# Numbers¶

Numbers are ubiquituous in computing, and according to the Guile reference there is a whole lot of aspects that can be discussed about them. However, for LilyPond users there is not that much required information to begin with. Scheme supports many different number types, but for LilyPond use, integers and real numbers are usually sufficient.

To check if a value is a number we can use the predicate `number?`

``````guile> (number? 4.2)
#t

guile> (number? "Hi")
#f
``````

## Integers¶

Integers, or whole numbers, are written as they are, `5`, `-12` or `1234`. If written in a LilyPond file they can be prepended with the `#` sign or written literally, so `#13` is equivalent to `13`

``````\paper {
min-systems-per-page = 5
max-systems-per-page = #7
}
``````

If arithmetic operations are performed on integers the results are integers as well:

``````guile> (+ 123 345)
468

guile> (* 4 5)
20
``````

The predicate for integers is `integer?`

## Real and Rational Numbers¶

Real numbers are all the non-integer numbers, in computing often referred to as floating-point numbers. Rational numbers are a subset thereof, namely all numbers that can be expressed as a fraction of integers. Fractions are written as two integers separated by a slash, without any whitespace in between.

``````guile> (real? 1.20389175)
#t

guile> (fraction? 5/4)
#t

guile> (fraction? 1.25)
#f
``````

#### Mixing Reals, Rationals and Integers¶

Above we said that arithmetic operations on integers produce integers again. However, if only one operand is a real number the whole expression gets converted to reals:

``````guile> (+ 3 1.0)
4.0
``````

When integers are divided Scheme will express the result as a fraction that is shortened as much as possible, but as soon as one real number is involved the fraction is converted to a floating point number:

``````guile> (/ 4 2)
2

guile> (/ 10 4)
5/2

guile> (/ 4 3)
4/3

guile> (/ 4 3.0)
1.33333333333333
``````

#### Exact and Inexact Numbers¶

Integers and fractions are always exact values that can be recalculated as often as desired, giving always the same result. Real numbers on the other hand are inexact as they are subject to an arbitrary precision as implemented by the programming language system. This means that when performing mathematical operations with real numbers one has to expect the possibility of rounding errors. Generally this is not an issue when using Scheme in LilyPond, but it should be noted that there is this issue.

## Calculations With Numbers¶

In order to learn what operations can be done with numbers in Scheme it may be a good idea to familiarize oneself with the documentation on integers and reals. Diving into these pages now may also be a good test on how to handle the reference style of the GNU Guile Manual.

Last update: November 3, 2022