Bits and Bytes

What is a bit

Bit is one binary digit which is the smallest unit of information on a machine.

0 or 1 is one bit.

Distinct Values

Number of distinct values that can be represented by a n-bit is calculated by 2^(n-bit)

0 bit does not exist, but represents distinct value `0` (2^0 = 1)
1 bit | 2^1 = 2 | 2 distinct values (0, 1)
2 bit | 2^2 = 4 | 4 distinct values (00, 01, 10, 11)
4 bit | 2^4 = 16 | 16 distinct values
8 bit | 2^8 = 256 | 256 distinct values (1 byte)
16 bit | 2^16 = 65,536 | 65,536 distinct values (2 bytes)

Max Value

Max value a bit can represent is 2^n-1 because n=1 bit is reserved for 0 representation

What is n-bit

When we see common terms 8-bit, 16-bit, 32-bit and 64-bit all refer to a processor's word size

word is a native size of information processor can place into a register and process without special instructions

  • also refers to the size of the memory address space

The word size of any chip is the most defining aspect of it's design

What is a word size

Word size refers to the amount of data a CPU's internal data registers can hold and process at one time

  • size of the internal functional units in the CPU itself

The difference in word size has a dramatic impact on the capabilities and performance of a given chip

  • Once you get up to 32-bits, the differences mainly become those of refinement (unless you are running a really big application, like genetic analysis or counting all the stars in the galaxy big)

What is Bit-Depth

Bit-depth is determined by the number of bits used to define each pixel

  • The greater the bit depth, the greater the number of tones (grayscale or color) that can be represented on the screen

Common bit Sizes

4-bit = 1-nibble (one hexadecimal)

Can represent 16 (2^4) different values.

Decimal representation from 0 to 15.

  • 16-1 since "distinct values" includes 0.

Written in binary as 0000 (0) to 1111 (15)

  • decimal value: 15

Represents one Hexadecimal character

8-bit (1-byte)

Can represent 256 (2^8) distinct values

Decimal representation from 0 to 255

  • 256-1 since "distinct values" includes 0

Written in binary as 0000 0000 (0) to 1111 1111 (decimal value: 255)

  • 128 (2^7) + 64 (2^6) + 32 (2^5) + 16 (2^4) + 8 (2^3) + 4 (2^2) + 2 (2^1) + 1 (2^0) = 256
  • those are 8 bits or binary digits

16-bit (2-bytes)

Can represent 65,536 distinct values.

Can represent decimal value from 0 to 65,535

  • 2^16 - 1 = 65,536 - 1 = 65,535

32-bit (4-bytes)

Can represent decimal value from 0 to 4,294,967,295

  • 2^32 - 1 = 4,294,967,296 - 1 = 4,294,967,295

64-bit (8 bytes)

Signed 64-bit

Max size: 19-digit number

Bit of highest significance is reserved for the sign

  • If this bit is 1, the number is negative

Because the last digit can represent 0 and first digit always represents signedness, it is possible to have both -0 and +0 in decimal representation

  • In ordinary arithmetic, this doesn't matter. But some particular operations could have different behaviors.

Which means the decimal representation can go as low as -2^63 and as high as 2^63 - 1:

  • -9,223,372,036,854,775,807
    • (sign reserved for first digit)
  • 9,223,372,036,854,775,807
    • (1 reserved for last digit)

Unsigned 64-bit

Max size: 20-digit number

  • can go as low as 0 and as high as 2^64 - 1.
  • unsigned integers cannot present negative values
    • technicality it is possible to represent the numeric aspect of a quantity with uint64 and represent it's positive/negative quality in a separate bool

Number from 0 to 18,446,744,073,709,551,615

Why subtract the one -1

Since the computer has to store the number 0 in an unsigned int, it is actually starting to count with 0, then 1 and so on

  • That means that the n-th number for the computer is, in fact, n-1

byte

Unit of digital information consisting of 8 bits.

Why byte instead of bits?

Hardware-level memory is naturally organized into addressable chunks

  • Small chunks means that you can have fine grained things like 4 bit numbers
  • Large chunks allow for more efficient operation (typically a CPU moves things around in chunks or multiple chunks)

Large Bytes

kilobyte (Kb), megabyte (Mb), gigabyte (Gb), and terabyte (Tb) are calculated two to the power of n * 10

Kilobyte = 2^10 = 1,024 bytes
Megabyte = 2^20 = 1,048,576 bytes
Gigabyte = 2^30 = 1,073,741,824 bytes
Terabyte = 2^40 = 1,099,511,627,776 bytes