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Least Significant Bit

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LeastSignificantBit

The least significant bit is the rightmost bit in a binary representation of a nonnegative integer, with place value 2^0. For the sequence of numbers 1, 2, 3, 4, ..., the least significant bits are therefore the alternating sequence 1, 0, 1, 0, 1, 0, ... (OEIS A000035). It can be represented as

a_n=1/2[1-(-1)^n]
(1)
=1/2(-1)^n[-1+(-1)^n]
(2)

or

a_n=n (mod 2).
(3)

It is also given by the linear recurrence equation

a_n=1-a_(n-1)
(4)

for n>=2, with a_1=1 (Wolfram 2002, p. 128).

Analogously, the "most significant bit" is the leftmost bit in an n-bit representation, with place value 2^(n-1).

The least significant bit has Lambert series

sum_(n=1)^inftylsb(n)(x^n)/(1-x^n)=(ln(1-x^2)+psi_(x^2)(1/2))/(ln(x^2))
(5)

for 0<x<1, where psi_q(x) is a q-polygamma function.

See also

Binary, Bit, Significant Digits

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References

Sloane, N. J. A. Sequence A000035/M0001 in "The On-Line Encyclopedia of Integer Sequences."Whitford, A. K. "Binet's Formula Generalized." Fib. Quart. 15, 21, 24, and 29, 1977.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 128, 2002.

Referenced on Wolfram|Alpha

Least Significant Bit

Cite this as:

Weisstein, Eric W. "Least Significant Bit." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LeastSignificantBit.html

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