Summary
Highlights
Normalization ensures a unique representation for each number and maximizes accuracy. A normalized positive number always starts with '01', while a normalized negative number starts with '10'.
This video is part two in a three-part series on floating-point binary, specifically covering normalization. Floating-point binary was introduced in the previous video, which is recommended to watch first.
The video demonstrates how the same decimal number (e.g., 1) can have multiple floating-point binary representations. Using five bits for the mantissa and three for the exponent, examples show different binary forms resolving to one. This variability is problematic, necessitating a consistent storage method through normalization.
The video illustrates converting 6.5 into normalized floating-point binary. First, 6.5 is represented in standard binary. Then, the binary point is moved to achieve the '01' leading sequence, and the exponent is set to reflect the shift. The mantissa and exponent are then stored according to the defined bit lengths. The process is verified by converting back to decimal.
Another example, 2.25, is converted similarly. The binary representation is established, the binary point is moved until the mantissa starts with '01', and the corresponding exponent is determined. The conversion back to decimal confirms the accuracy.
For negative numbers like -2.5, the positive version is first converted to binary. Then, the two's complement method is used to get the negative binary. The binary point is moved to ensure the mantissa starts with '10' (for negative normalized numbers), and the exponent is adjusted accordingly. Verification is done by converting back.
A more complex negative example, -0.25, is normalized. After converting the positive 0.25 to two's complement, the binary point is moved. This example involves moving the point to the right, resulting in a negative exponent. Backfilling with ones for negative numbers and verification confirm the conversion.
The video concludes by posing key questions for viewers to consider: How does a computer store fractions/real numbers? What is a normalized floating-point binary number?