the units place (four significant figures).
770. is accurate to the units place (note the decimal point).
10 is accurate to the tens place (one significant figure).
770 is accurate to the tens place (two significant figures).
700 is accurate to the hundreds place (one significant figure).
1000.0 is accurate to the tenths place (five significant figures).
450.0 is accurate to the tenths place (four significant figures).
3.14159 is accurate to the hundred-thousandths place.
0.00023 is accurate to the hundred-thousandths place.
0.000230 is accurate to the millionths place (note the extra zero).

The rules for significant figures are:

  • All nonzero digits are significant.
  • All zeroes embedded between significant digits are significant.
  • Leading zeros are NOT significant.
  • Trailing zeros of a non-decimal number are NOT significant.

Now that the concept of the significant figures is clear, we can round numbers to specific number of significant digits. The method is the same as discussed for rounding a number to a named place.

Example 1: 356892 rounded to four significant digits is 356900.

Example 2: 0.08274 rounded to two significant digits is 0.083.

Example 3: 120.82 rounded to four significant digits is 120.8.

Example 4: 100.82 rounded to two significant digits is 10. (note the decimal point after zero).

Reporting the results

The result of a calculation should be rounded as instructed and reported along with its unit. Example:

  • Wrong: The reaction force is 2.0.
  • Right: The reaction force is 2.0\rm N.

Recommendations:

Use a decimal point or a decimal point and zeros to indicate the accuracy whenever is applicable. For example 2.00\rm m, 100. \rm N.

Use engineering notation to report numbers having leading or trailing zeros. For example: write 1000 as 1\times 10^3, or 0.0045 as 0.45\times 10^{-4}.