CHAPTER 1
Introduction to some basic techniques
Topics covered in this chapter:
• Order of operations, rounding off numbers: rounding off numbers to the nearest whole number, truncation, significant figures, decimal places
• Estimation of answers, absolute and relative errors
• Indices
1.1 Introduction
In this chapter some of the basic techniques are explained. Most of these do not involve any complexity, but it is very important to solve a number of problems and to show the answer in an acceptable format.
1.2 Order of operations
In mathematics and other analytical subjects, the calculations may involve one or more of:
multiplication, division, addition, subtraction and brackets.
There is a definite order of operations in algebra that needs to be followed to get accurate solutions. For example, in the following calculation, the answers could be 51, 13 or 19:
The correct answer is 19, which results from following the right procedure. This procedure, known as the order of precedence of operations, involves dealing with brackets (B) first and then of (O), division (D), multiplication (M), addition (A), subtraction (S), in that order. This is usually remembered as BODMAS.
In the above example, 20 − 4 + (1 × 3) = 20 − 4 + 3 = 20 − 1 = 19 (−4 + 3 = −1)
Example 1.1
Solve: a) 16 + 5 − 2 × 1.5
b) 20 − 2 + (2 × 3)
c) 25 + 5 − 3(1.5 × 4) − 3 × 2
Solution:
a) 16 + 5 − 2 × 1.5 = 16 + 5 − 3 = 21 − 3 = 18
b) 20 − 2 + (2 × 3) = 20 − 2 + 6 = 20 + 4 = 24 (−2 + 6 = + 4)
c) 25 + 5 − 3(1.5 × 4) − 3 × 2 = 25 + 5 − 3(6) − 6 = 25 + 5 − 18 − 6
= 30 − 18 − 6 = 6
1.3 Rounding
1.3.1 To the nearest whole number
The convention is to round 0.5 and above to the next highest whole number, and 0.499 (recurring) to the next lowest whole number. For example, 8.5 will become 9.0 when rounded to the nearest whole number. Similarly, 8.499 will become 8.0 when rounded to the nearest whole number.
1.3.2 Truncation
Truncation involves the omission of the unwanted digits at the end of a number. For example, 25.3458 truncated to four figures becomes 25.34.
1.3.3 Significant figures
3462, 346200 and 0.03462 all have four significant figures, not counting zeros at the beginning or end of the number. To write these numbers to 3 significant figures, the last figur...
