1.1: Binary Systems:
"Binary" means that something can be in 2 states, such as electrical switches that the processors use
to store data. Binary in computers can be called "Base 2", for the mathematical bases.
00000001 = 2 as the powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256 (up to 2 to the power of 8)
Therefore,
01001011 = 150
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Bits and Bytes
Binary numbers are made up of 1 and 0. This is also known as a bit, and one bit is either 1 or 0. This
means that a bit is the smallest piece of memory in a computer and means an on/off switch in the
computer's electronic memory.
These bits tend to be put into groups of 8 that are called "Bytes", and these Bytes are used within
the computer therefore we write it that way as well. To complete a Byte, just add 0s at the end.
Measuring memory
The area if the computer processor that stores data as on/off signals is called RAM or random-access
memory. RAM is measured by how many bytes it can hold.
Kilobyte (KB) is about 1024 bytes. This is equal to half a page of text.
Megabyte (MB) is 1024 kilobytes/1048576 bytes. A picture is about 3 MB.
Gigabyte (GB) is 1024 megabytes/1073741824 bytes. A movie is about 4 gigabytes.
Terabyte (TB) is 1024 gigabytes/1099511627776 bytes.
RAM and computer speed
The computer processor makes changes to the data in the RAM. It is easy for the processor to access
the RAM, and the processing will be quick. If the RAM does not have any space, then the data will be
allocated outside of the RAM, and the processor is slower to reach this, and the computer will go
slower.
For example, in Adobe Photoshop or in RPG games, it requires a lot of RAM to manipulate the pixels
to edit, or to render a 3D world.
The reason why a bigger RAM means a faster speed, is that there is more data that the RAM can take
from the hard drive, take from a register to give to the processor, etc.
Registers
In the main memory, about 8 GB of RAM is common. However, processing does not happen in the
main memory. Data that is ready to be processed is copied into a much smaller area of memory
called a register. The register allows the processor to navigate to the correct area faster.
,Registers are normally measured by the number of bits they hold, such as an 8-bit register or a 32-bit
register. A processor often contains several kinds of register for different purposes.
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Binary and Denary
Denary
The denary system is also known as "base 10” and is the everyday number system. This base 10 can
represent every number using:
0123456789
Something that can be in 10 different forms is known as denary, so our number system is denary.
Another name is called "decimal".
The positions of the digits change its meaning:
702
720
207
The 2 in the numbers mean 2, 20 and 200 (respectively). The difference is the position of the number
in the number.
1000 100 10 1
0 7 2 0
The denary system was invented in India, around 300 BC. It was the best for doing complex math at
the time. It spread to the Arabic world, and around 850 ADS, a Persian mathematician named Al-
Khwarizmi and an Arab mathematician named Al-Kindi wrote a book explaining the denary number
system. Then, at some point the Europeans learnt of this system and adopted it and used to call it
Arabic Numerals.
Binary
Binary numbers only consist of 1 and 0 (bits). Every number can also be represented with only 1 and
0. The position of the bit tells its value, like if you start at the right of the number and move left, the
value of the number increases by 2x. The Binary number 101 is...
8421
0101
In this, there is 1 four and 1 one, meaning that it means 5.
,A German philosopher called Gottfried Leibniz set out the idea of Binary numbers in 1689, and then
in the 1800s, a British mathematician called Ada Lovelace wrote notes about how this could be used
in digital processing.
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Example Questions:
Q) What is the difference between a bit and a byte?
1) A bit is only a 1 or a 0, whereas a byte is a group of 8 of bits.
How many bytes are there in a kilobyte?
2) 1024.
How many pictures can be stored in a Gigabyte?
3) 341?
How can you increase the speed of a computer?
4) More RAM, and bigger Registers.
5) Because denary means it can be in 10 forms.
6) 4
7) 2
8) The table has eight values since a byte has 8 bits.
9) The position values
10) 37
Conversions:
128 64 32 16 8 4 2 1
1st 1 0 0 0 0110
2nd 0 0 1 0 1010
3rd 1 1 0 0 0101
4th 0 1 0 1 0001
8 4 2 1 Total
00011
00102
00113
01004
01015
01106
01117
10008
, 10019
1 0 1 0 10
1 0 1 1 11
1 1 0 0 12
1 1 0 1 13
1 1 1 0 14
1 1 1 1 15
1.2: Hexadecimal:
Hexadecimal digits:
Hexadecimal uses 16 different digits, which means that it is also known as "base 16".
The 16 hexadecimal digits are:
0 1 2 3 4 5 6 7 8 9 A B C D E F.
These digits stand for 0 – 15, NOT 16!
The digits 0 – 9 stands for the numbers they indicate, however the letters ABCDEF stand for the
numbers 10 – 15.
Such as the other number systems, the position value dictates how the value. Many hexadecimal
numbers will only have 2 digits, the right-hand-most digit being 1, and if you continue left it
increases by 16x.
16 1
B 3
This number is equal to 16 * 11 (176) + 3 = 179.
Therefore, the hexadecimal number B3 stands for 179 in denary terms.
Why do we use hexadecimal?
Binary numbers have disadvantages as well as advantages:
It is hard for people to read and understand
When you write a binary number, it becomes easy to make a mistake
It is hard to spot and fix mistakes in the binary numbers
Writing binary numbers take a lot of space
Since denary to binary and vice versa was difficult, the hexadecimal notation was used instead:
It is easy to convert hexadecimal into binary, and vice versa.
Hexadecimal is easier to read than binary.
