1. What is a Floating-Point Number?

Imagine you have a long number like 123.456. Sometimes, numbers are very big like 1000000.0, or very small like 0.000001. A floating-point number is a way for the computer to store these numbers, no matter how big or small they are.

2. Parts of a Floating-Point Number

A floating-point number is stored in three parts:

• Sign (S): Tells if the number is positive or negative. (1 bit)
• Exponent (E): Tells where the decimal point is. (A few bits)
• Mantissa (M): Tells the actual digits of the number. (A lot of bits)

Think of it like a scientific notation:

3. What Does This Mean?

• Sign (S): If S is 0, the number is positive. If S is 1, the number is negative.
• Mantissa (M): The actual digits, but usually between 1 and 2.
• Exponent (E): Tells us how to move the decimal point.

4. Example: Storing 5.75

Let’s see how to store the number 5.75:

• Step 1: Write in binary:
  • 5.75 in binary is 101.11.
  • This is 1.0111 × 2^2.
• Step 2: Break it down:
  • Sign (S): 0 (because it’s positive).
  • Mantissa (M): 0111 (we drop the leading 1.).
  • Exponent (E): The exponent is 2.
• Step 3: Adjust the Exponent:
  • We need to add a bias to store the exponent. For simplicity, let’s say the bias is 127 (common for 32-bit float).
  • New Exponent E = 2 + 127 = 129.
• Step 4: Put It Together:
  • Sign (S): 0
  • Exponent (E): 129 in binary is 10000001.
  • Mantissa (M): 01110000000000000000000.
• Memory Representation:

  • The number 5.75 in memory is stored as:

    0 | 10000001 | 01110000000000000000000
    
    

  • This is a 32-bit floating-point number.