Understanding Engineering Mathematics

of us find inequalities difficult to handle and they require a lot of practice. However,
in this book we will need only the basic properties of inequalities. We will say more
about inequalities in Section 3.2.6.
Often we wish to refer to thepositiveorabsolutevalue of a numberx(for example in
a rectified sine wave). We denote this by themodulus ofx,|x|. For example

|− 4 |= 4

By definition|x|is never negative, so|x|≥0. Also, note that|x|<ameans−a<x<a.
For example:

|x|< 3

means
− 3 <x< 3

Solution to review question 1.1.

(i) ‘xis a positive non-zero number’ is expressed byx> 0

(ii) ‘xlies strictly between 1 and 2’ is expressed by 1<x< 2

(iii) ‘xlies strictly between−1 and 3’ is expressed by− 1 <x< 3

(iv) ‘xis equal to or greater than−2 and is less than 2’ is expressed by

− 2 ≤x< 2

(v) If the absolute value ofxis less than 2 then this means that ifx

is positive then 0≤x<2, but ifxis negative then we must have

− 2 <x≤0. So, combining these we must have− 2 <x<2. This

can also be expressed in terms of the modulus as|x|<2.

1.2.3 Highest common factor and lowest common multiple ➤
328 ➤

Aprime numberis a positive integer which cannot be expressed as a product of two or
more smaller distinct positive integers. That is, a prime number cannot be divided exactly
by any integer other than 1 or itself. From the definition, 1 is not a prime number. 6, for
example, is not a prime, since it can be written as 2×3. The numbers 2 and 3 are called
its (prime)factors. Another way of defining a prime number is to say that it is has no
integer factors other than 1 and itself.
There are an infinite number of prime numbers:

2 , 3 , 5 , 7 , 11 , 13 ,...

but no formula for thenth prime has been discovered. Prime numbers are very important
in the theory of codes and cryptography. They are also the ‘building blocks’ of numbers,
since any given integer can be written uniquely as a product of primes:

12 = 2 × 2 × 3 = 223

This is calledfactorisingthe integer into its prime factors. It is an important operation,
for example, in combining fractions.