Table of Contents

1. Introduction {#introduction}

When people first hear that computers work using only 1s and 0s, it can sound strange. At first, it seems impossible that everything on a computer like text, pictures, videos, games, and apps could come from something so simple. But at the core, that is exactly how computers work. They use number systems to store, process, and display information.

This research looks at three important numbering systems: Binary, Decimal, and Hexadecimal.

These systems are all ways of representing values, but each one is useful in a different way. Binary is the language computers use internally, decimal is the number system people use in everyday life, and hexadecimal is a shorter and more practical way for humans to read binary data in computing.

Positional number systems work by giving each place value a weight based on the base being used. In binary the base is 2, in decimal it is 10, and in hexadecimal it is 16.

System          Base    Symbols
─────────────────────────────────
Binary           2      0, 1
Decimal         10      0–9
Hexadecimal     16      0–9, A–F

2. Binary {#binary}

Binary is a base-2 numbering system, which means it uses only two digits: 0 and 1. Every position in a binary number represents a power of 2. Moving from right to left, the place values go 1, 2, 4, 8, 16, 32, and so on.

Binary is extremely important in computing because electronic hardware can easily represent two states such as on and off, high and low voltage, or true and false. That makes binary a practical fit for digital systems. A single binary digit is called a bit, and a group of 8 bits is called a byte.

Place value chart powers of 2:

Bit position:   7     6     5     4     3     2     1     0
Place value:   128    64    32    16     8     4     2     1

Example: 1011₂ → Decimal

Position   Bit   Weight   Contribution
────────────────────────────────────────
   3        1      8           8
   2        0      4           0
   1        1      2           2
   0        1      1           1
                         ──────────
                  Total =    11

So: 1011₂ = 11₁₀

Expanded example: 10001001₂ → Decimal

Position   Bit   Weight   Contribution
────────────────────────────────────────
   7        1     128         128
   6        0      64           0
   5        0      32           0
   4        0      16           0
   3        1       8           8
   2        0       4           0
   1        0       2           0
   0        1       1           1
                         ──────────
                  Total =    137

So: 10001001₂ = 137₁₀

Binary is used everywhere in computing to store numbers, letters, colors, machine instructions, memory values, and more. Even though people usually do not work directly with long binary strings, computers rely on binary at the hardware level.

3. Why We Use Binary {#why-binary}

The main reason computers use binary is because it is reliable and simple for electronic systems. A device only needs to tell the difference between two states, not ten or more. That makes circuits easier to design and makes signals less likely to be misunderstood because of small electrical changes. This is one of the reasons binary became the standard system in digital computing.

Binary is also very useful for logic operations. Since computer systems depend heavily on true or false decisions, binary matches that kind of processing naturally. Even though binary numbers can become long, the system itself is stable, efficient, and easy for machines to work with.

Why binary works for hardware:

  Decimal requires   →   10 distinct voltage levels   (complex, error-prone)
  Binary requires    →   2 distinct states             (simple, reliable)

  State 0 = low voltage  = OFF = false
  State 1 = high voltage = ON  = true

4. Decimal {#decimal}

Decimal is the numbering system people use in normal daily life. It is a base-10 system, which means it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Each position in a decimal number represents a power of 10.

Example: 482₁₀

Position   Digit   Weight   Contribution
──────────────────────────────────────────
   2         4      100         400
   1         8       10          80
   0         2        1           2
                           ──────────
                    Total =    482

Compared with binary, decimal is much easier for people to read and use. That is why most interfaces show values in decimal unless there is a technical reason to use another system.

5. Hexadecimal {#hexadecimal}

Hexadecimal is a base-16 numbering system. It uses sixteen symbols: 0–9 and A–F.

Hex digit   Decimal value
─────────────────────────
    0             0
    1             1
    ...          ...
    9             9
    A            10
    B            11
    C            12
    D            13
    E            14
    F            15

Hexadecimal is very common in computing because it is a compact way to represent binary. One hexadecimal digit maps exactly to 4 binary bits (called a nibble). That makes binary values much shorter and easier for humans to read.

Binary vs Hex same value, different length:

Binary      →   1 1 1 1   1 1 1 1
                └───┘     └───┘
Hex group   →     F         F

Full value:  11111111₂  =  FF₁₆  =  255₁₀

That is why hex is used in: memory addresses, machine code, hex dumps, debugging, color codes (#FF5733), and networking.

6. Conversion {#conversion}

Binary → Decimal

Example: 10001001₂

Assign place values to each bit, multiply, and sum:

Bit position:  7    6    5    4    3    2    1    0
Bit value:     1    0    0    0    1    0    0    1
Weight:       128   64   32   16    8    4    2    1
Contribution: 128    0    0    0    8    0    0    1

128 + 8 + 1 = 137

10001001₂ = 137₁₀

Decimal → Binary

Example: 137₁₀

Divide by 2 repeatedly, record remainders, read bottom to top:

Step      Operation    Quotient   Remainder
───────────────────────────────────────────
  1       137 ÷ 2        68           1
  2        68 ÷ 2        34           0
  3        34 ÷ 2        17           0
  4        17 ÷ 2         8           1
  5         8 ÷ 2         4           0
  6         4 ÷ 2         2           0
  7         2 ÷ 2         1           0
  8         1 ÷ 2         0           1
                                ↑
                       Read upward: 10001001

137₁₀ = 10001001₂

Binary → Hexadecimal

Example: 10001001₂

Split into groups of 4 bits from right to left, convert each group:

Binary:    1000   1001
            ↓      ↓
Hex:        8      9

10001001₂ = 89₁₆

4-bit conversion reference:

0000 = 0    0100 = 4    1000 = 8    1100 = C
0001 = 1    0101 = 5    1001 = 9    1101 = D
0010 = 2    0110 = 6    1010 = A    1110 = E
0011 = 3    0111 = 7    1011 = B    1111 = F

Hexadecimal → Binary

Example: 2F₁₆

Convert each hex digit into its 4-bit binary equivalent:

Hex:     2        F
         ↓        ↓
Binary: 0010     1111

2F₁₆ = 00101111₂

Decimal → Hexadecimal

Example: 729₁₀

Divide by 16 repeatedly, record remainders, read bottom to top:

Step      Operation    Quotient   Remainder
───────────────────────────────────────────
  1       729 ÷ 16       45           9
  2        45 ÷ 16        2          13 → D
  3         2 ÷ 16        0           2
                                ↑
                       Read upward: 2D9

729₁₀ = 2D9₁₆

Hexadecimal → Decimal

Example: 3F₁₆

Multiply each digit by its power of 16:

Position   Digit   Weight   Contribution
──────────────────────────────────────────
   1         3      16¹ = 16     48
   0         F      16⁰ = 1      15

48 + 15 = 63

3F₁₆ = 63₁₀

7. 4-Bit Binary to Hex Lookup Table {#lookup}

Binary   Hex   Decimal      Binary   Hex   Decimal
──────────────────────      ──────────────────────
 0000     0       0          1000     8       8
 0001     1       1          1001     9       9
 0010     2       2          1010     A      10
 0011     3       3          1011     B      11
 0100     4       4          1100     C      12
 0101     5       5          1101     D      13
 0110     6       6          1110     E      14
 0111     7       7          1111     F      15

Every possible 4-bit combination maps to exactly one hex digit. This is why hex is so useful, two hex digits represent one full byte (00–FF), which covers the range 0–255 in decimal.

8. Conclusion {#conclusion}

Binary, decimal, and hexadecimal are different ways of writing values, but they are closely connected. Binary is the foundation of digital systems because computers and electronic hardware work well with two-state logic. Decimal is the system people use every day because it is familiar and easy to read. Hexadecimal is especially useful in computing because it makes long binary values much shorter and easier for humans to understand.

Same value three representations:

  Binary       →   10001001
  Decimal      →   137
  Hexadecimal  →   89

Understanding these three numbering systems is important for anyone studying information technology, cybersecurity, programming, networking, or computer hardware. Once you understand how they relate to each other, it becomes much easier to understand how computers store and process information.

9. Sources {#sources}

  • Encyclopaedia Britannica: Binary number system
  • Encyclopaedia Britannica: Positional numeral system
  • Encyclopaedia Britannica: Numeral system
  • Encyclopaedia Britannica: Binary code
  • NIST CSRC Glossary: bit
  • NIST CSRC Glossary: byte
  • University of Waterloo ECE: Binary and hexadecimal
  • University of Illinois Chicago: Number Systems course notes