L1. Introduction to Bit Manipulation | 1's 2's Compliment | Bit Operators

Summary

The video introduces number representations and bitwise operators used in bit manipulation. It covers converting between decimal and binary, signed integers, and the operators AND, OR, XOR, shifts, and NOT.

Decimal–binary conversion

• Decimal to binary: Repeatedly divide the number by 2 and record each remainder. Read the remainders from bottom to top to get the binary representation.
  • Example: (13_{10} = 1101_2).
  • One suggested approach is to store the remainders in a string and reverse it at the end.
  • Complexity: (O(\log n)) time and space for a positive number (n).
• Binary to decimal: Starting at the rightmost bit, multiply each bit by (2^{\text{position}}), numbering the rightmost position 0, then add the results.
  • Example: (1101_2 = 1 \times 2^0 + 0 \times 2^1 + 1 \times 2^2 + 1 \times 2^3 = 13_{10}).
  • Complexity: (O(L)) time for a binary string of length (L), and (O(1)) extra space.

The presenter notes that binary and decimal conversion are the main conversions needed for the bit-manipulation problems covered later.

How integers are represented

Computers represent values internally with bits rather than storing their decimal-written form. The lesson uses a 32-bit integer as its working assumption and explains signed values using a sign bit and two’s-complement representation.

• One’s complement: Flip every bit.
• Two’s complement: Flip every bit, then add 1. This is the standard method described for forming a negative value in two’s-complement representation.

A signed 32-bit integer usually ranges from (-2^{31}) through (2^{31} - 1). The presenter notes that the same concepts apply to 64-bit integers, with a different width.

Bitwise operators

Bitwise operators act on corresponding bits in their operands:

• AND (&): A result bit is 1 only when both input bits are 1.
  • Example: (13 \& 7 = 5).
• OR (|): A result bit is 1 when at least one input bit is 1.
  • Example: (13 \mid 7 = 15).
• XOR (^): A result bit is 1 when the input bits differ. Equivalently, it is 1 when there is an odd number of 1s in that bit position.
  • Example: (13 \mathbin{\char94} 7 = 10).
• Right shift (>>): Moves bits to the right. For nonnegative values in the examples, shifting right by (k) places corresponds to integer division by (2^k).
  • Examples: (13 \gg 1 = 6), (13 \gg 2 = 3).
• Left shift (<<): Moves bits to the left. When no overflow occurs, shifting left by (k) places corresponds to multiplying by (2^k).
  • Example: (13 \ll 1 = 26).
  • Shifting can overflow the integer type’s representable range.
• NOT (~): Flips every bit. In the usual two’s-complement integer model, ~5 is (-6); the exact interpretation depends on the integer width and language.

Important implementation caveat

The video uses simplified bit diagrams and assumes 32-bit integers. In real code, integer width, signed representation, shift behavior, and overflow rules can vary by language. Signed values should be understood using the full fixed-width representation, not only the few low-order bits shown in an illustration. The decimal-to-binary loop should also handle 0 and 1 as special cases.

Speakers and sources featured

• Take U Forward presenter/instructor: Delivers the lesson; the subtitles do not identify the presenter by name.
• Unidentified outro vocal/music: A brief sung line is audible at the end; the subtitles do not identify its source.