For the first time we land behind the point about the age of ten. It’s not a particularly fascinating experience because it takes place on boring math. I obviously mean a decimal point and decimal fractions, which we first meet when paying for a lollipop or weighing “a kilo and two hundred” of flour by grandma. And this article is about precision.

Here goes an example:

## How to steal a fortune by penny shaving?

Penny shaving is a way of stealing tiny sums of money, thousands of times, from thousands of people. In banking computer systems, transferring amounts **less** than 1 cent can be possible with appropriate authorization. So, a thief goes just one step further beyond the point than the others (to $0.001 instead of $0.01). Sometimes these thefts were lasting for years, without anyone noticing because nobody cared about such details.

Did a question pop up with this example?

## So, how many decimal places matter?

Here’s the problem! Because the answer is: it depends. Sadly, in every field it depends on something else.

### Currencies

In Polish (English, US, and many other) shops we’ll go just two steps past the point (a cent is $0.01), but in Czechia… it varies. Although a heller is 0.01 of a koruna, there are no heller coins in circulation, you can’t pay with them. If you buy two items for 10.40 CZK both, you won’t pay 20.80 CZK, but 21 CZK – you go back from the second place! On the other hand, while exchanging currency, we often go four places forward, for example 1 Czech koruna can cost 0,1677 Polish zloty. It matters because quite often large amounts of money are exchanged. When buying 10 000 CZK, it’s better not to lose by spending 1700 PLN, but pay 1677 PLN!

### Furniture

Yes, something completely different now. Arranging a furniture set requires measuring the space in a room first. So, three steps past the point (a millimeter is 0.001 m!) are taken at once. If that part gets screwed up – problems might pile up on all the later parts. So please, measure to the third decimal place, even several times – the walls will almost certainly turn out to be curvy.

Anyone with a carpenter friend knows that from time to time even the most ordinary pieces of furniture receive some improvements. Some enable leveling something out here and there in case it doesn’t fit perfectly (adjustable hinges, legs, hangers). Such contraptions are usually more costly than those without regulation, but they still pretty often become a standard. So, you can make a profit by selling stuff that corrects flaws of floors and walls that are ‘beyond the decimal point’!

### Olympic record of all times

In 2008 in Beijing, Michael Phelps had a chance to win 8 gold medals as he started in that many events. Nobody has ever achieved that feat before. In the first six races, he won undisputedly. In the seventh run, he was first before the Serb Milorad Čavić by just **0.01 s!** The Serbs protested against the result, hoping for gold. Aside the automatic measurement from the touch panels in the pool, the video footage was reviewed. It has confirmed the American’s victory, but still, there is a conspiracy theory about falsifying the result. **But**: in this case, there is no point in measuring time more precisely because the pools had 3 cm of length tolerance back then (according to Olympic regulations). That’s what Phelps swam in **0.15 s!**

### The outer space! Down to Earth.

Do you know how does GPS work?

It doesn’t measure distance directly. Really. It measures **time!** Calculating distance from it is easy. It is the time that a signal takes to travel from a satellite to, say, a smartphone, which is about 0.12 s. We go past the point in this approximation, but far too close. To reach accuracy of 10 meters, we need to go 9 steps (precision of 0.000000033 s). It is achieved (simply saying) thanks to redundant satellites and super-precise clocks mounted inside them.

### A mathematical absurd called **π**

The π number (imprecisely equal 3.14) has its various quirky properties. One of them is: it’s used in memory competitions. The record holder – reportedly – has ventured 100,000 places beyond the decimal point with his **memory**. He was just reciting the digits for hours! Of course, it’s a child’s play for the computers, which have gone billions of steps further. This is used, for example, to calculate the volume of a sphere with increasing accuracy. Specifically, if you take the not-so-spherical Earth, then making the 24th step past the decimal point will increase the accuracy by about **1 mm ^{3}**. Well, that’s so precise that nobody does it practically. It’s not worth the effort. And a hundred decimal places is ridiculously many because nobody will ever need to measure the Universe with an atom-sized accuracy…

## So, what are these travels for?

Because the devil is in the details. One could list many practical examples, just don’t go nuts. Don’t worry about travels too extensive to be practical! Two steps in a shop, two in a swimming pool too, a few in a bank, up to about ten in a GPS. This calls for more interesting examples!

But… WHAT FOR were the billions of places in π? This is a place for an entirely different story…