R Markdown:  "Why 10 * 0.1 is rarely 1.0"
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Why "10 * 0.1 is rarely 1.0"
============================
The above is a citation of one of the early books in computer science,
[The Elements of Programming Style](http://en.wikipedia.org/wiki/The_Elements_of_Programming_Style) by Kernighan and Plauger, 1974, where it was the title of section 36 (of 56).

Actually, it *is* in today's R, typically:
```{r}
10 * 0.1 == 1.0
```
Still, as you will hopefully come to see, it is rather *lucky coincidence*, because we will see that `0.1` is not exactly the same as the mathematical $1/10$.
Let's explore some of the numbers in R:
```{r}
x1 <- seq(0, 1, by = 0.1)
(x2 <- 0:10 / 10)
```
So far, nothing special. Both `x1` and `x2` are the
the same, 0, 0.1, 0.2, ..., 1.0,  right ?
```{r}
x1 == x2
```
Oops, what happened?
```{r}
n <- 0:10
rbind(x1 * 10 == n,
      x2 * 10 == n)
```

So it seems, that  `x2` is exact, but `x1` is not?
Not true:
```{r}
(d2  <- diff(x2))
unique(d2)
```
Aha...  So, one `0.1` differs from some other `0.1` ??
Is R computing nonsense?

No.  The computer can only compute with finite precision, **and** it uses _binary_ representations of all the "numeric" (_double_) numbers. 
```{r}
print(x2, digits=17)
```


On one hand, we learned that $(a + b) - a = b$ or similarly $(a/b) \times b = a$, 
or  $\frac a n + \frac b n = \frac{a+b}{n}$
in high school -- or earlier.   
As you can see from above, this is not always true in computer arithmetic:
```{r}
1/10  + 2/10 ==  3/10
```
Why? ... ...  
Well if they are not equal, let's look at the difference
```{r}
1/10  + 2/10  - 3/10
```
Aha... So what happened above?
```{r}
(u2 <- unique(d2))
u2 * 10
u2 - 1/10
```

**Note**: Among the fractions $n/m$ only those are exact in (usual) computer arithmetic where  
$n = 2^k, k \in \{0,1,\dots\}$,
e.g.  1/2,  3/4,  13/16,   but not 1/10, 2/10, 3/10, 4/10 !