Format numbers

Types of notation

Notation to represent large and small numbers depends on the mode of communication. In a computer script, for example, we might encode the Avogadro constant as N_A = 6.0221*10^23. The asterisk (*) and caret (^) in this expression, however, communicate instructions to a computer, not syntactical mathematics. And while scientific E-notation (6.0221E+23) has currency in some discourse communities, power-of-ten notation, e.g., \(\small N_A = 6.0221 \times 10^{23}\), is the conventional format for professional technical communication.

Power-of-ten notation is expressed,

\[ \small a \times 10^n, \]

where \(\small a\) is the coefficient in decimal form and the exponent \(\small n\) is an integer. Two formats are in common use (Chase, 2021, pp. 63–67):

• scientific:   \(\small 1\leq{|a|} < 10\), e.g., \(\small N_A =\) \(\small 6.0221 \times 10^{23}\).

• engineering:   \(\small 1\leq{|a|} < 1000\) and \(\small n\) is a multiple of 3, e.g., \(\small N_A =\) \(\small 602.21 \times 10^{21}\).

The utility of the engineering form follows from the SI prefixes for physical units such as “mega-”, “kilo-”, “milli-”, etc., corresponding to powers of 10 that are integer multiples of three.

Notes on syntax.   Programming symbols are not necessarily mathematical symbols:

• asterisk (*). Programming symbol for multiplication, e.g., x = a * b or y = a * (b + c). In the grammar of mathematics, multiplication is indicated by the symbol (\(\times\)) when needed.

• caret (^). Programming symbol for exponentiation, e.g., x = y^2 or z = 10^-3. In the grammar of mathematics, exponents are typeset as superscripts, e.g., \(\small x = y^2\) or \(\small z = 10^{-3}\).

• multiplication (\(\small\times\)). Mathematical symbol for multiplication, not to be confused with the letter “x”. Generally omitted when the meaning is clear, e.g., \(\small x=ab\) or \(\small y=a(b+c)\), but conventionally included in power-of-ten notation, e.g., \(\small 6.0221 \times 10^{23}\). When a comma is used as the decimal marker, multiplication may be indicated by the half-high dot, e.g., \(\small 6,0221 \cdot 10^{23}\).

Decimal subsets.   In a vector of numbers formatted in power-of-ten form, the decimal form may be preferred for any subset of values with exponents near zero, e.g., \(\small n \in \{-1, 0, 1, 2\}\).

Decimal columns.   A table of numeric information can include columns formatted in both power-of-ten notation and decimal notation. For example, a table of atmospheric properties shown below has altitude in integer form, temperature in decimal form, and density in power-of-ten engineering notation (except for those values with exponents near zero).

The purpose of the decimal format in formatdown is to match the font face and size of decimal columns to those of the power-of-ten columns. If no power-of-ten columns are used, of course, decimal columns can be displayed as-is or formatted using other R tools.

Packages.   If you are writing your own script to follow along, we use the following packages in this vignette. Data frame operations are performed with data.table syntax. Some users may wish to translate the examples to use base R or dplyr syntax.

library("formatdown")
library("data.table")
library("knitr")

Markup

We format numbers as inline math expressions delimited by $ ... $ or the optional \( ... \). For example, the Avogadro constant is marked up as

    $6.0221 \times 10^{23}$, 

where the \times macro creates the multiplication symbol (\(\small\times\)). This math markup, as an inline equation in an R markdown document, renders as: \(\small 6.0221 \times 10^{23}\). To program the markup, however, we enclose it in quote marks as a character string, that is,

    "$6.0221 \\times 10^{23}$", 

which requires us to “escape” the backslash in \times by adding an extra backslash. When the optional font size argument is assigned, formatdown adds a LaTeX-style sizing macro such as \small or \large, for example,

    "$\\small 6.0221 \\times 10^{23}$", 

where again the markup includes an extra backslash.

format_sci()

Converts numbers to character strings in power-of-ten form,

    "$a \\times 10^{n}$" 

where \(\small a\) is the coefficient and \(\small n\) is the exponent. format_sci() is a wrapper for the more general function format_numbers(). For a subset of values with exponents near zero, e.g., \(\small n \in \{-1, 0, 1, 2\}\), the output is in decimal form,

    "$a$"

Usage.  

format_sci(x,
           digits = 4,
           ...,
           omit_power = c(-1, 2),
           set_power = NULL,
           delim          = formatdown_options("delim"),
           size           = formatdown_options("size"),
           decimal_mark   = formatdown_options("decimal_mark"),
           small_mark     = formatdown_options("small_mark"),
           small_interval = formatdown_options("small_interval"), 
           whitespace     = formatdown_options("whitespace"))

• Arguments before the dots do not have to be named if argument order is maintained.
• Arguments after the dots (...) must be named.
• The arguments assigned via formatdown_options() can be reset by the user locally in a function call or globally using formatdown_options().

Examples.   These early examples are shown with default arguments. Arguments are explored more fully starting with Numeric input section.

# 1. Avogadro constant
L <- 6.0221e+23
format_sci(L)
#> [1] "$6.022 \\times 10^{23}$"

# 2. Elementary charge
e <- 1.602176634e-19
format_sci(e)
#> [1] "$1.602 \\times 10^{-19}$"

Examples 1 and 2 (in inline code chunks) render as,

1. The Avogadro constant is \(\small L =\) \(\small 6.022 \times 10^{23}\) \(\small \mathit{mol}^{-1}\).
2. The elementary charge constant is \(\small e =\) \(\small 1.602 \times 10^{-19}\) \(\small C\).

format_engr()

Similar to format_sci() except using engineering notation, i.e., exponents are multiples of 3.

Usage.

format_engr(x,
            digits = 4,
            ...,
            omit_power = c(-1, 2),
            set_power = NULL,
            delim          = formatdown_options("delim"),
            size           = formatdown_options("size"),
            decimal_mark   = formatdown_options("decimal_mark"),
            small_mark     = formatdown_options("small_mark"),
            small_interval = formatdown_options("small_interval"), 
            whitespace     = formatdown_options("whitespace"))

Examples.   (with default arguments)

# 3. Avogadro constant
format_engr(L)
#> [1] "$602.2 \\times 10^{21}$"

# 4. Elementary charge
format_engr(e)
#> [1] "$160.2 \\times 10^{-21}$"

Examples 3 and 4 render as,

3. The Avogadro constant is \(\small L =\) \(\small 602.2 \times 10^{21}\) \(\small \mathit{mol}^{-1}\).
4. The elementary charge constant is \(\small e =\) \(\small 160.2 \times 10^{-21}\) \(\small C\).

format_dcml()

A wrapper for the more general function format_numbers(); converts numbers to character strings in decimal form,

    "$a$"

where \(\small a\) is the decimal value.

Usage.

format_dcml(x,
            digits = 4,
            ...,
            size           = formatdown_options("size"),
            delim          = formatdown_options("delim"),
            decimal_mark   = formatdown_options("decimal_mark"),
            big_mark       = formatdown_options("big_mark"),
            big_interval   = formatdown_options("big_interval"),
            small_mark     = formatdown_options("small_mark"),
            small_interval = formatdown_options("small_interval"), 
            whitespace     = formatdown_options("whitespace"))

Examples.   (with default arguments)

# 5. Speed of light in a vacuum
c <- 299792458
format_dcml(c)
#> [1] "$299800000$"

# 6. Molar gas constant
R <- 8.31446261815324
format_dcml(R)
#> [1] "$8.314$"

Examples 5 and 6 render as,

5. The speed of light in a vacuum is \(\small c =\) \(\small 299800000\) \(\small\mathit{m/s}\).
6. The molar gas constant is \(\small R =\) \(\small 8.314\) \(\small\mathit{J}\cdot\mathit{K}^{-1}\mathit{mol}^{-1}\).

format_numbers()

format_numbers() is the general-purpose formatting function called by format_sci(), format_engr(), and format_dcml(). The general function can be used instead of the convenience functions simply by setting its format argument to "sci", "engr" (default), or "dcml".

Usage.

format_numbers(x,
               digits = 4,
               format = "engr",
               ...,
               omit_power = c(-1, 2),
               set_power = NULL,
               delim          = formatdown_options("delim"),
               size           = formatdown_options("size"),
               decimal_mark   = formatdown_options("decimal_mark"),
               big_mark       = formatdown_options("big_mark"),
               small_mark     = formatdown_options("small_mark"),
               big_interval   = formatdown_options("big_interval"),
               small_interval = formatdown_options("small_interval"), 
               whitespace     = formatdown_options("whitespace"))

Examples.   Reproducing some of the earlier examples using format_numbers().

# 7. Scientific
format_numbers(L, format = "sci")
#> [1] "$6.022 \\times 10^{23}$"

# 8. Engineering
format_numbers(e, format = "engr")
#> [1] "$160.2 \\times 10^{-21}$"

# 9. Decimal
format_numbers(R, format = "dcml")
#> [1] "$8.314$"

Examples 7–9 render as,

7. The Avogadro constant is \(\small L =\) \(\small 6.022 \times 10^{23}\) \(\small \mathit{mol}^{-1}\).
8. The elementary charge constant is \(\small e =\) \(\small 160.2 \times 10^{-21}\) \(\small C\).
9. The molar gas constant is \(\small R =\) \(\small 8.314\) \(\small\mathit{J}\cdot\mathit{K}^{-1}\mathit{mol}^{-1}\).

Numeric input

This section begins our detailed discussion of arguments.

Scalar input.   Generally used with inline R code. For example, the following R markdown sentence, which includes some math markup and some inline R code,

    The Avogadro constant is $L = $ `r format_sci(L)` $\mathit{mol}^{-1}$. 

renders as: The Avogadro constant is \(\small L =\) \(\small 6.022 \times 10^{23}\) \(\small\mathit{mol}^{-1}\).

Vector.   A vector of numbers (or a data frame column) is marked up as follows,

# 10. Sample vector
x <- c(2.3333e-05, 0.00034444, 0.052222, 0.63333, 81.111, 922.22, 24444, 311110,
    4222200)
format_engr(x)
#> [1] "$23.33 \\times 10^{-6}$" "$344.4 \\times 10^{-6}$"
#> [3] "$52.22 \\times 10^{-3}$" "$0.6333$"               
#> [5] "$81.11$"                 "$922.2$"                
#> [7] "$24.44 \\times 10^{3}$"  "$311.1 \\times 10^{3}$" 
#> [9] "$4.222 \\times 10^{6}$"

In a table, the output renders as,

DT <- data.table(x, format_engr(x))
knitr::kable(DT, align = "r", col.names = c("Unformatted", "Engr notation"), caption = "Example 10.")

For values with exponents \(\small n\in\{-1, 0, 1, 2\}\), the default format is decimal; see Excluding exponents.

Units input

The units R package (website: Measurement Units for R) provides measurement units for R vectors, converting vectors of class “numeric” to class “units” (Pebesma et al., 2016). For example

# Number
x <- 10320
class(x)
#> [1] "numeric"

# Convert to units class
units(x) <- "m"
x
#> 10320 [m]
class(x)
#> [1] "units"

# Operations are reflected in the values and its units
y <- x^2
y
#> 106502400 [m^2]

# Unit conversion is supported
z <- y
z
#> 106502400 [m^2]
units(z) <- "ft^2"
z
#> 1146382293 [ft^2]

If an input argument to format_numbers() (or its convenience functions) is of class “units”, formatdown attempts to extract the units character string, format the number in the expected way, and append a units character string to the result. For example,

# 11. Units-class inputs
format_sci(x)
#> [1] "$1.032 \\times 10^{4}\\>\\mathrm{m}$"
format_sci(y)
#> [1] "$1.065 \\times 10^{8}\\>\\mathrm{m^{2}}$"
format_sci(z)
#> [1] "$1.146 \\times 10^{9}\\>\\mathrm{ft^{2}}$"

Example 11 renders as,

• \(\small 1.032 \times 10^{4}\>\mathrm{m}\)
• \(\small 1.065 \times 10^{8}\>\mathrm{m^{2}}\)
• \(\small 1.146 \times 10^{9}\>\mathrm{ft^{2}}\)

More complicated units can be managed. For example the Newtonian gravitational constant could be formatted as follows, where the exponents in the units definition are given in “implicit” form, that is, where \(\small m^3 kg^{-1} s^{-2}\) is represented by "m3 kg-1 s-2".

    G        <- 6.6743e-11
    units(G) <- "m3 kg-1 s-2"
    format_sci(G)

Applying a similar procedure to several physical constants and collecting the results in a data frame yields,

This table is constructed simply to illustrate how formatdown returns a variety of units-class values with units appended to the formatted number.

In a typical application, however, the numbers in a column have the same physical units and are formatted as a vector. For example,

# Example 12
DT <- air_meas[, .(temp, pres, sp_gas, dens)]

# Examine data
DT[]
#>     temp   pres sp_gas  dens
#>    <num>  <num>  <int> <num>
#> 1: 294.1 101100    287 1.198
#> 2: 294.1 101000    287 1.196
#> 3: 294.6 101100    287 1.196
#> 4: 293.4 101000    287 1.200
#> 5: 293.9 101100    287 1.199

# Assign units
units(DT$temp) <- "K"
units(DT$pres) <- "Pa"
units(DT$sp_gas) <- "J kg-1 K-1"
units(DT$dens) <- "kg m-3"

# Format one column at a time
DT$temp <- format_dcml(DT$temp)
DT$pres <- format_engr(DT$pres)

# Or format multiple columns in one pass
cols <- c("sp_gas", "dens")
DT[, (cols) := lapply(.SD, format_dcml), .SDcols = cols]

knitr::kable(DT, align = "r", caption = "Example 12.")

Significant digits

Significant digits are applied to the input argument using the base R function signif() before additional formatting is applied. For example,

# 13. Significant digits
format_sci(e, digits = 5)
#> [1] "$1.6022 \\times 10^{-19}$"
format_sci(e, digits = 4)
#> [1] "$1.602 \\times 10^{-19}$"
format_sci(e, digits = 3)
#> [1] "$1.60 \\times 10^{-19}$"

Example 13 renders as,

• \(\small 1.6022 \times 10^{-19}\)
• \(\small 1.602 \times 10^{-19}\)
• \(\small 1.60 \times 10^{-19}\)

Formats

The format argument appears in format_numbers() only. The default is “engr”. The format is preset in the format_dcml(), format_engr(), and format_sci() convenience functions.

To compare the effects across many orders of magnitude, we display the same vector in different formats.

# 14. Comparing formats
x <- c(2.3333e-05, 0.00034444, 0.052222, 0.63333, 81.111, 922.22, 24444, 311110,
    4222200)
dcml <- format_numbers(x, 3, format = "dcml")
sci <- format_numbers(x, 3, format = "sci")
engr <- format_numbers(x, 3, format = "engr")
DT <- data.table(dcml, sci, engr)
knitr::kable(DT, align = "r", col.names = c("decimal", "scientific", "engineering"),
    caption = "Example 14.")

The values displayed without powers-of-ten notation in the scientific and engineering columns are determined by the omit_power argument discussed next.

Excluding a range of exponents

When specifying power-of-ten notation, numbers with exponents lying within the range of the omit_power argument are typeset in decimal form. In engineering notation, the exponent is checked for lying within the range before and after the conversion to multiple-of-3 exponents.

To illustrate, we compare two omit_power settings in both scientific and engineering formats. In some columns, we set omit_power = NULL, which imposes power-of-ten notation on the entire vector.

# 15. Effects of omit_power
DT <- atmos[3:12, .(pres)]
DT[, sci_all := format_sci(pres, 3, omit_power = NULL)]
DT[, sci_omit := format_sci(pres, 3, omit_power = c(-1, 0))]
DT[, engr_all := format_engr(pres, 3, omit_power = NULL)]
DT[, engr_omit := format_engr(pres, 3, omit_power = c(-1, 0))]
knitr::kable(DT, align = "r", col.names = c("Unformatted", "all scientific", "scientific w/ omit",
    "all engineering", "engineering w/ omit"), caption = "Example 15.")

Comments:

• In the “scientific with omit” column, the three values \(\small (0.184, 1.05, 5.22)\) are in decimal form because their exponents lie within the omit_power range \(\small \{-1, 0\}\).
• The same is true in the “engineering with omit” column because these exponents are in the omit range before converting to engineering form.
• The “engineering with omit” column also displays the values \((\small 22, 80, 287)\) in decimal form because their exponents are in the omit range after converting to engineering notation.

If a single value is assigned, e.g., omit_power = 0, the argument is interpreted as c(0, 0).

# 16. Omit power used for a single value of exponent
DT <- atmos[3:12, .(pres)]
DT[, sci_all := format_sci(pres, 3, omit_power = NULL)]
DT[, sci_omit := format_sci(pres, 3, omit_power = 0)]
DT[, engr_all := format_engr(pres, 3, omit_power = NULL)]
DT[, engr_omit := format_engr(pres, 3, omit_power = 0)]
knitr::kable(DT, align = "r", col.names = c("Unformatted", "all scientific", "scientific w/ omit",
    "all engineering", "engineering w/ omit"), caption = "Example 16.")

Setting omit_power = c(-Inf, Inf) yields the same decimal result as format = "dcml" and overrides any other format setting. For example,

# 17. Different ways of creating a decimal format
(y <- 0.00678)
#> [1] 0.00678

(p <- format_numbers(y, 3, "sci", omit_power = c(-Inf, Inf)))
#> [1] "$0.00678$"

(q <- format_numbers(y, 3, "dcml"))
#> [1] "$0.00678$"

(r <- format_dcml(y, 3))
#> [1] "$0.00678$"

all.equal(p, q)
#> [1] TRUE
all.equal(p, r)
#> [1] TRUE

Example 17 (all cases) renders as,

• \(\small 0.00678\)

Enforcing a specific exponent

When values in a table column span only a few orders of magnitude, an audience is sometimes better served by setting the notation to a constant power of ten. For example, here we show numbers in scientific format and compare to columns in which the exponents are set to fixed values. Assigning a value to set_power overrides omit_power and format.

# 18. set_power argument
DT <- atmos[alt <= 40, .(alt, pres, dens)]
DT[, sci_pres := format_sci(pres, 3, omit_power = c(-1, 2))]
DT[, set_pres := format_sci(pres, 3, omit_power = c(-1, 2), set_power = 3)]
DT[, sci_dens := format_engr(dens, 3, omit_power = c(-1, 2))]
DT[, set_dens := format_engr(dens, 3, omit_power = c(-1, 2), set_power = -2)]
DT[, pres := NULL]
DT[, dens := NULL]
knitr::kable(DT, align = "r", col.names = c("Altitude (km)", "Pressure (Pa)", "with set_power",
    "Density (kg/m$^{3}$)", "with set_power"), caption = "Example 18.")

Options

Arguments assigned using formatdown_options() are described in the Global settings article.