We have a byte, which consists of 8 bits. What integers can be represented using a byte? The answer depends on how we interpret the bits. With 8 bits, we have \( 2^8 = 256 \) different possible combinations, so 256 different values can be stored.

For example, one everyday use of these 8 bits is to represent unsigned integers in the range [0..255]. These numbers are all non-negative; this data type is typically referred to as a byte or an unsigned short int in many programming languages.

Example: how to add two binary numbers?

   10110110   (182)
+  01101101   (109)
------------
  100100011   (291)

But what happens if we need to store negative numbers as well? How can we represent both positive and negative integers?

One common solution is reserving the highest-order bit (the most significant bit, also known as the 7th bit, since counting starts from zero) as a sign bit. This bit indicates the sign of the number, and it is often denoted as the sign bit, 's'.

• If the sign bit (s) is 0, the number is positive or zero.
• If the sign bit (s) is 1, the number is negative.

The remaining 7 bits are used to represent the magnitude of the number, either positive or negative. This method allows us to represent integers in the range [-128..+127]. In other words:

• If the sign bit is 0, the number is in the range [0 .. 127].
• If the sign bit is 1, the number is in the range [-1 .. -128].

This is a simple form of sign-magnitude representation.

In modern computing, two's complement is a more commonly used method for representing signed integers. In this system, the highest-order bit is also used as the sign bit, but the way negative numbers are stored differs. Here’s how it works:

• The most significant bit is still used as the sign bit, where 0 indicates a positive number, and 1 indicates a negative number.
• However, negative numbers are represented by taking the two’s complement of their absolute value. This involves inverting all the bits and then adding 1 to the result.

Example: Encode -5 with two's complement method with 8 bits.

• The number 5 in binary (in an 8-bit system) is represented as 00000101
• Invert the bits: 11111010
• Add one: 11111011

Example: Add -5 + 7 in 8 bits binary format

Now we can add the two binary numbers:

  11111011   (-5 in two's complement)
+ 00000111   (7)
------------
  00000010   (2)

This is the reason why we are using the special two's complement form. Without using this encoding, we cannot add negative and positive numbers.

Let’s convert the decimal number 156 into binary.

Start by dividing the decimal number (156) by 2 and keep track of the quotient and remainder. The remainder will be either 0 or 1, which forms the binary digits from bottom to top.

To get the binary representation, take the remainders from bottom to top. The binary equivalent of 156 is:

10011100₂

To verify, we can convert the binary number back to decimal:

(1 × 2⁷) + (0 × 2⁶) + (0 × 2⁵) + (1 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
= 128 + 0 + 0 + 16 + 8 + 4 + 0 + 0
= 156

Hence, the binary representation of 156 is correct.