Topics from the Cambridge IGCSE (9-1) Computer Science syllabus.

we’re going to dive into three ways to represent positive and negative numbers in binary: Sign and Magnitude, One’s Complement, and Two’s Complement. Let’s break down each method and see how they work. Part 1: Sign and Magnitude Let’s start with Sign and Magnitude. This is one of the simplest ways to represent positive and negative numbers in binary. Here’s how it works: In Sign and Magnitude, we use the leftmost bit – known as the sign bit – to indicate whether the number is positive or negative: If the sign bit is 0, the number is positive. If the sign bit is 1, the number is negative. The remaining bits represent the actual value, or magnitude. For example: In an 8-bit system, the binary number 0000 1010 has a sign bit of 0, meaning it’s positive. The remaining bits, 0000 1010, represent the number 10 in decimal. Now, if we change that sign bit to 1, making it 1000 1010, the number becomes -10 in decimal. The rest of the bits stay the same, representing the value. However, Sign and Magnitude has limitations because it can make binary arithmetic more complex, especially when dealing with subtraction. Part 2: One’s Complement Next, we have One’s Complement. This method also uses the leftmost bit as a sign bit, but it represents negative numbers differently. Here’s the rule for One’s Complement: To find the One’s Complement of a binary number, invert all the bits – that means changing 0s to 1s and 1s to 0s. For example: If we want to represent -10 in an 8-bit system: Start with the positive binary representation of 10, which is 0000 1010. Invert each bit to get the One’s Complement: 1111 0101. So, 1111 0101 represents -10 in One’s Complement. One’s Complement simplifies some operations, but there’s a problem: it has two representations of zero. 0000 0000 represents positive zero, and 1111 1111 represents negative zero, which can make arithmetic operations tricky. Part 3: Two’s Complement Finally, let’s look at Two’s Complement. This is the most widely used method because it solves the issue of having two zeros and makes binary arithmetic easier. Here’s how to find the Two’s Complement of a number: Start with the positive version of the number in binary. Invert all the bits to get the One’s Complement. Add 1 to the result. For example: Let’s represent -10 in Two’s Complement. Start with the positive version of 10 in binary: 0000 1010. Invert all the bits to get 1111 0101. Add 1 to the inverted result: 1111 0101 + 1 = 1111 0110. So, 1111 0110 represents -10 in Two’s Complement. Two’s Complement only has one representation of zero, which is 0000 0000. It also makes addition and subtraction much simpler for computers because they can treat both positive and negative numbers the same way during calculations. Part 4: Summary To recap: Sign and Magnitude uses a sign bit to indicate positive or negative, but it can make arithmetic tricky. One’s Complement inverts all the bits to represent negative numbers, but it has two zeros. Two’s Complement inverts all the bits and adds 1, making binary arithmetic easier and solving the two-zero issue. And that’s it! These three methods – Sign and Magnitude, One’s Complement, and Two’s Complement – are fundamental to understanding how computers represent both positive and negative numbers in binary. Thanks for listening, and I hope this explanation helps you see how each method works!

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