Moore / Robinson Mathematics for the General Course in Engineering

The Commonwealth and International Library of Science, Technology, Engineering and Liberal Studies: General Engineering Division, Volume 1
1. Auflage 2013
ISBN: 978-1-4831-5117-5
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 146 Seiten

Mathematics for the General Course in Engineering, Volume I covers the syllabus in mathematics for the G.1 year of the general course in engineering. Provided in this text are 31 unworked examples, which form a comprehensive revision course that students are recommended to work through toward the end of the G.l year. Answers to the text examples are provided at the end. The subjects covered in this book are arithmetic; indices, logarithms, and the use of tables; length, area, and volume; algebra; geometry; and trigonometry. This volume provides students taking mathematics for the G.1 year in engineering a sound basis for the work of the G.2 year.

Autoren/Hrsg.

Moore, John C

Robinson, Robert

Leseproben

CHAPTER ONE

Arithmetic

Publisher Summary

This chapter discusses proper fractions or fractions. By multiplying both the numerator and the denominator of a fraction by the same number, the new fraction would come out exactly equal to the old. However, by dividing both the numerator and the denominator of a fraction by the same number, the new fraction would come out exactly equal to the old. It is easy to add fractions whose denominators are equal. However, if one takes 10 pounds weight of pure water at 62°F—the air pressure is also specified as 30 inches of mercury—the volume that it occupies would be called a standard gallon. If a standard gallon is divided into four equal parts, each part would be called a quart. However, if a quart is divided into two equal parts, each part would be called a pint. This is the British system for measuring capacity. Whenever measured values are used in a calculation, the final answer must always be inaccurate as it is based on measured values that are themselves inaccurate.

FRACTIONS

Suppose we take the number 1 and break it into smaller pieces. These smaller pieces are called PROPER FRACTIONS or simply FRACTIONS.

When 1 is divided into 2 equal parts each part is called a HALF and is written .

DIAGRAM 1

Further fractions are illustrated below.

DIAGRAM 2

Names of the Fundamental Fractions

The above list can be extended indefinitely and we shall call these fractions the FUNDAMENTAL fractions. Any other fraction can be expressed in terms of them.

When we meet a fraction such as the number at the top has a special name. It is called the NUMERATOR. The number at the bottom also has a special name. It is called the DENOMINATOR.

An important principle

If we take a fraction and MULTIPLY both the numerator and the denominator by the same number then the new fraction is exactly equal to the old.

If we take a fraction and DIVIDE both the numerator and the denominator by the same number then the new fraction is exactly equal to the old.

Addition of Fractions

It is easy to add fractions whose DENOMINATORS are equal.

If we are called upon to add fractions whose DENOMINATORS are not equal then we use the first part of the principle printed above in red.

Ex. Calculate

Think of the smallest number into which 4, 6 and 8 will divide exactly. It is 24.

Using the first part of the principle each fraction can be expressed with a 24 in the DENOMINATOR.

So

Method of setting out

The previous example is usually set out as follows.

If we write 1 this means 1 +

Note that

A fraction such as in which the NUMERATOR is larger than the DENOMINATOR is called an IMPROPER FRACTION.

Ex. Express 1 as an IMPROPER FRACTION using a quick method.

Quick method

Multiply the 1 by the 4 and then add in the 3. Put this number over 4.

Ex. Express 3 as an IMPROPER FRACTION using a quick method.

Ex. Calculate

Subtraction of Fractions

Ex. Calculate

Multiplication of Fractions

Ex. Calculate of using a diagram.

DIAGRAM 3

If the total area represents 1 then the portion enclosed by the thick black lines represents

of will be the shaded area.

This shaded area is made up of 10 small rectangles and each rectangle represents (there are 21 rectangles altogether).

So the shaded area is 10 × or

We usually use the multiplication symbol (×) and write

Quick method

The student will notice that there is a quick way of working out ×

We simply say 5 × 2 = 10 and 7 × 3 = 21.

The answer is

Division of Fractions

Ex. Calculate ÷ using a diagram.

of the total area in Diagram 3 comes to 15 small rectangles.

of the total area in Diagram 3 comes to 14 small rectangles.

So ÷ = 15 small rectangles ÷ 14 small rectangles

Quick method

We can obtain the correct answer by turning the upside-down and multiplying.

In some examples where fractions are multiplied together it is necessary to CANCEL.

Ex. Calculate ×

When we CANCEL we are dividing both the numerator and the denominator by the same number. This does not alter the value of the fraction.

Ex. Express as a fraction in its LOWEST TERMS.

No more cancelling is possible and the fraction is in its LOWEST TERMS.

Ex. Calculate

DECIMALS

When we write 26 we mean 2 TENS and 6 UNITS.

Similarly 347 means 3 HUNDREDS + 4 TENS + 7 UNITS. When we write 568.342 we mean 5 HUNDREDS + 6 TENS + 8 UNITS + 3 TENTHS + 4 HUNDREDTHS + 2 THOUSANDTHS.

DIAGRAM 4

Addition of Decimals

Ex. Calculate 42.049 + 2.0072

The decimal points must be placed under one another.

Subtraction of Decimals

Ex. Calculate 262.07 - 29.395

The decimal points must be placed under one another.

Multiplication of Decimals

Ex. Calculate 2.48 × 6.1

Place the extreme right-hand figures of each number under one another.

The position of the decimal point in the answer is obtained by ADDING the number of decimal places in each of the multiplied numbers.

Division of Decimals

Ex. Calculate

MAKE THE DENOMINATOR A WHOLE NUMBER by moving the decimal point in the denominator 1 place to the right. The decimal point in the numerator must also be moved 1 place to the right.

Relation between Fractions and Decimals

Ex. Express as a decimal giving the answer correct to 2 decimal places.

The answer is nearer to 0.43 than 0.42 and so

= 0.43 (correct to 2 decimal places).

Ex. Express 0.57 as a fraction.

0.57 means 5 TENTHS + 7 HUNDREDTHS

Ex.

(i) Express as a decimal correct to 4 decimal places.

(ii) Express 0.036 as a fraction in its lowest terms.

(iii) Calculate Correct to 3 decimal places.

LENGTH

The STANDARD YARD is the distance, at 62°F, between two plugs of gold sunk in a certain bar of bronze kept in London. If we divide the STANDARD YARD into 3 equal parts each part is called a FOOT. If we divide the FOOT into 12 equal parts each part is called an INCH. A distance equal to 1760 STANDARD YARDS is called a MILE. This is the BRITISH SYSTEM for measuring length.

In the METRIC SYSTEM which is in operation in many parts of the world the basic unit of length is the METRE and the STANDARD METRE is defined as the distance, at 0°C, between two fine lines engraved on a certain platinum—indium bar kept at Sèvres in France. Other units of length are obtained from this basic unit by means of the following prefixes.

The student should note in particular

DIAGRAM 5

Ex. Given that 1 INCH = 2.54 CENTIMETRES, express a YARD in METRES.

The student should look at Diagram 5 and note that we have to...