RealDigits

✖
`RealDigits`

✖

RealDigits[x]

gives a list of the digits in the approximate real number x, together with the number of digits that are to the left of the decimal point.

✖

RealDigits[x,b]

gives a list of base‐b digits in x.

✖

RealDigits[x,b,len]

gives a list of len digits.

✖

RealDigits[x,b,len,n]

gives len digits starting with the coefficient of bⁿ.

Details

RealDigits gives the most significant digits first, as in standard positional notation.
RealDigits[x] normally returns a list of digits of length Round[Precision[x]].
RealDigits[x] and RealDigits[x,b] normally require that x be an approximate real number, returned for example by N. RealDigits[x,b,len] also works on exact numbers.
For integers and rational numbers with terminating digit expansions, RealDigits[x] returns an ordinary list of digits. For rational numbers with non‐terminating digit expansions, it yields a list of the form {a₁,a₂,…,{b₁,b₂,…}} representing the digit sequence consisting of the a_i followed by infinite cyclic repetitions of the b_i. »
If len is larger than Precision[x]/Log[10,b], then remaining digits are filled in as Indeterminate.
RealDigits[x,b,len,n] starts with the digit which is the coefficient of bⁿ, truncating or padding with zeros as necessary. »
RealDigits[x,b,len,-1] starts with the digit immediately to the right of the base‐b decimal point in x.
RealDigits[x,b,Automatic,n] gives as many digits as it can in a fixed-precision number.
The base b in RealDigits[x,b] need not be an integer. For any real b such that b>1, RealDigits[x,b] successively finds the largest integer multiples of powers of b that can be removed while leaving a non‐negative remainder.
RealDigits[x] discards the sign of x.
RealDigits[0.] gives {{0},-Floor[Accuracy[0.]]}.
FromDigits can be used as the inverse of RealDigits.

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Give the list of digits and exponent:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-m3r

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-cfd

Out[2]=2

Give 25 digits of in base 10:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-uu7

Out[1]=1

Give 25 digits of 19/7:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-e0i

Out[1]=1

Give an explicit recurring decimal form:

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-t5e

Out[2]=2

Scope (3)Survey of the scope of standard use cases

Base-2 digits:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-wd5

Out[1]=1

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-jan

Out[1]=1

20 digits starting with the coefficient of :

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-mof

Out[2]=2

20 digits starting with the coefficient of :

In[3]:=3

✖

https://wolfram.com/xid/0bdpp9kq6-yyg

Out[3]=3

Noninteger bases are allowed:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-e9

Out[1]=1

Generalizations & Extensions (2)Generalized and extended use cases

RealDigits gives Indeterminate if more digits than the precision are requested:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-ky7lli

Out[1]=1

Start at the coefficient of :

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-bgib90

Out[1]=1

Include only digits that are determined by the precision available:

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-fgrkxp

Out[2]=2

Applications (6)Sample problems that can be solved with this function

The 10000 digit of is an 8:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-w3o

Out[1]=1

Number of 1s in the first million base-2 digits of :

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-pge

Out[1]=1

Distribution of first 100000 digits of in base 47:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-c6pzme

Out[1]=1

Fibonacci representations of integers:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-l5u

Out[1]=1

Binary representation of a machine number:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-co587u

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-bbvkja

Out[2]=2

is equal to the number of bits times :

In[3]:=3

✖

https://wolfram.com/xid/0bdpp9kq6-evwl7o

Out[3]=3

Get the next larger machine number:

In[4]:=4

✖

https://wolfram.com/xid/0bdpp9kq6-ecjdda

Out[4]=4

The spacing between these numbers is 2^(e-1) $MachineEpsilon:

In[5]:=5

✖

https://wolfram.com/xid/0bdpp9kq6-ga784

Out[5]=5

Find the error in representing 1/10 with a machine number:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-c4lnzw

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-ivlpsc

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bdpp9kq6-k0835p

Out[3]=3

The next smaller machine number is farther away:

In[4]:=4

✖

https://wolfram.com/xid/0bdpp9kq6-b4uuy6

Out[4]=4

Properties & Relations (1)Properties of the function, and connections to other functions

The default number of digits returned is determined by the precision of the number:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-3ceqe

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-16mpx

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bdpp9kq6-edn1n2

Out[3]=3

Possible Issues (2)Common pitfalls and unexpected behavior

Digits unknown at the available precision are filled in as Indeterminate:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-ka5

Out[1]=1

For non-binary bases, the digits given may not be enough to reconstruct the number exactly:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-c5m99n

Out[1]=1

More than Round[MachinePrecision] decimal digits are required to separate x from 1:

In[2]:=2

✖

https://wolfram.com/xid/0bdpp9kq6-gs29vu

Out[2]=2

InputForm uses a sufficient number of digits to uniquely reconstruct the number:

In[3]:=3

✖

https://wolfram.com/xid/0bdpp9kq6-0f92l

In[4]:=4

✖

https://wolfram.com/xid/0bdpp9kq6-cm9vty

Out[4]=4

Neat Examples (1)Surprising or curious use cases

A hundred digits of starting with the millionth digit:

In[1]:=1

✖

https://wolfram.com/xid/0bdpp9kq6-pou7i7

Out[1]=1

Top