Converting to Decimal Fractions 2

The above method gives a clear understanding of how any rational fraction can be written as a sum of rational fractions with denominators in powers of 10.

An easier method would be to actually divide 1 (or 1.00000000..) by 7 as per normal rules of division.

By either method we will see that 1/7 = 0.142857142857142847142…….. We can see that the sequence of numerals 142857 keeps on repeating infinitely. By convention this is written as 0. 142857 and read out as “zero point one four one eight five seven recurring“. Recurring means this sequence repeats infinitely.

By the same process 1/6 = 0.1666666.. = 0.16 (zero point 1 and 6 recurring)

We find that a majority of fractions in rational number representation, when converted to decimal fractions, end up with an infinite, but recurring series of digits.

The culprit here is the base, number 10, that we have chosen. 10 has only 2 factors, 2 and 5. Hence most fractions cannot be written as fractions with denominators of 10 or powers of 10. Hence, they end up as recurring decimals. The digits recur since the remainders can be only numbers between 1 and 9 and will repeat after a certain number of steps yielding the same series of numbers again and again.

Decimal Representation of 1

There is an interesting equation in arithmetic which appears in higher math.

1 = 0.9999999999999999999999.....

Mathematician Steven Strogatz has declared it as his favorite equation. He justifies it as follows.

"I love how simple it is — everyone understands what it says — yet how provocative it is. Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity."

This can also be proved with math. If we assume x = 0.9999999...... then 10x = 9.99999999......

Then 9x = 9, since the decimal portions are equal as they are i9nfinite series of 9s. Hence x=1.

In the next chapter, we would see the reverse process - converting decimal fractions to rational form.