`bc`

is a quirky but useful calculator. It is a standard Unix utility and is also available for Windows.

One nice feature of `bc`

is that you can set the parameter `scale`

to indicate the desired precision. For example, if you set `scale=100`

, all calculations will be carried out to 100 decimal places.

**The first surprise** is that the default value of `scale`

is 0. So unless you change the default option, 1/2 will return 0. This is not because it is doing integer division: 1.0/2.0 also returns 0. `bc`

is computing 1/2 as 0.5 and displaying the default number of decimal places, i.e. none! Note also that `bc`

doesn’t round results; it truncates.

`bc`

has one option: `-l`

. This option loads the math library and sets the default value of `scale`

to 20. I always fire up `bc -l`

rather than just `bc`

.

**The second surprise** with `bc`

is that its math library only has five elementary functions. However, you can do a lot with these five functions if you know a few identities.

The **sine** and **cosine** of `x`

are computed by `s(x)`

and `c(x)`

respectively. Angles are measured in radians. There is no tangent function in `bc`

. If you want the tangent of `x`

, compute `s(x)/c(x)`

. (See here for an explanation of how to compute other trigonometric functions.) As minimal as `bc`

is, it did make a minor concession to convenience: it could have been more minimal by insisting you use sin(π/2 – *x*) to compute a cosine.

The only inverse trigonometric function is `a(x)`

for **arctangent**. This function can be bootsrapped to compute other inverse functions via these identities:

arcsin(*x*) = arctan(*x* / sqrt(1 – *x*^{2}))

arccos(*x*) = arctan(sqrt(1 – *x*^{2} )/ *x*)

arccot(*x*) = π/2 – arctan(*x*)

arcsec(*x*) = arctan(sqrt(*x*^{2} – 1))

arccsc(*x*) = arctan(1/sqrt(*x*^{2} – 1))

The functions `l(x)`

and `e(x)`

compute (natural) **logarithm** and **exponential** respectively. `bc`

has a power operator `^`

but it can only be used for integer powers. So you could compute the fourth power of *x* with `x^4`

but you cannot compute the fourth root of *x* with `x^0.25`

. To compute *x*^{y} for a floating point value *y*, use `e(l(x)*y)`

. Also, you can use the identity log_{b}(*x*) = log(*x*) / log(*b*) to find logarithms to other bases. For example, you could compute the log base 2 of *x* using `l(x)/l(2)`

.

Not only is `bc`

surprising for the functions it *does not* contain, such as no tangent function, it is also surprising for what it *does* contain. **The third surprise** is that in addition to its five elementary functions, the `bc`

math library has a function `j(n,x)`

t0 compute the *n*th Bessel function of *x* where *n* is an integer. (You can pass in a floating point value of *n* but `bc`

will lop off the fractional part.)

I don’t know the history of `bc`

, but it seems someone must have needed Bessel functions and convinced the author to add them. Without `j`

, the library consists entirely of elementary functions of one argument and the names of the functions spell out “scale.” The function `j`

breaks this pattern.

If I could include one advanced function in a calculator, it would be the gamma function, not Bessel functions. (Actually, the *logarithm* of the gamma function is more useful than the gamma function itself, as I explain here.) Bessel functions are important in applications but I would expect more demand for the gamma function.

**Update**: Right after I posted this, I got an email saying `bc -l`

was following me on Twitter. Since when do Unix commands have Twitter accounts? Well, Hilary Mason has created a Twitter account bc_l that will run `bc -l`

on anything you send it via DM.

**Update** (November 14, 2011): The account `bc_l`

is currently not responding to requests. Hilary says that the account stopped working when Twitter changed the OAuth protocol. She says she intends to repair it soon.

**Related posts**:

How many trig functions are there?

Diagram of Bessel function relationships