Information Representation and Number systems

Welcome, class! Today we will delve into number systems, which are fundamental to how computers perceive and process data. Can anyone tell me what a number system is?

Isn’t it just a way to represent numbers?

Exactly! A number system provides a consistent way of expressing values. For instance, we will discuss the decimal, binary, octal, and hexadecimal systems.

What’s the difference between these systems?

Great question! Each system has a different base or radix. Decimal is base 10, binary is base 2, octal is base 8, and hexadecimal is base 16. Let’s remember this with the acronym 'DBOH' - Decimal, Binary, Octal, Hexadecimal.

So how does a computer use these different systems?

Computers primarily use binary, as they function on bits. Each digit represents a power of 2. At the end of our session, I will summarize the significance of these systems.

Now, let's look at how to convert a decimal number to binary. Can someone provide an example of a decimal number?

How about 75?

Excellent choice! To convert 75 to binary, we repeatedly divide by 2, recording the remainders. Who can outline this process with me?

First, we divide 75 by 2, which gives us 37 remainder 1.

Then, 37 divided by 2 gives us 18 remainder 1.

Continuing, 18 divided by 2 gives us 9 remainder 0.

Correct! If we keep going until we reach 0, we compile our binary number in reverse. So what would be the binary of 75?

It’s 1001011!

Exactly! Fantastic teamwork! Remember, this conversion process is vital in understanding how computers process data.

Next, let’s focus on how integers are represented in computers. What do you think a 'bit' represents?

A bit is a binary digit, either 0 or 1.

Correct! A collection of bits represents an integer. The representation depends on the number of bits used. Can you recall how bits affect the range of values?

More bits means a larger range of numbers, right?

Exactly right! For example, with 8 bits, you can represent values from 0 to 255. Let's summarize that: if n is the number of bits, the maximum integer is 2^n - 1.

What about negative numbers?

Good question! We can represent negative integers using methods such as sign magnitude and two's complement, which we’ll cover next.

Before we conclude, let's talk about how negative numbers are represented in binary. Who can explain the two's complement method?

Isn't it where you invert the bits and add one?

Exactly! This allows computers to perform arithmetic operations seamlessly. Let's see how we can convert -3 to two's complement using 4 bits.

So we start with 3, which is 0011, then invert to 1100 and add 1. So it becomes 1101!

Great job! Remember, two's complement allows easy addition of positive and negative numbers. Let’s recap what we’ve discussed today.

Today we’ve highlighted various number systems, learned to convert decimal to binary, discussed integer representations, and methods for representing negative values. Any questions before we end?