16. Rounding Crooked Numbers:

0.9999…

The Question:

It is often easier to deal with nice round figures like six or eighteen, rather than with crooked numbers like 6.01356837 or 17.986757321. This is why we often round numbers up or down, and 6.01356837 becomes six, and 17.986757321 becomes eighteen. We realize that a small error arises when rounding but very often the resulting ease of computation is worth that error. But are there crooked numbers that can be rounded up without incurring any error at all?

The Paradox:

Yes, there are.

The never-ending number 0.999999…., for example, with an infinite number of 9s after the decimal point, turns out to be not only nearly equal to 1.0, but actually is exactly equal to 1.0.

How can that be? Surely, even with an infinite number of 9s after the decimal point, the resulting number must be a teeny-weeny, itsy-bitsy less than 1.0. Well, no, it’s not, as we shall see below.

And what about numbers like 3.19999…. or 7.63529999….?

Background:

Decimal fractions were invented by the Arab mathematician Abu'l-Hasan al-Uqlidisi in the ninth century, and re-invented by the Persian scholar Jamshid al-Kashi in the fifteenth. In the 16th century the Flemish physicist, mathematician and engineer Simon Stevin represented numbers by unending decimals. And in 1758, the Swiss mathematician, astronomer and philosopher Johann Heinrich Lambert proved that the decimal representation of Pi never ends, i.e., it has infinitely many digits after the decimal point.

Another Swiss, Leonhard Euler (1707-1783), was one of the foremost mathematicians of the eighteenth century. Together with the Bernoulli brothers, cousins and uncles – his close friends and mentors (all of them Swiss) – he was largely responsible for the development of the infinitesimal calculus on which modern mathematics and engineering is based. But Euler also dealt with algebra, as it is nowadays taught in high schools. In 1770, he published a two-volume textbook on the subject. The first volume contains 562 paragraphs, numbered consecutively throughout the book. The second volume adds another 250.

The book was not directed at mathematicians but was meant to be readable by anybody. In fact, Euler – who had lost his eyesight four years earlier and by that time was nearly totally blind – dictated the text to a tailor who wrote everything down as Euler spoke. It is said that the great teacher’s explanations were so clear that this young man, being of only mediocre intellect, not only understood everything that Euler dictated but was soon able to solve algebraic problems himself.

It is in the chapter on infinite decimal fractions, in paragraph 524 to be exact, that the number 0.9999… made its first appearance. (See below.)

* * *

In the booklet Mathematics: A Very Short Introduction, Timothy Gowers claims that “0.9999… equals 1.0” is a convention, albeit an indispensible one in conventional mathematics. Because if 0.9999…. were unequal to 1.0, then what would the difference between the two numbers be? If it is zero, then 0.9999… cannot be unequal to 1.0. (Whenever the difference between two numbers is zero, the two numbers must be equal to each other.) If it is not zero, then it must be something infinitesimally small, but nevertheless larger than zero. And what could that be? 0.0000…..1? For such a number to exist, a completely new, non-conventional mathematics would have to be invented.

Dénouement:

Let’s denote the number 0.9999…. by X:

X = 0.9999….

Multiply both sides by ten and we get

10·X = 9.9999….

Now let’s deduct X, respectively 0.9999… from both sides of the equation, and we have,

9·X = 9,

and now it’s obvious: X = 1. QED.

Another way to see that 0.9999… is equal to 1.0 is as follows:

The fraction one third (1/3) is written in decimal notation as 0.3333… If we multiply this fraction by three we get 0.9999… On the other hand, three times one third is equal to 1.0. So, voilà, 0.9999… is equal to 1.0.

Technical Supplement:

0.9999…. is not the only number that is equal to another number. For example, the number Z = 3.19999….. is equal to 3.2. Why? Well, multiply both sides by ten, deduct Z, respectively 3.19999… from both sides, and you get 9·Z = 28.8. Now divide 28.8 by nine, and there you have it: Z = 3.2.

As an exercise, do the same with, say, 7.63529999….

The argument also works the other way around. Hence, the number 4.57 is identical to the number 4.569999….

* * *

Leonhard Euler arrived at this conclusion in a more sophisticated way. He was not really interested in numbers ending in 9999…. Rather, his interest was kindled by so-called ‘progressions’. Progressions are mathematical constructs of the form

1 + k + k2 + k3 + k4 + ….

(Nowadays, we call Euler’s ‘progression’ a geometric series.) Can such a never-ending progression be summed? Euler showed that it can, as long as k is smaller than one. In fact, he showed that in this case the sum of such an infinite series is equal to 1/(1 – 1/k).

Guess what Euler used to illustrate his result. Yes … it was the number 0.9999… since it can be written as an infinite progression:

0.9999… = 0.9 (1 + 1/10 + 1/102 + 1/103 + 1/104 + ….)

With k = 1/10, the sum of the progression inside the parentheses becomes 1/(1 – 1/10) = 10/9,

Hence, 0.9999… = 0.9·(10/9) = 1.0