Binary Systems
Binary Systems
• The positive integers represented by the sequence of 1s and 0s are known as the binary numbers or the base 2 numbers.
• Whereas the base 10 numbers are known as denary numbers.
Base 10 Headings:
Base 2 Headings:
Why We Use Binary Numbers?
• Transistors are used by the microprocessors that identify voltage levels instead of decimal numbers.
• High voltage in a microprocessor is identified as an ON state or a '1' whereas, a low voltage is identified as an OFF state or a '0'.
• The low voltage level is usually 0V while high voltage varies from 3.3V to 5V.
• Different hardware uses different ways to identify the binary number system e.g. in CD ROMs, tiny black spots are considered as ‘0’ and shiny spot represents a ‘1’.
What is a Bit?
• The smallest representation of binary numbers can be broken down into ‘bits.
• Bits in binary numbers can be considered as ‘digits’ in the denary numbers.
• In denary numbers, a digit is a place that holds values between 0 to 9 whereas in binary numbers a bit is a place that can hold value ‘0’ or ‘1’ only.
• In denary numbers, a group of digits form larger numbers e.g. 67 has two digits, 984 has three digits, etc.
• In 67, ‘7’ holds the ‘1s’ place and ‘6’ holds ‘10s’ place.
• In 984, ‘4’ holds the ‘1s’ place, ‘8’ holds ‘10s’ place and ‘9’ holds ‘100s’ place.
Number Representation in Denary:
• Let’s take 984 for understanding the number representation.
1st Method:
(9 x 100) + (8 x 10) + (4 x 1) = 900 + 80 + 4 = 984 |
• In this method we simply multiply each digit with the place it fills.
2nd Method:
(9 x 10^{2}) + (8 x 10^{1}) + (4 x 10^{0}) = 900+ 80 + 4 = 984 |
• In this method, starting from right to left, each digit is multiplied by 10x.
• The power of 10 is least i.e. zero at the R.H.S and highest on the L.H.S.
Number Representation in Binary:
• In binary, we use the 'raised to the power' method to find out the value of the binary number.• This method is exactly the same, however, instead of 10x, we will use 2x.
Example:
• Let’s find out the value of binary number 1010.• Using the raised to the power method,(1 x 2^{3}) + (0 x 2^{2}) + (1 x 2^{1}) + (1 x 2^{0})= 8 + 0 + 2 + 0 = 10 |
Binary to Denary Conversion:
• Converting a binary number to denary is simple and we already discussed it in the previous heading.• It involves multiplication of each bit with 2x where ‘x’ is lowest at R.H.S and highest at L.H.S.• Just to revise it, let’s take a binary number 10111.(1 x 2^{4}) + (0 x 2^{3}) + (1 x 2^{2}) + (1 x 2^{1})+ (1 x 2^{0}) = 16 + 0 + 4 + 2 + 1 = 23 |
Denary to Binary Conversion:
• There are two methods to convert a denary number to its binary representation.
• The first method is 'trial and error' while the second method involves the successive division of the number by 2.
• Let’s understand both with an example.
Method 1 - Trial & Error Method:
• This method involves placing ‘1’ and ‘0’ at correct places so that the sum equals the denary value.
• Let’s see the conversion of 107.
• Do not feel confused with the numbers above ‘1’ and ‘0’ as it represents 2x.
• From R.H.S, 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64 and 27 = 128.
• The binary number representation contains 8 bits (standard) that’s why used 2 raised to the power 0 till 7.
• If you see a representation that contains less than 8 bits e.g. 1011. The rest 4 bits on the left side were '0s'. 00001011 and 1011 represent the same number and both ways are correct.
Method 2 – Successive Division Method:
Let’s use 107 to understand this method.
In this method 107 is divided again and again until the value becomes 0.
• We divided 107 by 2, it gave us 53 and remainder 1.
• 53 divided by 2, gave us 26 and remainder 1.
• 26 divided by 2, gave us 13 and remainder 0.
• 13 divided by 2, gave us 6 and remainder 1.
• 6 divided by 2, gave us 3 and remainder 0.
• 3 divided by 2, gave us 1 and remainder 1.
• 1 divided by 2, gave us 0 and remainder 1.
• Now that the number became 0, we can add 0s on the L.H.S to make it an 8-bit number.
What is a Byte & Memory Size?
• One binary digit is known as a ‘bit’ while 8 binary bits make a ‘byte.
• It is the smallest unit of memory in a computer.
• Some systems use larger bytes e.g. 16-bit, 32-bit, etc. but memory size is always a multiple of 8.
• A single letter or character uses 1 byte of memory but many special characters and characters from foreign languages use more than one byte.
Example Use of Binary System:
• The computer system uses registers to control devices like robotics, digital instruments, and power systems.
• The registers are made up of binary bits i.e. ‘0’ and ‘1’.
• Following is an example of a robot vacuum cleaner that will help you understand how registers are used to control devices.
• As shown in the figure above, there are three wheels in a robot vacuum cleaner.
• Wheel ‘A’ rotates on a spindle while Wheels ‘B’ and ‘C’ are fixed and attached to a motor.
• Wheel ‘A’ also facilitates in direction changes.
• An 8-bit register is used to control the vacuum cleaner.
• If the register contains the value 10101010 then Motor B and C both are on and the vacuum cleaner will move forward.