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Shifts Obsidian::STEM binary
binary

Overview

Left shift operations (<<) drop the k most significant bits and fills the right end of the result with k zeros. Right shift operations (>>) are classified in two ways:

  • Logical
    • Drops the k least significant bits and fills the left end of the result with k zeros.
    • This mode is always used when calling >> on unsigned data.
    • Sometimes denoted as >>> to disambiguate from arithmetic right shifts.
  • Arithmetic
    • Drops the k least significant bits and fills the left end of the result with k copies of the most significant bit.
    • This mode is usually used when calling >> on signed data.

%%ANKI Basic How is 2^n written in binary? Back: As 1 followed by n zeros. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

END%%

%%ANKI Basic How is 2^n written using bitwise shift operators? Back: 1 << n Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tag: c

END%%

%%ANKI Basic What decimal value does 1 << n translate to? Back: 2^n Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tag: c

END%%

%%ANKI Basic What kinds of left shift operations are there? Back: Just logical. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

END%%

%%ANKI Basic How many most significant bits are dropped on a left shift by k (e.g. << k)? Back: k Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic How many zeros exist in the result of a left shift by k (e.g. << k)? Back: At least k. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic What kinds of right shift operations are there? Back: Logical and arithmetic Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

END%%

%%ANKI Basic What is a logical right shift operation? Back: One that fills the left end of the result with k zeros. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

END%%

%%ANKI Basic What is an arithmetic right shift operation? Back: One that fills the left end of the result with k copies of the most significant bit. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

END%%

%%ANKI Basic How many least significant bits are dropped on a right shift by k (e.g. >> k)? Back: k Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic How many zeros exist in the result of a logical right shift by k (e.g. >> k)? Back: At least k. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic How many zeros exist in the result of an arithmetic right shift by k (e.g. >> k)? Back: Zero or more. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic What kind of right shift operation is usually applied to signed numbers? Back: Arithmetic. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic What kind of right shift operation is applied to unsigned numbers? Back: Logical. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic What portability issue do shift operations introduce? Back: There is no standard on whether right shifts of signed numbers are logical or arithmetic. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Cloze {1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

In C, it is undefined behavior to shift by more than the width w of an integral type. Typically though, only the last w bits are considered in the computation. For example, given int32_t x, (x << 32) = (x << 0).

%%ANKI Basic According to the C standard, what is the result of shifting an int32_t value by 32? Back: It is undefined behavior. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic According to the C standard, what is the result of shifting an int32_t value by a negative value? Back: It is undefined behavior. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic Ignoring UB, what typically happens when shifting an int32_t by k ≥ 32 bits? Back: The shift value is interpreted as k mod 32. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

%%ANKI Basic How is x \bmod 2^k equivalently written as a bit mask? Back: x & (k - 1) Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c

END%%

Arithmetic

Shifting left by a single bit is equivalent to multiplication by 2. Likewise, logically shifting right is equivalent to floor division by 2. This is most clearly seen by examining the decimal expansion of a binary value. For example, consider $001101_2 = 2^3 + 2^2 + 2^0$

Multiplying by 2 yields $2 \cdot (2^3 + 2^2 + 2^0) = 2^4 + 2^3 + 2^1 = 011010_2$

Dividing by 2 yields (2^3 + 2^2 + 2^0) / 2 = 2^2 + 2^1 = 000110_2$$

%%ANKI Basic Multiplication by 2 is equivalent to what bitwise shift operation? Back: Left shifting by 1 bit. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic Floor division by 2 is equivalent to what bitwise shift operation? Back: Logical right shifting by 1 bit. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic Ceiling division by 2 is equivalent to what bitwise shift operation? Back: None. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Cloze {1:Multiplication} by 2 is to {2:left shifts} whereas {2:floor division} by 2 is to {1:logical right shifts}. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic What arithmetic operation is equivalent to shifting left by n bits? Back: Multiplication by 2^n. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic When is 2n + 1 equivalent to a one-bit cyclic shift left? Back: When n's leading bit is 1. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic What arithmetic operation is equivalent to a one-bit cyclic shift left? Back: Given n with no leading zeros, 2n + 1. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic What arithmetic operation is equivalent to logically shifting right by n bits? Back: Floor division by 2^n. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic Given n = 0110_2, what is the binary value of 2n? Back: 1100_2 Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic Given n = 0110_2, what is the binary value of 2n + 1? Back: 1101_2 Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

%%ANKI Basic Given n = 0110_2, what is the binary value of \lfloor n / 2 \rfloor? Back: 0011_2 Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

END%%

References

  • Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
  • Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).