Decimal to Binary/Hexadecimal Converter

How to Use

1. Run Binary_Hexadecimal_Converter.py .
2. Input a decimal number between 0 to 255.

Example

For example, if you input the decimal number "10", the program will output:

You entered: 10
Binary representation: 1010
Hexadecimal representation: A

Conversion Principle

Info

• Draft and explanation can be found in Notes.ipynb。
• Conversion methods are defined as Convert_2_Bin() and Convert_2_Hex() in Binary_Hexadecimal_Converter.py and Binary_Hexadecimal_Converter_GUI.py .

In the decimal system, when a number exceeds ten, it will carry over to the next digit. For example, when the number is 9, the next number is ten, so 1 is carried over to the "tens" digit, and the "ones" digit returns to 0.

$$9+1 = 1 \times 10^1 + 0 \times 10^0 = 10$$

When a number exceeds 10 in the decimal system, we divide it by 10 to obtain a quotient ($\textbf{Q}$) and a remainder ($\underline{r}$). The quotient is then added to the next digit, and the remainder remains in the current digit.

$$a \div b = Q ... r$$

$$a = (Q \times b) + r$$

For example：

$$10 \div 10 = \textbf{1} ... \underline{0}$$

$$10 = (\textbf{1} \times 10) + \underline{0}$$

This process is repeated until the entire number has been converted.

$$\begin{align*} \underline{42310} &= 42310\times 1\\ &= \textbf{4231} \times 10 + \underline{0}\times 1\\ &= \Big(\textbf{423} \times 10 + \underline{1}\Big) \times 10 + \underline{0}\times 1\\ &= \bigg(\Big(\textbf{42} \times 10 + \underline{3}\Big) \times 10 + \underline{1}\bigg) \times 10 + \underline{0}\times 1\\ &=\ ...\\ &= \underline{4} \times 10^4 + \underline{2} \times 10^3 + \underline{3} \times 10^2 + \underline{1} \times 10^1 + \underline{0} \times 10^0 \end{align*}$$

The process of dividing by the base and taking the quotient and remainder is a fundamental principle of positional notation, which is used in all numbering systems. Therefore, the same conversion principle applies to binary, hexadecimal, octal, and other numbering systems as well.

Representation of Binary and Hexadecimal

• In binary notation, as mentioned earlier, each digit represents a remainder obtained by dividing the number by 2, and therefore, there are only two possible values: 0 and 1.

  Decimal	$r \times 2^n$	Binary
  0	$0 \times 2^0$	0
  1	$1 \times 2^0$	1
  2	$1 \times 2^1 + 0 \times 2^0$	10
  3	$1 \times 2^1 + 1 \times 2^0$	11
  4	$1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0$	100
  5	$1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$	101
  6	$1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0$	110
  7	$1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$	111
  8	$1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0$	1000

• In hexadecimal notation, each digit can have a value from 0 to 15. To represent the values greater than 9, letters A through F are used.

  Decimal	Hexadecimal	Decimal	Hexadecimal
  0	0	8	8
  1	1	9	9
  2	2	10	A
  3	3	11	B
  4	4	12	C
  5	5	13	D
  6	6	14	E
  7	7	15	F

Bonus：GUI version (Experimental)

How to Use

1. Run Binary_Hexadecimal_Converter_GUI.py.

2. Enter an integer within the range of 0 to 255 in the text box.

3. Click the Convert button and the output will be displayed below.

   Binary representation: <bin_num>
   Hexadecimal representation: <hex_num>