• Basic Oscilloscope Operation
• Binary and Hexadecimal Numbers
• Programming the Microprocessor

What are binary numbers and why do we use them?

The Binary System

Decimal to Binary Conversion:

The remainder 1 resulting from the last division is the MSB, while the first remainder is the LSB of the conversion. From this example we see that the decimal number 132 is equal to the binary number 10000100.

Hexadecimal (Hex) Representation:

The rules for addition of binary numbers are straightforward:

0 + 0 = 0, 0 + 1 = 1, and 1 + 1 = 0 with a carry of 1, i.e. 1 + 1 = 10₂.

Until now, we have discussed only positive numbers. These numbers were called "unsigned 8-bit integers". In an 8-bit byte, we can represent a set of 256 positive numbers in the range 0₁₀-255₁₀. However, in many operations it is necessary to also have negative numbers. For this purpose, we introduce "signed 8-bit integers". Since we are limited to 8-bit representation, we remain also limited to a total of 256 numbers. However, half of them will be negative (-128₁₀ through -1₁₀) and half will be positive (0₁₀ through 128₁₀).

The representation of signed (positive and negative) numbers in the computer is done through the so-called 8-bit 2's complement representation. In this representation, the 8^th bit indicates the sign of the number (0 = +, 1 = -).

The signed binary numbers must conform to the obvious laws of signed arithmetic. For example, in signed decimal arithmetic, -3₁₀ + 3₁₀ = 0₁₀. When performing signed binary arithmetic, the same cancellation law must be verified. This is assured when constructing the 2's complement negative binary numbers through the following rule:

Example 1: Use the above rule to represent in 8-bit 2's complement the number -3₁₀

Solution: Subtract the 8-bit binary representation of 3₁₀ from the 8-bit binary representation of 0₁₀ using 8-bit arithmetic (8-bit arithmetic implies that you can liberally take from, or carry into the 9^th bit, since only the first 8 bits count!).

BINARY DECIMAL 00000000 - 0₁₀ - 00000011 3₁₀ 11111101 -3₁₀

Note that, in this operation, a 1 was liberally borrowed from the 9^th bit and used in the subtraction!Verification We have establish that -3₁₀ = 11111101₂. Verify that -3₁₀ + 3₁₀ = 0₁₀ using 8-bit arithmetic. BINARY DECIMAL 11111101 - -3₁₀ - 00000011 3₁₀ 00000000 0₁₀Note that, in this operation, a carry of 1 was liberally lost in the 9^th bit!

Example 2: Given the binary number 00110101, find it's 2's complement.

The rule outlined above can be applied to both binary and hex numbers.

Example 3: Given the hex number 6A, find its 8-bit 2's complement.

Solution: Subtract the number from 00₁₆ using 8-bit arithmetic: