#informatique S1 07 Les opérations arithmétiques en binaire (partie 2)

Introduction

This tutorial covers the operations of multiplication and division in binary numbers, essential for understanding computer arithmetic. By mastering these operations, you can enhance your programming and computational skills, particularly in areas related to computer science and digital logic.

Step 1: Understanding Binary Basics

Before diving into multiplication and division, ensure you are familiar with the binary number system:

• Binary uses only two digits: 0 and 1.
• Each digit represents a power of 2.
• For instance, the binary number 1011 represents:
  • 1*(2^3) + 0*(2^2) + 1*(2^1) + 1*(2^0) = 8 + 0 + 2 + 1 = 11 in decimal.

Practical Tip

Practice converting between binary and decimal numbers to solidify your understanding.

Step 2: Multiplication in Binary

Binary multiplication follows a process similar to decimal multiplication but uses simpler rules:

1. Align the numbers: Write the binary numbers one below the other.
2. Multiply each bit:
   • If the bit is 1, write down the other number shifted left (like adding a zero in decimal).
   • If the bit is 0, write down a row of zeros.
3. Add the results: Sum all the rows to get the final product.

Example

To multiply 101 (5 in decimal) by 11 (3 in decimal):

    101
  x  11
  ------
    101   (this is 101 multiplied by 1)
   1010   (this is 101 multiplied by 1, shifted one position left)
  ------
   1111   (add the two results)

The result is 1111, which is 15 in decimal.

Step 3: Division in Binary

Binary division is similar to long division in decimal:

1. Set up the division: Write the dividend (number being divided) and the divisor (number you're dividing by).
2. Divide step by step:
   • Start from the leftmost bit of the dividend.
   • Compare the divisor with the current bits of the dividend. If the divisor fits, write down 1; otherwise, write down 0.
   • Subtract the divisor from the dividend if you wrote down 1.
3. Bring down the next bit: Repeat the process until all bits are processed.

Example

To divide 1101 (13 in decimal) by 11 (3 in decimal):

     11
   ------
11 | 1101
      11   (3 fits into 13)
   ------
       01  (subtract 3 from 3)
        1   (bring down the next bit)
   ------
       01   (3 does not fit into 1)
        0   (bring down the next bit)
   ------
        10   (3 fits into 10)
       -11
   ------
         1

The result is 11 with a remainder of 1, which is 4 with a remainder of 1 in decimal.

Conclusion

Mastering binary multiplication and division is crucial for anyone interested in computer science and digital technology. Practice these operations using various binary numbers to gain confidence. Explore more complex topics, such as binary addition and subtraction, to build a solid foundation in binary arithmetic. Keep practicing, and soon you'll handle binary operations with ease!