Numbers

Types of Numbers	Arbitrary‐Precision Calculations
Complex Numbers	Arbitrary‐Precision Numbers
Numeric Quantities	Machine‐Precision Numbers
Digits in Numbers	Interval Arithmetic
Exact and Approximate Results	Indeterminate and Infinite Results
Numerical Precision	Controlling Numerical Evaluation

Types of Numbers

Four underlying types of numbers are built into the Wolfram System.

Integer	arbitrary‐length exact integer
Rational	integer/integer in lowest terms
Real	approximate real number, with any specified precision
Complex	complex number of the form number+number I

Intrinsic types of numbers in the Wolfram System.

Rational numbers always consist of a ratio of two integers, reduced to lowest terms:

Approximate real numbers are distinguished by the presence of an explicit decimal point:

An approximate real number can have any number of digits:

Complex numbers can have integer or rational components:

They can also have approximate real number components:

123	an exact integer
123.	an approximate real number
123.0000000000000	an approximate real number with a certain precision
123.+0.I	a complex number with approximate real number components

Several versions of the number 123.

You can distinguish different types of numbers in the Wolfram System by looking at their heads. (Although numbers in the Wolfram System have heads like other expressions, they do not have explicit elements which you can extract.)

The object 123 is taken to be an exact integer, with head Integer:

The presence of an explicit decimal point makes the Wolfram System treat 123. as an approximate real number, with head Real:

NumberQ[x]	test whether x is any kind of number
IntegerQ[x]	test whether x is an integer
EvenQ[x]	test whether x is even
OddQ[x]	test whether x is odd
PrimeQ[x]	test whether x is a prime integer
Head[x]===type	test the type of a number

Tests for different types of numbers.

NumberQ[x] tests for any kind of number:

5. is treated as a Real, so IntegerQ gives False:

If you use complex numbers extensively, there is one subtlety you should be aware of. When you enter a number like 123., the Wolfram System treats it as an approximate real number, but assumes that its imaginary part is exactly zero. Sometimes you may want to enter approximate complex numbers with imaginary parts that are zero, but only to a certain precision.

When the imaginary part is the exact integer 0, the Wolfram System simplifies complex numbers to real ones:

Here the imaginary part is only zero to a certain precision, so the Wolfram System retains the complex number form:

The distinction between complex numbers whose imaginary parts are exactly zero, or are only zero to a certain precision, may seem like a pedantic one. However, for example, when the interpretation of powers and roots of complex numbers is discussed in "Functions That Do Not Have Unique Values", the distinction becomes significant.

One way to find out the type of a number in Wolfram System is just to pick out its head using Head[expr]. For many purposes, however, it is better to use functions like IntegerQ that explicitly test for particular types. Functions like this are set up to return True if their argument is manifestly of the required type, and to return False otherwise. As a result, IntegerQ[x] will give False unless x has an explicit integer value.

Complex Numbers

You can enter complex numbers in the Wolfram Language just by including the constant I, equal to

. Make sure that you type a capital I.

If you are using notebooks, you can also enter I as

by typing EsciiEsc (see "Mathematical Notation in Notebooks: Numerical Calculations"). The form

is normally what is used in output. Note that an ordinary i means a variable named

, not

This gives the imaginary number result

This gives the ratio of two complex numbers:

Here is the numerical value of a complex exponential:

x+I y	the complex number
Re[z]	real part
Im[z]	imaginary part
Conjugate[z]	complex conjugate or
Abs[z]	absolute value
Arg[z]	the argument in

Complex number operations.

Numeric Quantities

NumberQ[expr]	test whether expr is explicitly a number
NumericQ[expr]	test whether expr has a numerical value

Testing for numeric quantities.

Pi is a symbol, so Pi+3 is not explicitly a number:

It does however have a numerical value:

This finds the explicit numerical value of Pi+3:

The Wolfram Language knows that constants such as Pi are numeric quantities. It also knows that standard mathematical functions such as Log and Sin have numerical values when their arguments are numerical.

Log[2+x] contains x, and is therefore not a numeric quantity:

Many functions implicitly use the numerical values of numeric quantities:

In general, the Wolfram Language assumes that any function which has the attribute NumericFunction will yield numerical values when its arguments are numerical. All standard mathematical functions in the Wolfram Language already have this attribute. But when you define your own functions, you can explicitly set the attribute to tell the Wolfram Language to assume that these functions will have numerical values when their arguments are numerical.

Digits in Numbers

IntegerDigits[n]	a list of the decimal digits in the integer n
IntegerDigits[n,b]	the digits of n in base b
IntegerDigits[n,b,len]	the list of digits padded on the left with zeros to give total length len
IntegerLength[n]	the number of decimal digits in n
IntegerLength[n,b]	the number of base b digits in n
IntegerExponent[n,b]	the number of zeros at the end of n in base b
RealDigits[x]	a list of the decimal digits in the approximate real number x, together with the number of digits to the left of the decimal point
RealDigits[x,b]	the digits of x in base b
RealDigits[x,b,len]	the first len digits of x in base b
RealDigits[x,b,len,n]	the first len digits starting with the coefficient of bⁿ
FromDigits[list]	construct a number from its decimal digit sequence
FromDigits[list,b]	construct a number from its digit sequence in base b
FromDigits["string"]	construct an integer from a string of digits
FromDigits["string",b]	construct an integer from a string of digits in base b
IntegerString[n]	a string of the decimal digits in the integer n
IntegerString[n,b]	a string of the digits of n in base b

Converting between numbers and lists or strings of digits.

Here is the list of base 16 digits for an integer:

This gives a list of digits, together with the number of digits that appear to the left of the decimal point:

Here is the binary digit sequence for 56, padded with zeros so that it is of total length 8:

This reconstructs the original number from its binary digit sequence:

Here is 56 as a binary string:

This reconstructs the original number again:

b^^nnnn	a number in base b
BaseForm[x,b]	print with x in base b
IntegerString[n,b]	a string representing n in base b

Numbers in other bases.

When the base is larger than 10, extra digits are represented by letters a–z.

The number

in base 2 is

in base 10:

This prints 37 in base 2:

This gives the base 2 representation as a string:

Here is a number in base 16:

You can do computations with numbers in base 16. Here the result is given in base 10:

This gives the result in base 16:

You can give approximate real numbers, as well as integers, in other bases:

Here are the first few digits of

in octal:

This gives an explicit list of the first 15 octal digits:

This gives 15 octal digits starting with the coefficient of

"Output Formats for Numbers" describes how to print numbers in various formats. If you want to create your own formats, you will often need to use MantissaExponent to separate the pieces of real numbers.

MantissaExponent[x]	give a list containing the mantissa and exponent of x
MantissaExponent[x,b]	give the mantissa and exponent in base b

Separating the mantissa and exponent of numbers.

This gives a list in which the mantissa and exponent of the number are separated:

Exact and Approximate Results

A standard electronic calculator does all your calculations to a particular accuracy, say ten decimal digits. With the Wolfram Language, however, you can often get exact results.

The Wolfram Language gives an exact result for

, even though it has 31 decimal digits:

You can tell the Wolfram Language to give you an approximate numerical result, just as a calculator would, by ending your input with //N. The N stands for "numerical". It must be a capital letter. "Special Ways to Input Expressions" will explain what the // means.

This gives an approximate numerical result:

The Wolfram Language can give results in terms of rational numbers:

//N always gives the approximate numerical result:

expr//N

give an approximate numerical value for expr

Getting numerical approximations.

When you type in an integer like 7, the Wolfram Language assumes that it is exact. If you type in a number like 4.5, with an explicit decimal point, the Wolfram Language assumes that it is accurate only to a fixed number of decimal places.

This is taken to be an exact rational number, and reduced to its lowest terms:

Whenever you give a number with an explicit decimal point, the Wolfram Language produces an approximate numerical result:

Here again, the presence of the decimal point makes the Wolfram Language give you an approximate numerical result:

When any number in an arithmetic expression is given with an explicit decimal point, you get an approximate numerical result for the whole expression:

Numerical Precision

As discussed in "Exact and Approximate Results", the Wolfram Language can handle approximate real numbers with any number of digits. In general, the precision of an approximate real number is the effective number of decimal digits in it which are treated as significant for computations. The accuracy is the effective number of these digits which appear to the right of the decimal point. Note that to achieve full consistency in the treatment of numbers, precision and accuracy often have values that do not correspond to integer numbers of digits.

Precision[x]	the total number of significant decimal digits in x
Accuracy[x]	the number of significant decimal digits to the right of the decimal point in x

Precision and accuracy of real numbers.

This generates a number with 30‐digit precision:

This gives the precision of the number:

The accuracy is lower since only some of the digits are to the right of the decimal point:

This number has all its digits to the right of the decimal point:

Now the accuracy is larger than the precision:

An approximate real number always has some uncertainty in its value, associated with digits beyond those known. One can think of precision as providing a measure of the relative size of this uncertainty. Accuracy gives a measure of the absolute size of the uncertainty.

The Wolfram Language is set up so that if a number

has uncertainty

, then its true value can lie anywhere in an interval of size

from

. An approximate number with accuracy

is defined to have uncertainty

, while a nonzero approximate number with precision

is defined to have uncertainty

Precision[x]
Accuracy[x]

Definitions of precision and accuracy in terms of uncertainty.

Adding or subtracting a quantity smaller than the uncertainty has no visible effect:

N[expr,n]	evaluate expr to n‐digit precision using arbitrary‐precision numbers
N[expr]	evaluate expr numerically using machine‐precision numbers

Numerical evaluation with arbitrary‐precision and machine‐precision numbers.

The Wolfram Language distinguishes two kinds of approximate real numbers: arbitrary‐precision numbers, and machine‐precision numbers or machine numbers. Arbitrary‐precision numbers can contain any number of digits, and maintain information on their precision. Machine numbers, on the other hand, always contain the same number of digits, and maintain no information on their precision.

Here is a machine‐number approximation to

These are both arbitrary‐precision numbers:

As discussed in more detail below, machine numbers work by making direct use of the numerical capabilities of your underlying computer system. As a result, computations with them can often be done more quickly. They are however much less flexible than arbitrary‐precision numbers, and difficult numerical analysis can be needed to determine whether results obtained with them are correct.

MachinePrecision	the precision specification used to indicate machine numbers
$MachinePrecision	the effective precision for machine numbers on your computer system
MachineNumberQ[x]	test whether x is a machine number

Machine numbers.

This returns the symbol MachinePrecision to indicate a machine number:

On this computer, machine numbers have slightly less than 16 decimal digits:

When you enter an approximate real number, the Wolfram Language has to decide whether to treat it as a machine number or an arbitrary‐precision number. Unless you specify otherwise, if you give fewer than Ceiling[$MachinePrecision+1] digits, the Wolfram Language will treat the number as machine precision, and if you give more digits, it will treat the number as arbitrary precision.

123.4	a machine‐precision number
123.45678901234567890	an arbitrary‐precision number on some computer systems
123.45678901234567890`	a machine‐precision number on all computer systems
123.456`200	an arbitrary‐precision number with 200 digits of precision
123.456``200	an arbitrary‐precision number with 200 digits of accuracy
1.234*^6	a machine‐precision number in scientific notation (1.234×10⁶)
1.234`200*^6	a number in scientific notation with 200 digits of precision
2^^101.111`200	a number in base 2 with 200 binary digits of precision
2^^101.111`200*^6	a number in base 2 scientific notation (101.111₂×2⁶)

Input forms for numbers.

When the Wolfram Language prints out numbers, it usually tries to give them in a form that will be as easy as possible to read. But if you want to take numbers that are printed out by the Wolfram Language, and then later use them as input to the Wolfram Language, you need to make sure that no information gets lost.

In standard output form, the Wolfram Language prints a number like this to six digits:

In input form, the Wolfram Language prints all the digits it knows:

Here is an arbitrary‐precision number in standard output form:

In input form, the Wolfram Language explicitly indicates the precision of the number, and gives extra digits to make sure the number can be reconstructed correctly:

This makes the Wolfram Language not explicitly indicate precision:

InputForm[expr,NumberMarks->True]	use ` marks in all approximate numbers
InputForm[expr,NumberMarks->Automatic]
	use ` only in arbitrary‐precision numbers
InputForm[expr,NumberMarks->False]	never use ` marks

Controlling printing of numbers.

The default setting for the NumberMarks option, both in InputForm and in functions such as ToString and OpenWrite is given by the value of $NumberMarks. By resetting $NumberMarks, therefore, you can globally change the way that numbers are printed in InputForm.

This makes the Wolfram Language by default always include number marks in input form:

Even a machine‐precision number is now printed with an explicit number mark:

Even with no number marks, InputForm still uses *^ for scientific notation:

In doing numerical computations, it is inevitable that you will sometimes end up with results that are less precise than you want. Particularly when you get numerical results that are very close to zero, you may well want to assume that the results should be exactly zero. The function Chop allows you to replace approximate real numbers that are close to zero by the exact integer 0.

Chop[expr]	replace all approximate real numbers in expr with magnitude less than 10^-10 by 0
Chop[expr,dx]	replace numbers with magnitude less than dx by 0

Removing numbers close to zero.

This computation gives a small imaginary part:

You can get rid of the imaginary part using Chop:

Arbitrary‐Precision Calculations

When you use //N to get a numerical result, the Wolfram Language does what a standard calculator would do: it gives you a result to a fixed number of significant figures. You can also tell the Wolfram Language exactly how many significant figures to keep in a particular calculation. This allows you to get numerical results in the Wolfram Language to any degree of precision.

expr//N or N[expr]	approximate numerical value of expr
N[expr,n]	numerical value of expr calculated with n‐digit precision

Numerical evaluation functions.

This gives the numerical value of

to a fixed number of significant digits. Typing N[Pi] is exactly equivalent to Pi//N:

This gives

to 40 digits:

Here is

to 30 digits:

Doing any kind of numerical calculation can introduce small roundoff errors into your results. When you increase the numerical precision, these errors typically become correspondingly smaller. Making sure that you get the same answer when you increase numerical precision is often a good way to check your results.

The quantity

turns out to be very close to an integer. To check that the result is not, in fact, an integer, you have to use sufficient numerical precision:

Arbitrary‐Precision Numbers

When you do calculations with arbitrary‐precision numbers, the Wolfram Language keeps track of precision at all points. In general, the Wolfram Language tries to give you results which have the highest possible precision, given the precision of the input you provided.

The Wolfram Language treats arbitrary‐precision numbers as representing the values of quantities where a certain number of digits are known, and the rest are unknown. In general, an arbitrary‐precision number x is taken to have Precision[x] digits which are known exactly, followed by an infinite number of digits which are completely unknown.

This computes

to 10‐digit precision:

After a certain point, all digits are indeterminate:

When you do a computation, the Wolfram Language keeps track of which digits in your result could be affected by unknown digits in your input. It sets the precision of your result so that no affected digits are ever included. This procedure ensures that all digits returned by the Wolfram Language are correct, whatever the values of the unknown digits may be.

This evaluates

to 30‐digit precision:

The result has a precision of exactly 30 digits:

If you give input only to a few digits of precision, the Wolfram Language cannot give you such high‐precision output:

If you want the Wolfram Language to assume that the argument is exactly 142/1000, then you have to say so explicitly:

In many computations, the precision of the results you get progressively degrades as a result of "roundoff error". A typical case of this occurs if you subtract two numbers that are close together. The result you get depends on high‐order digits in each number, and typically has far fewer digits of precision than either of the original numbers.

Both input numbers have a precision of around 20 digits, but the result has much lower precision:

Adding extra digits in one number but not the other is not sufficient to allow extra digits to be found in the result:

The precision of the output from a function can depend in a complicated way on the precision of the input. Functions that vary rapidly typically give less precise output, since the variation of the output associated with uncertainties in the input is larger. Functions that are close to constants can actually give output that is more precise than their input.

Here is a case where the output is less precise than the input:

Here is

evaluated to 20‐digit precision:

The result you get by adding the exact integer 1 has a higher precision:

It is worth realizing that different ways of doing the same calculation can end up giving you results with very different precisions. Typically, if you once lose precision in a calculation, it is essentially impossible to regain it; in losing precision, you are effectively losing information about your result.

Here is a 40‐digit number that is close to 1:

Adding 1 to it gives another 40‐digit number:

The original precision has been maintained:

This way of computing 1+x loses precision:

The result obtained in this way has quite low precision:

The fact that different ways of doing the same calculation can give you different numerical answers means, among other things, that comparisons between approximate real numbers must be treated with care. In testing whether two real numbers are "equal", the Wolfram Language effectively finds their difference, and tests whether the result is "consistent with zero" to the precision given.

These numbers are equal to the precision given:

The internal algorithms that the Wolfram Language uses to evaluate mathematical functions are set up to maintain as much precision as possible. In most cases, built‐in Wolfram Language functions will give you results that have as much precision as can be justified on the basis of your input. In some cases, however, it is simply impractical to do this, and the Wolfram Language will give you results that have lower precision. If you give higher‐precision input, the Wolfram Language will use higher precision in its internal calculations, and you will usually be able to get a higher‐precision result.

N[expr]	evaluate expr numerically to machine precision
N[expr,n]	evaluate expr numerically trying to get a result with n digits of precision

Numerical evaluation.

If you start with an expression that contains only integers and other exact numeric quantities, then N[expr,n] will in almost all cases succeed in giving you a result to n digits of precision. You should realize, however, that to do this the Wolfram Language sometimes has to perform internal intermediate calculations to much higher precision.

The global variable $MaxExtraPrecision specifies how many additional digits should be allowed in such intermediate calculations.

variable	default value
$MaxExtraPrecision	50	maximum additional precision to use

Controlling precision in intermediate calculations.

The Wolfram Language automatically increases the precision that it uses internally in order to get the correct answer here:

Using the default setting $MaxExtraPrecision=50, the Wolfram Language cannot get the correct answer here:

This tells the Wolfram Language that it can use more digits in its internal calculations:

Now it gets the correct answer:

This resets $MaxExtraPrecision to its default value:

Even when you are doing computations that give exact results, the Wolfram Language still occasionally uses approximate numbers for some of its internal calculations, so that the value of $MaxExtraPrecision can thus have an effect.

The Wolfram Language works this out using bounds from approximate numbers:

With the default value of $MaxExtraPrecision, the Wolfram Language cannot work this out:

Temporarily resetting $MaxExtraPrecision allows the Wolfram Language to get the result:

In doing calculations that degrade precision, it is possible to end up with numbers that have no significant digits at all. But even in such cases, the Wolfram Language still maintains information on the accuracy of the numbers. Given a number with no significant digits, but accuracy a, the Wolfram Language can then still tell that the actual value of the number must be in the range {-10^-a,+10^-a}/2. The Wolfram Language by default prints such numbers in the form 0.×10^e.

Here is a number with 20‐digit precision:

Here there are no significant digits left:

But the Wolfram Language still keeps track of the accuracy of the result:

Adding the result to an exact 1 gives a number with quite high precision:

One subtlety in characterizing numbers by their precision is that any number that is consistent with zero must be treated as having zero precision. The reason for this is that such a number has no digits that can be recognized as significant, since all its known digits are just zero.

This gives a number whose value is consistent with zero:

The number has no recognizable significant digits of precision:

But it still has a definite accuracy that characterizes the uncertainty in it:

If you do computations whose results are likely to be near zero, it can be convenient to specify the accuracy, rather than the precision, that you want to get.

N[expr,p]	evaluate expr to precision p
N[expr,{p,a}]	evaluate expr to at most precision p and accuracy a
N[expr,{Infinity,a}]	evaluate expr to any precision but to accuracy a

Specifying accuracy as well as precision.

Here is a symbolic expression:

This shows that the expression is equivalent to zero:

N cannot guarantee to get a result to precision 20:

But it can get a result to accuracy 20:

When the Wolfram Language works out the potential effect of unknown digits in arbitrary‐precision numbers, it assumes by default that these digits are completely independent in different numbers. While this assumption will never yield too high a precision in a result, it may lead to unnecessary loss of precision.

In particular, if two numbers are generated in the same way in a computation, some of their unknown digits may be equal. Then, when these numbers are, for example, subtracted, the unknown digits may cancel. By assuming that the unknown digits are always independent, however, the Wolfram Language will miss such cancellations.

Here is a number computed to 20‐digit precision:

The quantity 1+d has about 34‐digit precision:

This quantity has lower precision, since the Wolfram Language assumes that the unknown digits in each number d are independent:

Numerical algorithms sometimes rely on cancellations between unknown digits in different numbers yielding results of higher precision. If you can be sure that certain unknown digits will eventually cancel, then you can explicitly introduce fixed digits in place of the unknown ones. You can carry these fixed digits through your computation, then let them cancel, and get a result of higher precision.

SetPrecision[x,n]	create a number with n decimal digits of precision, padding with base‐2 zeros if necessary
SetAccuracy[x,n]	create a number with n decimal digits of accuracy

Functions for modifying precision and accuracy.

This introduces 10 more digits in d:

The digits that were added cancel out here:

The precision of the result is now about 44 digits, rather than 34:

SetPrecision works by adding digits which are zero in base 2. Sometimes, the Wolfram Language stores slightly more digits in an arbitrary‐precision number than it displays, and in such cases, SetPrecision will use these extra digits before introducing zeros.

This creates a number with a precision of 40 decimal digits. The extra digits come from conversion to base 10:

variable	default value
$MaxPrecision	Infinity	maximum total precision to be used
$MinPrecision	0	minimum precision to be used

Global precision control parameters.

By making the global assignment $MinPrecision=n, you can effectively apply SetPrecision[expr,n] at every step in a computation. This means that even when the number of correct digits in an arbitrary‐precision number drops below n, the number will always be padded to have n digits.

If you set $MaxPrecision=n as well as $MinPrecision=n, then you can force all arbitrary‐precision numbers to have a fixed precision of n digits. In effect, what this does is to make the Wolfram Language treat arbitrary‐precision numbers in much the same way as it treats machine numbers—but with more digits of precision.

Fixed‐precision computation can make some calculations more efficient, but without careful analysis you can never be sure how many digits are correct in the results you get.

Here is a small number with 20‐digit precision:

With the Wolfram Language’s usual arithmetic, this works fine:

This tells the Wolfram Language to use fixed‐precision arithmetic:

The first few digits are correct, but the rest are wrong:

Restore the default value of $MinPrecision and $MaxPrecision:

Machine‐Precision Numbers

Whenever machine‐precision numbers appear in a calculation, the whole calculation is typically done in machine precision. The Wolfram Language will then give machine‐precision numbers as the result.

Whenever the input contains any machine‐precision numbers, the Wolfram Language does the computation to machine precision:

Zeta[5.6] yields a machine‐precision result, so the N is irrelevant:

This gives a higher‐precision result:

When you do calculations with arbitrary‐precision numbers, as discussed in "Arbitrary‐Precision Numbers", the Wolfram Language always keeps track of the precision of your results, and gives only those digits which are known to be correct, given the precision of your input. When you do calculations with machine‐precision numbers, however, the Wolfram Language always gives you a machine‐precision result, whether or not all the digits in the result can, in fact, be determined to be correct on the basis of your input.

This subtracts two machine‐precision numbers:

The result is taken to have machine precision:

Here are all the digits in the result:

The fact that you can get spurious digits in machine‐precision numerical calculations with the Wolfram Language is in many respects quite unsatisfactory. The ultimate reason, however, that the Wolfram Language uses fixed precision for these calculations is a matter of computational efficiency.

The Wolfram Language is usually set up to insulate you as much as possible from the details of the computer system you are using. In dealing with machine‐precision numbers, you would lose too much, however, if the Wolfram Language did not make use of some specific features of your computer.

The important point is that almost all computers have special hardware or microcode for doing floating‐point calculations to a particular fixed precision. The Wolfram Language makes use of these features when doing machine‐precision numerical calculations.

The typical arrangement is that all machine‐precision numbers in the Wolfram Language are represented as "double‐precision floating‐point numbers" in the underlying computer system. On most current computers, such numbers contain a total of 64 binary bits, typically yielding 16 decimal digits of mantissa.

The main advantage of using the built‐in floating‐point capabilities of your computer is speed. Arbitrary‐precision numerical calculations, which do not make such direct use of these capabilities, are usually many times slower than machine‐precision calculations.

There are several disadvantages of using built‐in floating‐point capabilities. One already mentioned is that it forces all numbers to have a fixed precision, independent of what precision can be justified for them.

A second disadvantage is that the treatment of machine‐precision numbers can vary slightly from one computer system to another. In working with machine‐precision numbers, the Wolfram Language is at the mercy of the floating‐point arithmetic system of each particular computer. If floating‐point arithmetic is done differently on two computers, you may get slightly different results for machine‐precision Wolfram Language calculations on those computers.

$MachinePrecision	the number of decimal digits of precision
$MachineEpsilon	the minimum positive machine‐precision number which can be added to 1.0 to give a result distinguishable from 1.0
$MaxMachineNumber	the maximum machine‐precision number
$MinMachineNumber	the minimum positive machine‐precision number that can be represented in normalized form
$MaxNumber	the maximum magnitude of an arbitrary‐precision number
$MinNumber	the minimum magnitude of a positive arbitrary‐precision number

Properties of numbers on a particular computer system.

Since machine‐precision numbers on any particular computer system are represented by a definite number of binary bits, numbers that are too close together will have the same bit pattern, and so cannot be distinguished. The parameter $MachineEpsilon gives the distance between 1.0 and the closest number that has a distinct binary representation.

This gives the value of $MachineEpsilon for the computer system on which these examples are run:

Although this prints as 1., the Wolfram Language knows that the result is larger than 1:

InputForm reveals that the result is not exactly 1:

Subtracting 1 gives $MachineEpsilon:

This prints as 1. also:

In this case, however, the result is not distinguished from 1. to machine precision:

Subtracting 1 from the result yields 0:

Machine numbers have not only limited precision, but also limited magnitude. If you generate a number that is larger in magnitude than $MaxMachineNumber, the Wolfram Language will automatically convert the number to arbitrary‐precision form.

This is the maximum machine‐precision number that can be handled on the computer system used for this example:

The Wolfram Language automatically converts any result larger than $MaxMachineNumber to arbitrary precision:

Here is another computation whose result is outside the range of machine-precision numbers:

As of Version 11.3, if a machine-precision computation generates a number that is smaller in magnitude than $MinMachineNumber, the Wolfram Language will issue a warning. However, it will retain the result as a subnormal machine number that has leading zeros in its binary significand.

This is the minimum machine‐precision number that can be put into normal form on the computer system used for this example:

The Wolfram Language issues warnings about computations that produce numbers smaller than $MinMachineNumber, but it does not automatically convert to arbitrary-precision arithmetic:

It is possible to lose all significant digits and end up with a machine zero answer:

Converting to arbitrary precision using SetPrecision can avoid this underflow:

Interval Arithmetic

Interval[{min,max}]	the interval from min to max
Interval[{min₁,max₁},{min₂,max₂},…]
	the union of intervals from min₁ to max₁, min₂ to max₂, …

Representations of real intervals.

This represents all numbers between

and

The square of any number between

and

is always between

and

Taking the reciprocal gives two distinct intervals:

Abs folds the intervals back together again:

You can use intervals in many kinds of functions:

Some functions automatically generate intervals:

IntervalUnion[interval₁,interval₂,…]	find the union of several intervals
IntervalIntersection[interval₁,interval₂,…]
	find the intersection of several intervals
IntervalMemberQ[interval,x]	test whether the point x lies within an interval
IntervalMemberQ[interval₁,interval₂]	test whether interval₂ lies completely within interval₁

Operations on intervals.

This finds the overlap of the two intervals:

You can use Max and Min to find the end points of intervals:

This finds out which of a list of intervals contains the point 7:

You can use intervals not only with exact quantities but also with approximate numbers. Even with machine‐precision numbers, the Wolfram Language always tries to do rounding in such a way as to preserve the validity of results.

This shows explicitly the interval treated by the Wolfram Language as the machine‐precision number 0.

This shows the corresponding interval around 100., shifted back to zero:

The same kind of thing works with numbers of any precision:

With ordinary machine‐precision arithmetic, this computation gives an incorrect result:

The interval generated here, however, includes the correct value of 0:

Indeterminate and Infinite Results

If you type in an expression like 0/0, the Wolfram Language prints a message, and returns the result Indeterminate:

An expression like 0/0 is an example of an indeterminate numerical result. If you type in 0/0, there is no way for the Wolfram Language to know what answer you want. If you got 0/0 by taking the limit of

, then you might want the answer 1. On the other hand, if you got 0/0 instead as the limit of

, then you probably want the answer 2. The expression 0/0 on its own does not contain enough information to choose between these and other cases. As a result, its value must be considered indeterminate.

Whenever an indeterminate result is produced in an arithmetic computation, the Wolfram Language prints a warning message, and then returns Indeterminate as the result of the computation. If you ever try to use Indeterminate in an arithmetic computation, you always get the result Indeterminate. A single indeterminate expression effectively "poisons" any arithmetic computation. (The symbol Indeterminate plays a role in the Wolfram Language similar to the "not a number" object in the IEEE Floating Point Standard.)

The usual laws of arithmetic simplification are suspended in the case of Indeterminate:

Indeterminate "poisons" any arithmetic computation, and leads to an indeterminate result:

When you do arithmetic computations inside Wolfram Language programs, it is often important to be able to tell whether indeterminate results were generated in the computations. You can do this by using the function Check discussed in "Messages" to test whether any warning messages associated with indeterminate results were produced.

You can use Check inside a program to test whether warning messages are generated in a computation:

Indeterminate	an indeterminate numerical result
Infinity	a positive infinite quantity
-Infinity	a negative infinite quantity ( DirectedInfinity[-1] )
DirectedInfinity[r]	an infinite quantity with complex direction r
ComplexInfinity	an infinite quantity with an undetermined direction
DirectedInfinity[]	equivalent to ComplexInfinity

Indeterminate and infinite quantities.

There are many situations where it is convenient to be able to do calculations with infinite quantities. The symbol Infinity in the Wolfram Language represents a positive infinite quantity. You can use it to specify such things as limits of sums and integrals. You can also do some arithmetic calculations with it.

Here is an integral with an infinite limit:

The Wolfram Language knows that

If you try to find the difference between two infinite quantities, you get an indeterminate result:

There are a number of subtle points that arise in handling infinite quantities. One of them concerns the "direction" of an infinite quantity. When you do an infinite integral, you typically think of performing the integration along a path in the complex plane that goes to infinity in some direction. In this case, it is important to distinguish different versions of infinity that correspond to different directions in the complex plane.

and

are two examples, but for some purposes one also needs

and so on.

In the Wolfram Language, infinite quantities can have a "direction", specified by a complex number. When you type in the symbol Infinity, representing a positive infinite quantity, this is converted internally to the form DirectedInfinity[1], which represents an infinite quantity in the

direction. Similarly, -Infinity becomes DirectedInfinity[-1], and I Infinity becomes DirectedInfinity[I]. Although the DirectedInfinity form is always used internally, the standard output format for DirectedInfinity[r] is r Infinity.

Infinity is converted internally to DirectedInfinity[1]:

Although the notion of a "directed infinity" is often useful, it is not always available. If you type in 1/0, you get an infinite result, but there is no way to determine the "direction" of the infinity. The Wolfram Language represents the result of 1/0 as DirectedInfinity[]. In standard output form, this undirected infinity is printed out as ComplexInfinity.

1/0 gives an undirected form of infinity:

Controlling Numerical Evaluation

NHoldAll	prevent any arguments of a function from being affected by N
NHoldFirst	prevent the first argument from being affected
NHoldRest	prevent all but the first argument from being affected

Attributes for controlling numerical evaluation.

Usually N goes inside functions and gets applied to each of their arguments:

This tells the Wolfram Language not to apply N to the first argument of f:

Now the first argument of f is left in its exact form:

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