Understanding Floating Point Numbers in Decimal and Binary Systems

What is the binary equivalent of 60.75 in decimal?

Using the subtraction method, what is the binary equivalent of 18.625 in decimal?

What is the binary equivalent of 190.5625 in decimal?

Using the multiplication and subtraction method, what is the binary equivalent of 37.25 in decimal?

What steps should be followed to convert a decimal number to binary using the subtraction method?

What do input devices do?

Which hardware devices are examples of input devices?

What is the process of sampling an image called?

How is an image stored using binary representation?

What does the sampling process of an image involve?

What is the color depth for an image with 256 available colors?

In a black and white image, how are the pixels represented in binary?

What does a higher resolution indicate about an image?

How many pixels are used to represent an image with a resolution of 300 pixels per inch?

What is the difference between a low resolution and a high resolution image?

Floating Point Numbers

A decimal point separates whole numbers from the fractional part in the decimal system.
Example: In 37.25, the whole part is 37 and the fractional part is 25/100.

Analysis of a Floating Point Number

The number 37.25 can be represented through digit analysis:
- 3 in the tens place: (3 \times 10)
- 7 in the units place: (7 \times 1)
- 2 in the tenths place: (2 \times \frac{1}{10})
- 5 in the hundredths place: (5 \times \frac{1}{100})
This representation aggregates to (37.25 = (3 \times 10) + (7 \times 1) + (2 \times \frac{1}{10}) + (5 \times \frac{1}{100})).

Binary Representation of Floating Point Numbers

The binary equivalent is calculated by converting whole and fractional parts separately.
For the whole part, methods include subtraction or division.
For the fractional part, use either subtraction or multiplication.

Fractional Part - Multiplication Method

Column values in binary representation are structured as: …, (2^5), (2^4), (2^3), (2^2), (2^1), (2^0); and fractional parts as (2^{-1}), (2^{-2}), (2^{-3}), etc.
Each power corresponds to specific fractional values (e.g., (1/2), (1/4), (1/8), …).

Example of Conversion using the Multiplication Method

To find the binary equivalent of 0.25:
- Multiply the fraction by 2 until reaching 0:
  - (0.25 \times 2 = 0.5)
  - (0.5 \times 2 = 1.0)
- Collect whole parts: the process yields (0.01) as the binary representation, where "1" appears after the binary point.

Summary of Conversion

Collect the whole parts in the order they appear after the radix point in binary representation.
The methodologies provided effectively convert both whole and fractional parts into binary equivalents.

Learn about how floating point numbers are represented in the decimal system with examples, and their binary equivalence. This quiz covers the basics of floating point numbers and their analysis in both decimal and binary systems.