1.1. Introduction
This chapter provides an overview of some mathematical notation and methods that are essential to engineering. All students should read through it, and many will be able to confirm that they have covered it all in previous studies. It is essential that any students who are not familiar with this material should study it further because, if it is not fully understood, many section of this book and many other aspects of a degree in engineering will be impossible to understand.
The main discussion uses metre–kilogramme–second (MKS) units; but the use of Imperial (US customary), and centimetre–gramme–second (CGS) units is also described. The purpose of this is not that students in Europe should read one section, and students in the United States should read the other. When searching for values for material properties on the Internet, they may be found in any units. All students should be fully familiar with all these types of units, and able to recognize them, and convert between them. Unit conversion macros are found easily online. The important concept is to recognize which system of units the data is in, and make sure that all data in an equation or calculation is in the same system.
1.2. Symbols
This book is about the properties of construction materials. Where possible, these properties are measured quantitatively; this means that values are assigned to them. For example, if a large block of concrete is considered which has one side 10-m long, this may be represented by equation
(1.1):
= 10m
(1.1)
where L is the variable we are using for the length of that side.
If the length of the block in other directions is considered and these are 8 and 12 m, this may be represented by equation
(1.2):
1 = 10m L2 = 8m L3 = 12m
(1.2)
To calculate the mass of the block, the density must be known. This could be given by the equation
(1.3):
= 2400 kg/m3
(1.3)
where ρ is the Greek letter rho that is often used as a variable for density. Greek letters are used because there are insufficient letters in the English alphabet (correctly called the Roman alphabet) for the different properties that are commonly measured.
The Greek letters that are commonly used are in
Table 1.1.
Table 1.1
Greek Letters in Common Use in Engineering
| Lower Case | Upper Case | Name |
1.3. Scientific notation
The mass of the block is given by equation
(1.4):
= L1 × L2×L3 × ρ = 2,304,000kg
(1.4)
The result has been given in kilogrammes. It may be seen that the number is large, and not easy to visualize or use. In order to make the number easier to use, it is expressed commonly in scientific notation as 2.304 × 106 kg.
It is important to express numbers correctly in scientific notation. The 2.304 should normally be between 1 and 10. To express this number as 0.2304 × 107 or 23.04 × 105 is mathematically the same, but should not normally be used. The number raised to a power must be 10, for example, 86 or 76 should not appear in this notation. The power must be a positive or negative integer, for example, numbers such as 104.5 or 106.3 should not appear, but negative powers such as 10−4 may be used.
Because many computer printers could not at one time print superscripts, an alternative notation is often used: 2.304 × 106 is written as 2.304E6. This notation may be found in books and papers, but it is not recommended for use in formal reports. It is, however, a convenient notation, and is used in spreadsheets. The E is represented by the “EXP” key on some calculators. Note that, for example, 108 is entered into a calculator as 1 EXP 8 not 10 EXP 8. 10 EXP 8 is 10 × 108, which is 109.
1.4. Unit prefixes
The alternative way of making the number easier to use is to change the units. For all metric units, the prefixes in
Table 1.2 are used.
Table 1.2
Metric Prefixes
Thus 2,304,000 kg = 2,304 Mg (Megagramme). It would be technically correct, but very unusual to express it as 2.304 Gg. One Mg is equal to a metric tonne, so the mass would commonly be expressed as 2304 tonnes.
1.5. Logs
Another method of expressing large numbers is the use of logarithms, called commonly logs. These are particularly useful on graphs because both small and large numbers can be represented on the same graph (see, e.g.,
Fig 15.1). The log function is available on many calculators, and in all spreadsheets.
Logs are always relative to a given base, and the log of a number
x to a base
a is written as log
a(
x), and is defined from equation
(1.5):
loga(x) = x
(1.5)
Some useful relationships with logs are:
a(xy)=loga(x)+loga(y)
(1.6)
axy=loga(x)−loga(y)
(1.7)
a(xy)=y×loga(x)
(1.8)
1.5.1. Logs to base 10
Logs to base are the logs that were used for calculations before calculators were invented. The procedure for multiplying two numbers together was to obtain the log of each, and then add these together (as in equation
(1.6)), and obtain the inverse log (shown as 10
x on calculators) of the result.
It may be seen...
PDF (Adobe DRM)
EPUB (Adobe DRM)
