Question

how do you prove algebraically that 0.5 recurring is 5/9. use x in your answer?

Answer 1

Rational Numbers

If we subtract after recurring digits match each other, it will be eliminated.

So, let's say $x=0.555...$.

When we multiply 10 to $x$, the decimal point shifts right next to itself.

Equations:-

1. $10x=5.555...$
2. $x=0.555...$

Now we subtract a greater number by a smaller one.

$\implies 10x - x = 5.\cancel{555...} - 0.\cancel{555...}$

$\implies 9x = 5$

$\therefore x = \dfrac{5}{9}$

Irrational Numbers

This method works because we eliminate the recurring decimal.

Let's try another number. For example, $x = 0.123456789101112131415...$.

But the decimal doesn't recur. Now we cannot express numbers with fractions. These number groups are called 'irrational numbers.'

Random Facts:

In the ancient Greeks myths, the Delos people are asked to make a new altar, which has 2 times the volume and the same shape as the previous one, which was a cube.

However, one side of the cube should get multiplied by $\sqrt[3]{2}$, which gives 2 after multiplying itself three times, because $\mathrm{Volume = Side \times Side \times Side}$.

This is an irrational number, which in the 19th Century it was proven impossible to make a new altar using a ruler and a compass.

In the end, the Delos people couldn't measure $\sqrt[3]{2}$, they couldn't drive the disease out.