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Number Systems & Units

Number System Notations​

  • $11_2=3_{10}$: 11 binary equals 3 in decimal

  • $7_8$ (using subscript) means a 7 digit in base-8 number NOT the number seven in decimal ($7_{10}$)

  • 0d10 = ten decimal

  • 0x10 = ten hexadecimal = 16

  • 0b0110 = binary

  • In C programming language (and Java, Python, C++):

    • The prefix 0x signify that the number is hexadecimal literal. Example: 0xFF = 255.
    • 0 is the prefix for Octal. Example: 077 = 63.

Decimal​

Use this online Base Converter

Decimal is called "base-10" because it uses ten digits (0–9).

Example: The number 253 means (2 × 100) + (5 × 10) + (3 × 1). Each position is a power of 10

  • 1 means "on / yes / true"; a truthy value in programming languages
  • 0 means "off / no / false"
  • Linux exit code 0 is successful. Any non-zero exit code (from 1 to 255) signifies error => Hiểu 0 la2 "no error"

DEC -> BIN​

  1. The subtraction method
  2. Successive division

75 = 64 + 8 + 2 + 1 => 100 1011

142 = 128 + 8 + 4 + 2 => 1000 1110

339 = 256 + 64 + 16 + 2 + 1 => 1 0101 0011

  • 75 / 2 = 37 R 1 <- Least Significant Bit

  • 37 / 2 = 18 R 1

  • 18 / 2 = 9 R 0

  • 9 / 2 = 4 R 1

  • 4 / 2 = 2 R 0

  • 2 / 2 = 1 R 0

  • 1 / 2 = 0 R 1 <- Most Significant Bit

  • => 75 = 100 1011 (write it from bottom -> top)

  • 142/2 = 71 R 0

  • 71/2 = 35 R 1

  • 35/2 = 17 R1

  • 17/2 = 8 R1

  • 8/2 = 4

  • 4/2 = 2

  • 2/2 = 1

  • 1/2 = 0 R1

  • => 142 = 1001110

DEC -> HEX​

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Binary​

Binary is "base-2" because it only uses two digits (0 and 1).

Example: The number $1101_2$ means (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 13 in decimal. Each position is a power of 2.

0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, ...

  • $2^0=1$
  • $2^1=2$
  • $2^2=4$
  • $2^3=8$
  • 2^4 = 16
  • 2^5 = 32
  • 2^6 = 64
  • 2^7 = 128
  • 2^8 = 256

4 bits max 1111 = 15 => Nhớ $2^4=16$ rồi trừ đi một là ra 15.

1 byte max 1111 111 = 255, 2^8 = 256

1 0000 + 1111 = 1 1111 or 16 + 15 = 31

The rightmost digit of a binary number is called the least-significant bit, because it has the least value. The leftmost digit is called the most-significant bit, because it has the greatest value.

You should be able to convert between binary and decimal in your head, without writing down the value of each bit.

Add & Subtract Binaries​

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BIN -> HEX​

  • 1011 0111

    • The first thing is you separate the binary in group of four, here we have two groups of four.
    • Convert to decimal: 1011 0111 = 11 & 7
    • 11 & 7 = 0xB7

  • $1101.1011.0010.1111_2$

    • Split the number into four-bit groups: 1101, 1011, 0010, 1111
    • Convert each four-bit group into decimal: 13, 11, 2, 15.
    • Convert each decimal number into hexadecimal: D, B, 2, F.

Memorizing the decimal values of hexadecimal A–F is very helpful: 0d10 = 0xA, 0d11 = 0xB, 0d12 = 0xC, 0d13 = 0xD, 0d14 = 0xE, 0d15 = 0xF.

Octal​

Octal is base-8. Digits used: 0, 1, 2, 3, 4, 5, 6, 7

One octal digit represents exactly three binary digits (7 = 111, 7 = 4+2+1). This is because 2^3 = 8.

Example: The number 37 means (3 × 8) + (7 × 1) = 31 in decimal.

0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20, ...

Octet​

In networking, an octet is a unit of size means "a group of 8 bits" (one byte). It is 0-255 in decimal & 00000000-11111111 in binary.

We use the word "octet" instead of "byte" because, historically, a "byte" could mean different sizes (like 6 or 7 bits) depending on the computer architecture. "Octet" is unambiguous.

The decimal range 0-255 represents all the possible values that can be stored in one 8-bit byte (256 different values).

OCT -> BIN​

  • 056
    • Separate the 5 & 6
    • 05 = 101
    • 06 = 110
    • 056 = 101 110

Linux File Permissions​

The most common place you'll see the octal system today is in Linux/macOS file permissions. When you use the chmod command, you're often using octal numbers:

Three bits represent three boolean flags: read-write-execute.

Cách viết rw-r--r-- thực chất là thể hiện 6 bits boolean flags chứ không phải 6 characters.

  • chmod 755

    • 07 = 111 => Read + Write + Execute (rwx)
    • 05 = 101 => Read + Execute (r-x)
    • 05 = 101 => Read + Execute

  • 06 = 110 => read, write (rw-)

  • 04 = 100 => read (r--)

The octal number system is NOT used for IPv4 (it use decimal from 0-255 to represent 1 byte). One octal digit can only represents exactly three binary digits.

Hexadecimal​

Hexadecimal is base-16. Digits used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (10), B (11), C (12), D (13), E (14), F (15).
Because more digits are available, hexadecimal can express large values in fewer characters.

Hexadecimal is a a very compact way to represent binary data. It's commonly used for colors (e.g., #FF0000 for red), memory addresses, and MAC addresses.

  • One "hex" digit = four binary digits because 2^4 = 16. Example: F = 1111 = 15, 15 = 8 + 4 + 2 + 1)
  • 2 hex digit = 8 bits (1 byte, an octet): FF = 11111111 = 255

Example: The number F3 means (15 × 16) + (3 × 1) = 243 in decimal

0 1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C, ...

  • $16^2=256=2^8$
  • $16^3=4096$
DecimalBinaryHexadecimal
000000
100011
200102
300113
401004
501015
601106
701117
810008
910019
101010A
111011B
121100C
131101D
141110E
151111F

HEX -> BIN​

Base 16 & 2 are related while base 10 is not. The relationship is: 16 = 2^4.

To convert 0x75 to decimal, just convert 7 to binary and 5 to binary and stick the results next to each other. This obviously doesn't work with converting to decimal, as 0x75 is 117, not 75. You have to do some multiplication by powers of 16 to get the correct decimal result.

Why is the process of converting hex to decimal completely different than hex to binary? Why does the 1st digit in 0x75 literally mean 0111 (7) in binary, but it doesn't literally mean 7 in decimal?

Base 16 is a power of 2, so you can group 4 bits to a hex, or vice versa. Base ten is not a power of two so conversion is not easy.

For example, if you had base 27 and base 3, the conversion is also straightforward because 27 is a power of 3.

This is exactly why Hexadecimal is a useful base in computer science.

Binary (base 2) and Hexadecimal (base 16) are both even powers of 2: 2^1 and 2^4 respectively. Decimal (base 10) is not an even power of 2 (2^3.32193).

  • A9

    • A = 10 = 8 + 2 = 1010
    • 9 = 8 + 1 = 1001
    • A9 = 1010 1001 = 169

  • 3B7

    • 3 = 0011
    • B = 1011
    • 7 = 0111
    • 3B7 = 0011 1011 0111

HEX -> DEC​

  • 23E = 2 x 16^2 + 3x16 + 14x1 = 574
  • F7D = 15x16^2 + 7x16 + 13x1 = 3965

Bitwise Operations​

These are logical operations performed on individual bits. The most important one in networking is bitwise AND.

Bitwise operators work on the individual bits of numbers, while logical operators work on true/false (boolean) values.

Logical operators are AND &&, OR ||, and NOT !.

  • Bitwise operator:
    • & AND
    • | OR
    • ~ NOT
    • ^ XOR
    • << / >> Bitwise Shift

One byte can store 8 boolean value.

  • The bitwise AND:
    • If both bits are 1, the resulting bit is 1.
    • If either bit is 0 (or both are 0), the resulting bit is 0.

The Truth Table for AND

xyx & y
000
010
100
111

Binary Arithmetic​

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Binary, Octal, Decimal, Hexadecimal number system​

Binary numbers only use the digits 0 and 1.

Octal Number System has a base of eight and uses the numbers from 0 to 7.

The sequence 1 2 4 10 20 40 100 200 400 corresponds to the powers of 2 in decimal (base-10).

  • 1 (octal) = 1 (decimal) = $2^0$
  • 2 (octal) = 2 (decimal) = $2^1$
  • 4 (octal) = 4 (decimal) = $2^2$
  • 10 (octal) = $1 \times 8^1 + 0 \times 8^0 = 8$ (decimal) = $2^3$
  • 20 (octal) = $2 \times 8^1 + 0 \times 8^0 = 16$ (decimal) = $2^4$
  • 40 (octal) = $4 \times 8^1 + 0 \times 8^0 = 32$ (decimal) = $2^5$
  • 100 (octal) = $1 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 64$ (decimal) = $2^6$
  • 200 (octal) = $2 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 128$ (decimal) = $2^7$
  • 400 (octal) = $4 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 256$ (decimal) = $2^8$