6 Chapter 1Numbers, variables, and units
EXAMPLE 1.5Division of fractions
The number 10 2 12 can be simplified by ‘dividing top and bottom’ by the common
factor 2: 10 2121 = 152 6 (see Section 1.3).
0 Exercises 18–
Every rational number is the solution of a linear equation
mx 1 = 1 n (1.10)
where mand nare integers; for example, 3x 1 = 1 2 has solutionx 1 = 122 3. Not all numbers
are rational however. One solution of the quadratic equation
x
21 = 12
is , the positive square root of 2 (the other solution is ), and this number
cannot be written as a rational numberm 2 n; it is called an irrational number.
Other irrational numbers are obtained as solutions of the more general quadratic
equation
ax
21 + 1 bx 1 + 1 c 1 = 10
where a, b, and care arbitrary integers, and of higher-order algebraic equations; for
example, a solution of the cubic equation
x
31 = 12
is , the cube root of 2. Irrational numbers like and are called surds.
The rational and irrational numbers obtained as solutions of algebraic equations
of type
a
01 + 1 a
1x 1 + 1 a
2x
21 + 1 a
3x
31 +1-1+ 1 a
nx
n1 = 10 (1.11)
wherea
0, a
1, =, a
nare integers, are calledalgebraic numbers; these numbers can
be expressed exactly in terms of a finite number of rational numbers and surds.
There existalso other numbers that are not algebraic; they are not obtained as
solutions of any finite algebraic equation. These numbers are irrational numbers
called transcendental numbers; ‘they transcend the power of algebraic methods’
2
3x= 2 2
3x= 2 − 2
2
3
4
5
2
3
5
4
10
12
÷=×=