Binary and Hex refresher course

[b3lha]

When we see a number like 1234 in our normal base10 number system, this actually represents

(1 x 1000) + (2 x 100) + (3 x 10) + (4 x 1)

As we used to say at primary school: Thousands, Hundreds, Tens and Units. Each column is a power of 10.

1000 is 10³
100 is 10²
10 is 10¹
1 is 10⁰

1234=(1x10³) + (2x10²) + (3x10¹) + (4x10⁰)

The same principle is used in Binary (base 2) but we use powers of two rather than 10.

128 is 2⁷
64 is 2⁶
32 is 2⁵
16 is 2⁴
8 is 2³
4 is 2²
2 is 2¹
1 is 2⁰

A number like 154 (decimal) is encoded as follows in binary:

154(decimal)=10011010(binary)=(1x2⁷)+(0x2⁶)+(0x2⁵)+(1x2⁴)+(1x2³)+(0x2²)+(1x2¹)+(0x2⁰)
154(decimal)=10011010(binary)=(1x128)+(0x64)+(0x32)+(1x16)+(1x8)+(0x4)+(1x2)+(0x1)

Here is how you count in binary:

Decimal	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

At the lowest level, everything in a computer is encoded in binary. The cells in the memory chips are either "on" or "off" - representing "1" or "0". For convenience, the memory cells are grouped into blocks of 8 called bytes and each byte has an address to identifiy it.

Computers love binary, but typing in all thoses 1's and 0's can be a little tedious for programmers, especially for larger numbers. So programmers like to use Hexadecimal (base 16). 16 is a power of 2 and therefore it is much easier to convert between binary and hex than it is to use decimal.

In hex, we have six extra digits between 9 and 10. You count from 0 to 9, then continue A to F, then 10 to 19, 1A to 1F then 20 etc. When you get to 9F, the next number is A0 and so on. When you get to FF, the next number is 100.

Converting between hex and binary is very easy because each hex digit represents exactly 4 bits of binary.

Hex	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
A	1010
B	1011
C	1100
D	1101
E	1110
F	1111

For example, the value A531 in hex is 1010 0101 0011 0001 in binary.
Or, the other way around 0110 1001 1111 0010 in binary is 69F2 in hex.

[b3lha]

LEFT and RIGHT shifting.

One of the easiest operations in the decimal system is to multiply or divide by 10.

For example:

To multiply 20 by 10, you shift the digits one place left and add a zero so it becomes 200.
To divide 20 by 10, you shift the digits one place right and lose a zero so it becomes 2.

It is exactly the same in binary, except we use 2 rather than 10.

To multiply 00011000 by 2 you shift it one place left and add a zero so it becomes 00110000
To divide 00011000 by 2 you shift it one place right and lose a zero so it becomes 00001100

When you see a piece of code like:

Code:

lda     ax, 0x1122    ; suppose the value is 3200
lsr     ax                   ; divide it by 2, now it's 1600
lsr     ax                   ; divide it by 2, now it's 800
lsr     ax                   ; divide it by 2, now it's 400
lsr     ax                   ; divide it by 2, now it's 200
lsr     ax                   ; divide it by 2, now it's 100
sta     al, 0x130e     ; store the value

It is taking the value in location 0x1122 and dividing it by 32 and storing the result at 0x130e.