Rational Numbers
We can write numbers such as 8, 4½, 1/5, 0.211, √49/16, 0.3 as exact fractions or ratios:
8/1, 9/2, 1/5, 211/1 000, 7/4, 1/3.
Such numbers are called rational numbers.
Numbers which cannot be written as exact fractions are called non-fractional numbers, or irrational numbers. √7 is an example of a non-rational number. √7 = 2.645 751 …., the decimals extending without end and without recurring.
π is another example of a non-rational number. Π = 3.141 592 …., again extending forever without repetition. The fraction 22/7 is often used for the value of π. However, 22/7 is a rational number and is only an approximate value of π.
All recurring decimals are rational numbers. Read the following example carefully.
Example
Write 3.17 as a rational number
Let n = 3/17
i.e. n =3.17 17 17 ………… (1)
Subtract (1) from (2),
99n = (317.17 17 . . .) – (3.17 17 . . .)
99n = 314
Thus, 3.17 = 314/99, a rational number.
A non-rational number extends forever and is non-recurring.
Exercise
1. Which of the following are rational and which are non-rational?
a. 9 b. 1/9 c. √9 d. 0.9 e. 2 2/3
Square roots
Some square roots are rational:
√4 = 2, √6.25 = 2.5 = 5/2
Other square roots are non-rational :
√11 = 3.316 624 . . ., √3.6 = 1.897 366 . . .
The fact that many square roots are non-rational, was first discovered by Pythagoras around 500 BC. He tried to find the length of a diagonal of a ‘unit square’. The fig below is a unit square, a square with side 1 unit.
In ∆ABC, using Pythagoras’ rule,
BC2 = AB2 + AC2
BC2 = 12 + 12 = 2
BC = √2
Pythagoras was unable to find a rational value for √2. Thus, although it is possible to draw the diagonal of a unit square, it is possible to measure its length accurately! This troubled Pythagoras so much that he called non-rational numbers ‘unspeakables’.
It is possible to find the approximate value of non-rational square roots by using a ‘trial and improvement’ method. The example immediately below shows…
Read more here- https://passnownow.com/lesson/rational-non-rational-numbers/
