Number Literal Semantics

This section describes the meaning of number literal forms, as distinct from their syntax.

Approximate Numbers (Real)

When a Number Literal isReal

A number literal representation will be interpreted as a real number if and only if any of the following are true:

  1. There is a point . in the mantissa, as in 3.2.
  2. The precision pseudo-operator ` appears in the number literal, regardless of whether or not a numerical precision is given.
  3. The accuracy pseudo-operator `` appears in the number literal.

Note that a real number cannot be produced by virtue of the use of the scientific form pseudo-operator *^. See Scientific Form below.

Precision and Accuracy

A Real number is not a real number in the mathematical sense. Rather, a Real number is what is called a floating point number in other languages. The precision and accuracy of a Real number are functionally equivalent notions that can be understood as follows. Suppose \(x\) is a Real number which represents a real number \(\hat{x}\in\mathbb{R}\) up to an error of \(\delta_x\), that is, the true value \(\hat{x}\) of the number represented by \(x\) lies in the interval \((x - \delta_x/2, x + \delta_x/2)\).

  • If a Real number \(x\) has \(p\) digits in base \(b\) of precision, then \(\delta_x = |x|b^{-p}\).
  • Likewise, if a Real number \(x\) has accuracy \(a\) in base \(b\), then \(\delta_x = b^{-a}\).

It is clear from the above that precision depends on the magnitude of the number \(x\), while accuracy does not. Indeed, we have \(p = a + \log_{b}|x|\).

If the precision (or accuracy) of a number literal given with explicit base \(b\) is provided, then that precision (repectively accuracy) is interpreted to be the number of digits in the given base \(b\) of precision (respectively accuracy).

A number written without an explicit base has base \(10\). Therefore, whenever a number literal is written without ^^, the exponent represents decimal digits of precision/accuracy, that is, the number of digits in base \(10\).


The number of digits of precision/accuracy in base \(10\) is only approximate in general on a digital computer. This is why most other numerical computing systems only use base 2 corresponding to bits of precision/accuracy. If we desire \(b\) bits of accuracy, then we need \(b/\log_2(10)\) decimal digits of accuracy. Said another way, every decimal digit of accuracy requires \(\log_2(10) \approx 3.32193\) bits of accuracy.

Compatibility Warning

Syntactically, only undecorated decimal numbers can follow the *^ pseudo-operator, which is inconvenient if one wishes to represent a number in base \(10\) (or other nonbinary bases) having a precise number of bits of precision/accuracy.

A number decorated with the precision pseudo-operator ` that is not immediately followed by an explicit precision is interpreted as having $MachinePrecision, regardless of whether or not a point . appears in the mantissa, and regardless of how many digits are given explicitly in the number's representation.


Mathematica defines a $MachinePrecision Real number as a double precision floating point number ("double") as defined by the host platform. On systems with 64-bit IEEE doubles, the IEEE standard reserves 53 bits to the mantissa, which is 53 bits of accuracy or about 15.9546 decimal digits of accuracy.

Scientific Form

The pseudo-operator *^ multiplies the given digits by a power of the base. For a number literal of the form b^^m*^p with base b, mantissa m, and power p (which must be an integer), *^p has the effect of multiplying b^^m by b^p (b raised to the p power). If an explicit base is not given using ^^, then the base is \(10\) and *^p has the effect of multiplying the number by 10^p.

The type of the number literal is determined according to the following rules:

  1. If the number would be an Integer without the exponent *^p, then the number with the exponent *^p is either an Integer or a Rational according to whether the number is divisible by b^p.
  2. If the number would be a Real number without the exponent *^p, then the number with the exponent is a Real number.

The pseudo-operator *^ has no effect on the accuracy of a number literal.