The natural numbers consist of all positive integers 1, 2, 3, ... Often zero is also included, so that the natural numbers cover all non-negative integers 0, 1, 2, 3, ...
Natural numbers are usually represented in the decimal numeral system or base-10 positional numeral system:
703 = 7*10^2 + 0*10^1 + 3*10^0
= 7*100 + 0*10 + 3*1
But besides the human possessing 10 fingers on his two hands there is no special advantage in using 10 as a base for a positional numeral system. The generic positional notation for a number with an arbitrary base or radix is shown below.
Number = ... + Digit*Radix^Index + ... + Digit_1*Radix^1 + Digit_0*Radix^0
Actually for a computer built from electronic NAND and NOR gates implementing Boolean logic, the base-2 binary numeral system is a much more convenient number representation:
1010111111 = 1*2^9 + 0*2^8 + 1*2^7 + 0*2^6 + 1*2^5 + 1*2^4 + 1*2^3 + 1*2^2 + 1*2^1 + 1*2^0
1*512 + 0*256 + 1*128 + 0*64 + 1*32 + 1*16 + 1*8 + 1*4 + 1*2 + 1*1
For humans the many digits of a binary number representation are quite tiresome. Therefore the more compact base-8 octal numeral system
1277 = 1*8^3 + 2*8^2 + 7*8^1 + 7*8^0
= 1*512 + 2*64 + 7*8 + 7*1
or the very popular base-16 hexadecimal numeral system
2BF = 2*16^2 + B*16^1 + F*16^0
= 2*256 + 11*16 + 15*1
are extensively used in computer and software engineering.
The following table shows the symbols used by the hexadecimal, decimal, octal and binary numeral systems to encode the first 16 natural numbers including zero.
| Hex | Decimal | Octal | Binary |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 10 |
| 3 | 3 | 3 | 11 |
| 4 | 4 | 4 | 100 |
| 5 | 5 | 5 | 101 |
| 6 | 6 | 6 | 110 |
| 7 | 7 | 7 | 111 |
| 8 | 8 | 10 | 1000 |
| 9 | 9 | 11 | 1001 |
| A | 10 | 12 | 1010 |
| B | 11 | 13 | 1011 |
| C | 12 | 14 | 1100 |
| D | 13 | 15 | 1101 |
| E | 14 | 16 | 1110 |
| F | 15 | 17 | 1111 |
The next table lists the power of each digit position for the decimal, binary, octal and hexadecimal numeral systems.
| n | 10^n | Units | k | 2^k | δ | i | 8^i | m | 16^m |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | ||
| 1 | 2 | - | - | ||||||
| 2 | 4 | - | - | ||||||
| 3 | 8 | 1 | 8 | - | |||||
| 1 | 10 | - | - | - | |||||
| 4 | 16 | - | 1 | 16 | |||||
| 5 | 32 | - | - | ||||||
| 6 | 64 | 2 | 64 | - | |||||
| 2 | 100 | - | - | - | |||||
| 7 | 128 | - | - | ||||||
| 8 | 256 | - | 2 | 256 | |||||
| 9 | 512 | 3 | 512 | - | |||||
| 3 | 1000 | kilo | 10 | 1'024 | 2.4% | - | - | ||
| 11 | 2'048 | - | - | ||||||
| 12 | 4'096 | 4 | 4'096 | 3 | 4'096 | ||||
| 13 | 8'192 | - | - | ||||||
| 14 | 16'384 | - | - | ||||||
| 15 | 32'768 | 5 | 32'768 | - | |||||
| 16 | 65'536 | - | 4 | 65'536 | |||||
| 6 | 1'000'000 | Mega | 20 | 1'048'576 | 4.9% | - | 5 | 1'048'576 | |
| 24 | ≈16M | 8 | ≈16M | 6 | ≈16M | ||||
| 9 | 1G | Giga | 30 | ≈1G | 7.4% | 10 | ≈1G | - | |
| 32 | ≈4G | - | 8 | ≈4G | |||||
| 12 | 1T | Tera | 40 | ≈1T | 10.0% | - | 10 | ≈1T | |
| 48 | 16 | ≈256T | 12 | ≈256T | |||||
| 15 | 1P | Peta | 50 | ≈1P | 12.6% | - | - | ||
| 18 | 1E | Exa | 60 | ≈1E | 15.3% | 20 | ≈1E | 15 | ≈1E |
| 64 | ≈16E | - | 16 | ≈16E | |||||
| 21 | 1Z | Zetta | 70 | ≈1Z | 18.1% | - | - | ||
| 24 | 1Y | Yotta | 80 | ≈1Y | 20.9% | - | 20 | ≈1Y |
Thus a 32 bit word encodes 2^32 = 4'294'967'296 or approximately 4G different values whereas a 64 bit word represents 2^64 = 18'446'744'073'709'551'616 or about 16E different numbers.
The address space of an x86-6464 bit processor is currently 48 bits allowing it to address 2^48 = 281'474'976'710'656 bytes amounting to about 256TB of memory.
The relative deviation δ between the scaling factors in the decimal and binary number systems is given by 2^k/10^n - 1 = 1.024^(n/3) - 1 and according to the table above amounts to 2.4% forkilo and 20.9% for Yotta. Transmission speeds like e.g. 64 kbit/s = 64'000 bits/s use decimal scaling factors whereas storage capacities like 64 kbytes = 65'536 bytes use binary scaling factors.
Actually it seems that storage device manufacturers have a rather opportunistic way of switching between decimal and binary scaling factors as a footnote from the Western Digital Red drive datasheet shows:
As used for storage capacity, one megabyte (MB) = one million bytes, one gigabyte (GB) = one billion bytes, and one terabyte (TB) = one trillion bytes. As used for buffer or cache, one megabyte (MB) = 1'048'576 bytes. As used for transfer rate or interface, megabyte per second (MB/s) = one million bytes per second, and gigabit per second (Gb/s).
It is assumed that the Utah Data Center run by the National Security Agency (NSA) has a storage capacity of about 3 to 12 exabytes. And IDC predicts that the collective sum of the world’s data will grow from 33 zettabytes in 2018 to 175 ZB by 2025.
The ASCII graphics below show a 32 bit data word in so called network order as defined by RFC 1700. Network order is used to transmit data over communication links in a platform-independent way.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 32 bits
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 hex nibbles
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| byte 0 | byte 1 | byte 2 | byte 3 | 4 bytes
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| 32 bit word | 1 word
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
The left-most byte 0 which is the most significant byte (MSB) in the 32 bit word is transmitted first, followed by byte 1, then byte 2 and at last byte 3, the least significant byte (LSB). Within the left-most byte 0, the left-most bit 0 is the most significant bit that is transmitted first and bit 7, the least significant bit, is transmitted last.
Each byte can be represented by 2 hexadecimal nibbles so that a 32 bit word comprising 4 bytes totals 8 nibbles.
Python 1: Base conversions from decimal to binary, octal or hexadecimal format are a piece of cake
>>> bin(703)
'0b1010111111'
>>> oct(703)
'0o1277'
>>> hex(703)
'0x2bf'The following expression formats a binary number with a constant length of 16 bits
>>> format(703, '#018b')
'0b0000001010111111'In a similar fashion we can format a hexadecimal number with a constant length of 4 nibbles
>>> format(703, '#06x')
'0x02bf'And if we want to get rid of the base prefixes
>>> format(703, '016b')
'0000001010111111'
>>> format(703, '04x')
'02bf'The following expression stores the bits of a 16 bit integer in an array
>>> [int(x) for x in format(703, '016b')]
[0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1]that can be used for further processing
>>> bits = [int(x) for x in format(703, '016b')]
>>> print('msb:', bits[0], 'lsb:', bits[15])
msb: 0 lsb: 1We can also directly pick individual bits using the shift-right operator >> and the single bit mask 0x0001
>>> value = 703
>>> mask = 0x0001
>>> value & mask # lsb
1
>>> (value >> 6) & mask # bit at b*2^6
0
>>> (value >> 7) & mask # bit at b*2^7
1
>>> (value >> 8) & mask # bit at b*2^8
0
>>> (value >> 9) & mask # bit at b*2^9
1
>>> (value >> 15) & mask # msb
0Since a hex number contains the characters A to F we have to analyze the nibbles as a string
>>> nibbles = format(703, '04X')
>>> print(nibbles[0], nibbles[1], nibbles[2], nibbles[3])
0 2 B For we can convert each nibble in the string above into an integer
>>> [int(x,16) for x in nibbles]
[0, 2, 11, 15]Using the byte mask 0x00ff we can determine the decimal value of each byte
>>> value = 703
>>> mask = 0x00ff
>>> value & mask # lower byte
191
>>> (value >> 8) & mask # upper byte
2Python 2: In the reverse direction, binary, octal or hexadecimal constants can be displayed as decimal values
>>> print(0b1010111111)
703
>>> print(0o1277)
703
>>> print(0x2bf)
703Usually numbers in various formats are input as character strings which can be converted to an integer using the int(value, base) function
>>> int('0b1010111111', 2)
703
>>> int('0o1277', 8)
703
>>> int('0x2bf', 16)
703
>>> int('703', 10)
703
>>> int('703')
703Since the base is given, the base prefix is not required
>>> int('1010111111', 2)
703
>>> int('1277', 8)
703
>>> int('2bf', 16)
703Author: Andreas Steffen CC BY 4.0