Numbering System

Decimal System

Nowadays we use the decimal numbering system in almost everything that is related to numbers. We use it so often and in a natural way that we forget it's meaning. What is decimal system? Every decimal number, has only digits between zero and nine, making a total of 10 digits Note: how many fingers do you have? ...10. In fact the decimal system is bound to human anatomy. Ok, and what is the meaning of each digit? Consider the following numbers: 234 and 234,43

We do some transformations

-> 234 i.e. 200 + 30 + 4 i.e. 2 * 10^2 + 3 * 10^1 + 4 * 10^0

-> 234,43 = 2 * 10^2 + 3 * 10^1 + 4 * 10^0 + 0,43 = 2 * 10^2 + 3 *

10^1 + 4 * 10^0 + 4 * 10^-1 + 3 * 10^-2

Do you see the relation? Each digit appearing to the left of the decimal point represents a value between zero and nine times an increasing power of ten. Digits appearing to the right of the decimal point represent a value between zero and nine times a decreasing power of ten.

Binary system

Binary system uses only two digits, by convention the digits are 0 and 1.

This system is so widely used in computers... By coincidence or not this system adjusts perfectly to computers... Computers operate using binary logic. The computer represents values using two different voltage levels, in this way we can represent 0 and 1. Like I said before the same applies to binary system, it is well adjust to computer anatomy!


1.2.1. Converting a binary number to a decimal number

Apply the same rule we saw in 1.1, but with powers of two.

Example: 1010 -> 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 10

1.2.2. Converting a decimal number to a binary number

We have two ways to do it:

1.2.2.1 We consecutively divide the decimal value by a power

two(keeping the remainder), while the result of the division is different than zero. The binary representation is obtained by the sequence of remainders in the inverse order of the divisions.

Consider the number 10(in decimal):



10 / 2

0 5 / 2

1 2 / 2

0 1 / 2

1 0



So in binary we write 1010



1.2.2.2 You can try to find out the number by adding powers of two, that added will produce the decimal result.



Consider for example number 123... hmmm it's a number not less than 2^0 and not greater then 2^7. Cool…



2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0

0 1 1 1 1 0 1 1 because 1 * 2^6 + 1 * 2^5 + 1 *

2^4 + 1 * 2^3 + 0 * 2^2 + 1*2^1 + 1 * 2^0

= 123

Our result is 1111011.

Hexadecimal system

You saw how many digits took to represent the number 123 in binary. 7 digits! Imagine 1200, 10000,... it hurts. So programmers had to choose another numbering system, just to "talk" to the machine... and no... it's not the decimal system!! You saw the trouble we had to convert one simple number like 10 between decimal and binary... I think you don't want to spend half of your life doing that. Engineers thought on that and they elected the hexadecimal system... Hexadecimals is the "english" for computers. They have two special features:

- They're very compact

- it's simple to convert them to binary and vice-versa. A hexadecimal number has digits with a value between 0 and 15 times a certain power of sixteen. Because we only know digits between 0-9 we have to use six more digits! We can use the 6 first letters of the alphabet.

Let's see a example: FF = 15 * 16^1 + 15 * 16^0 = 255 (16) (10)

Converting between binary and hexadecimal is very easy! To convert binary to hexadecimal remember that every four digits correspond to a single hexadecimal digit... to convert back to binary just apply the inverse rule! Let's take a look at the next example:

110 1011 = 0110 1011 =6B

(2) (16)

It's very easy!! To make things easier, take a look at the following table:

################
# D # H # B #
################
# 0 # 0 # 0000 #
# 1 # 1 # 0001 #
# 2 # 2 # 0010 #
# 3 # 3 # 0011 #
# 4 # 4 # 0100 #
# 5 # 5 # 0101 #
# 6 # 6 # 0110 #
# 7 # 7 # 0111 #
# 8 # 8 # 1000 #
# 9 # 9 # 1001 #
#10 # A # 1010 #
#11 # B # 1011 #
#12 # C # 1100 #
#13 # D # 1101 #
#14 # E # 1110 #
#15 # F # 1111 #
################

Conventions

Programming in assembly, requires you to obey some rules when using numbers, because you can use three different numbering systems.

When writing a number:

- all numbers have to start with a decimal digit

- all numbers end with a letter, indicating the type of number:

. for hexadecimals the letter is h

. binary numbers end with b

. decimals end with t or d We will use the following notation:

Xn Xn-1 ... X2 X1 -> Xi represents a bit, and i<-[0,1,...,n] represents it's position.