num2bin

Expression of all numbers in a variety of binary forms.

num2bin is a tool that performs extraction of the bits of a number in MATLAB. It returns a struct that contains multiple fields, which show the number in all forms.

Single and double precision numbers are resolved, as well as negative numbers.

num2bin(single(-0.1))
ans =
      struct with fields:
  
                Class: 'single'
                 Sign: -1
             Exponent: -4
             Mantissa: '110011001100110011001101'
      BinaryExpansion: [-4 -5 -8 -9 -12 -13 -16 -17 -20 -21 -24 -25 -27]
                BiSci: '-1.10011001100110011001101 B-4'
              BiCimal: '-0.000110011001100110011001101'
  
  num2bin(1.23456789e6)
  ans = 
    struct with fields:
  
                Class: 'double'
                 Sign: 1
             Exponent: 20
             Mantissa: '10010110101101000011111100011110101110000101000111101'
      BinaryExpansion: [20 17 15 14 12 10 9 7 2 1 0 -1 -2 -3 -7 -8 -9 -10 -12 -14 -15 -16 -21 -23 -27 -28 -29 -30 -32]
                BiSci: '1.0010110101101000011111100011110101110000101000111101 B20'
              BiCimal: '100101101011010000111.11100011110101110000101000111101'
    
  

Integer classes are properly handled: uint8, uint16, uint32, uint64 as well as int8, int16, int32, int64, even logical.

  num2bin(uint32(123456))
  ans = 
    struct with fields:
  
                Class: 'uint32'
                 Sign: 1
             Mantissa: '11110001001000000'
             Exponent: 16
      BinaryExpansion: [16 15 14 13 9 6]
                BiSci: '1.1110001001000000 B16'
              BiCimal: '11110001001000000'

Zero has a sign of 0.

num2bin(false)
  ans = 
    struct with fields:
  
                Class: 'logical'
                 Sign: 0
             Exponent: 0
             Mantissa: '0'
      BinaryExpansion: []
                BiSci: '0'
              BiCimal: '0'  

eps

num2bin(eps)
  ans = 
    struct with fields:
  
                Class: 'double'
                 Sign: 1
             Exponent: -52
             Mantissa: '10000000000000000000000000000000000000000000000000000'
      BinaryExpansion: -52
                BiSci: '1.0000000000000000000000000000000000000000000000000000 B-52'
              BiCimal: '0.00000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000'
  

Denormal numbers, thus smaller than realmin, but not quite small enough to underflow into a zero are extracted. The smallest such number representable as a double would be realmin/2^52. Beyond that point we would see an underflow to zero.

num2bin(realmin/2^52)
  ans =
    struct with fields:
  
                Class: 'double'
                 Sign: 1
             Exponent: -1023
             Mantissa: '00000000000000000000000000000000000000000000000000001'
      BinaryExpansion: -1074
                BiSci: '0.0000000000000000000000000000000000000000000000000001 B-1022'
              BiCimal: '0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001'  

Exact powers of 2 are representable exactly.

num2bin(2^-100)
  ans = 
    struct with fields:
  
                Class: 'double'
                 Sign: 1
             Exponent: -100
             Mantissa: '10000000000000000000000000000000000000000000000000000'
      BinaryExpansion: -100
                BiSci: '1.0000000000000000000000000000000000000000000000000000 B-100'
              BiCimal: '0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000'

Note that 1/3 is a repeating decimal in base 10, but also a repeating decimal in base 2. In base 2, 1/3 repeats the binary "digits": '01' forever.

num2bin(1/3)
  ans = 
    struct with fields:
  
                Class: 'double'
                 Sign: 1
             Exponent: -2
             Mantissa: '10101010101010101010101010101010101010101010101010101'
      BinaryExpansion: [-2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54]
                BiSci: '1.0101010101010101010101010101010101010101010101010101 B-2'
              BiCimal: '0.010101010101010101010101010101010101010101010101010101'  

Similarly, 1/5 has a finite expansion in base 10 (thus 0.2), yet is infinitely repeating in base 2.

num2bin(1/5)
  ans = 
    struct with fields:
  
                Class: 'double'
                 Sign: 1
             Exponent: -3
             Mantissa: '11001100110011001100110011001100110011001100110011010'
      BinaryExpansion: [-3 -4 -7 -8 -11 -12 -15 -16 -19 -20 -23 -24 -27 -28 -31 -32 -35 -36 -39 -40 -43 -44 -47 -48 -51 -52 -54]
                BiSci: '1.1001100110011001100110011001100110011001100110011010 B-3'
              BiCimal: '0.0011001100110011001100110011001100110011001100110011010'

inf - an overflow, as a single, stored as 2^128. As a double, effectively 2^1024.

num2bin(single(-inf))
  ans = 
    struct with fields:
  
                Class: 'single'
                 Sign: -1
             Exponent: 128
             Mantissa: '100000000000000000000000'
      BinaryExpansion: 128
                BiSci: '-1.00000000000000000000000 B128'
              BiCimal: '-1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000'

NaN? what do you expect to see in this case? It is not a number.

num2bin(NaN)
  ans = 
    struct with fields:
  
                Class: 'double'
                 Sign: NaN
             Exponent: NaN
             Mantissa: '1000000000000000000000000000000000000000000000000000'
      BinaryExpansion: 'NaN'
                BiSci: 'NaN'
              BiCimal: 'NaN'