Assembly Basics

Binary and Hexadecimal

The binary system is fundamental to programming in assembly. Computers use the binary system, but before I explain this we'll look at the denary system, the number system we use in everyday life.

The Denary Number System

In the denary system, there are ten digits, which are labelled 0 to 9. In a multi-digit number the rightmost position is the 'ones' position. Each digit is multiplied by 1, or 10⁰. The next position over to the left from this one is the 'tens' position, and the numbers are multiplied by 10¹ to obtain their value. Similarly the next column is multiplied by 10². The 'n'th column would be multiplied by 10^n-1 to give their value.

The Binary Number System

The difference with the binary system is that we limit ourselves to just two digits rather than 10. The digits are 0 and 1. Now, instead of the rightmost position being multiplied by 10⁰, we multipy by 2⁰. The next position over to the left is multiplied by 2¹, and so on.

To get used to this we'll do an example. We'll find the denary value of the binary number 1001101. Starting at the right we'll take the first position. It contains a one, which we multiply by 2⁰. This is 1 * 1, which is 1. The second column contains a 0, which when multiplied by 2¹, gives 0. The third column gives 1 * 2², or four. After multiplying the remaining digits by their corresponding value, we will have:

⁰

⁴

⁵

⁶

Each of the positions for a digit in a binary number is a 'bit'. Therefore a number with 8 possible positions is called an 8 bit number. An 8 bit number has a maximum value of 255 (eight 'ones'). Other sizes which are used on computers are 16 and 32 bit, with maximum sizes of 65535 and 4,294,967,294 respectively.
In your assembler, to specify a binary number, use the notation ????????xB, where the ?'s are the binary digits.

The Hexadecimal system

The hexadecimal system uses 16 digits instead of 10 or 2. The reason it is used is as a shorthand for binary. The 16 digits which are used are '0', '1', ... '9', 'A', 'B', ... 'F'. The letters can be thought of as extra digits. The 'A' has a decimal value of 10, and the 'F' has a decimal value of 15. To convert a hexadecimal number to decimal, follow the same system as for the binary system but use instead 16⁰, 16¹, etc. The main advantage of using hexadecimal is that it is easily converted to binary and vice versa. To do the conversion we use the binary values of the hexadecimal digits, which are given below.

Hexadecimal Digit	Binary equivalent	Hexadecimal Digit	Binary equivalent
0	0	8	1000
1	1	9	1001
2	10	A	1010
3	11	B	1011
4	100	C	1100
5	101	D	1101
6	110	E	1110
7	111	F	1111

To convert from hexadecimal to binary, simply replace the hex(short for hexadecimal) digits with their binary equivalents, making sure to include leading zeroes to make the binary number 4-bit. For example: The hex number A5 is converted to binary by replacing the 'A' with 1010 and replacing the 5 with 0101, giving 10100101. An 8-bit number uses two hex digits, a 16-bit four etc.
To convert from binary to hex, divide the binary number into groups of four, starting at the right, and replace each group by its corresponding hex digit.
In assembly language, the notation ????xH is used to denote a hexadecimal number, where the ?'s are the hex digits.

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