Information Processing Session 5B Binary Arithmetic Slide

Objectives After studying this week’s work, you should: gain insight into how the processor deals with information at the bit level Understand numbers written in binary form Be able to convert numbers from binary to decimal notation and vice-versa Be able to add numbers in binary form Slide

Bits A bit is the smallest piece of information in the computer At a single location, the information is either One - current is ON Zero -current is OFF There are no in-between states ON OFF Slide

Bytes The way that information is coded is to use a sequence of zeros and ones It is usual to have a sequence of 8 bits collected together This is called a byte Slide

Bits and Bytes Depending upon the design of the computer, there could be 4, 8 16, 32, 64 (or even more!) bits processed by the computer at once For the next few slides we will look at a simple 4-bit device. Slide

A 4-bit register Reading from the right, each bit is worth double the one preceding it. The sequence, reading from the right is: 1,2,4,8,... If we had more bits, it would continue:... 16, 32, 64, etc is ON 1 is ON Slide

Binary Numbers The register shown on the right represents the binary number 0101 This has ones in the 1 and 4 cells, and zeroes in the others. The number represented is Slide = 5

Counting in Binary Counting is an automatic process Follow the sequence... Slide

Counting in Binary A pulse enters on the right Slide

Counting in Binary To begin with the first cell was OFF It is flipped to ON Slide

Counting in Binary Another pulse enters on the right Slide

Counting in Binary The first cell was ON. It is flipped to OFF The pulse moves to the next cell on the left Slide

Counting in Binary The next cell on the left was OFF That cell is flipped to ON Slide

Counting in Binary Another pulse enters from the right Slide

Counting in Binary The first cell was OFF It is flipped to ON Slide

Counting in Binary Another pulse enters from the right Slide

Counting in Binary The first cell was ON, and is flipped to OFF The pulse moves to the second cell Slide

Counting in Binary The second cell was ON, and is flipped to OFF the pulse moves to the third cell Slide

Counting in Binary The third cell was OFF and is flipped to ON Slide

Counting in Binary Follow the sequence on the right, and try to continue it. you will see that the switching creates a pattern off ON/OFF in each column Slide

Counting in Binary The 1’s column alternates 1,0,1,0 etc. The 2’s column starts at 2 and alternates two 1’s, two 0’s The 4’s column starts at 4, and alternates four 1’s, four 0’s The 8’s column starts at 8 and alternates eight 1’s eight 0’s Slide

Decimal Numbers By decimal, we simply mean that the numbers are written in powers of ten These are 1, 10, 100, 1000, etc. So that: 352 = Slide

Binary Numbers By Binary, we mean that numbers are written in powers of two These are 1, 2, 4, 8, 16 etc. So that: = Which is = Slide

Converting Binary to Decimal Example: Reading from right to left the columns are 1,2,4,8 etc. i.e So the number in decimal notation is: = 45 Slide

How do we convert Decimal to Binary? There is a specific technique which allows us to do this. It involves repeatedly dividing a number by two and noting the remainder. Slide

Converting Decimal to Binary: An example Convert 117 to binary: 117÷ 2 = 58 remainder 1 58 ÷ 2 = 29 remainder 0 29 ÷ 2 = 14 remainder1 14 ÷ 2 = 7 remainder0 7 ÷ 2 = 3 remainder1 3 ÷ 2 = 1 remainder1 1 ÷ 2 = 0 remainder1 In binary the number is: Slide

Adding In Binary Addition in binary is a direct counterpart of what happens at the processor level. First of all we will look at a numerical example Slide

Adding in Binary There are only four possible combinations. The first three are “obvious” The last one is special (remember = 2, which is 10 in binary) = = = = 0, carry 1 Slide

Adding in Binary Adding = 2 This is 10 in Binary Put 0 in the answer, carry 1 Slide

Adding in Binary Adding = 2 This is 10 in Binary Put 0 in the answer, carry 1 Slide

Adding in Binary Adding = 3 This is 11 in Binary Put 1 in the answer, carry 1 Slide

Adding in Binary Adding Carry on with this… Slide

Adding in Binary The answer: Slide

The Binary Adder We will add 0011(3) 0110(6) Slide

The Binary Adder Starting with the end column, top cell is ON This pulse enters into the bottom cell Slide

The Binary Adder The bottom cell was OFF The pulse causes it to flip to ON Slide

The Binary Adder The next top cell was ON The pulse enters into the bottom cell Slide

The Binary Adder The bottom cell was ON The pulse flips it to OFF The pulse moves to the next cell Slide

The Binary Adder The next cell is ON The pulse flips it to OFF The pulse moves to the next cell Slide

The Binary Adder The next cell is OFF The pulse flips it to ON Slide

The Binary Adder The bottom line now reads: 1001 This is 8 + 1= 9 Slide

Bits and Bytes We have seen that a 4-bit register can count from 0 [0000] to 15 [1111] This means that it has 16 different states. Slide

Bits and Bytes Each bit in the register can be ON or OFF. This means that there are two possibilities for each cell That is, altogether 2 x 2 x 2 x 2 = 16 states Slide x 2 x 2 x 2 = 2 4

Bits and Bytes The number of possible states of registers of other sizes can be worked out in the same way For example an 8-bit register (byte) has 2x2x2x2x2x2x2x2= 2 8 = 256 different states. BitsStates , ,294,987,296 Slide

Megabits and Kilobytes A Kilobyte is 2 10 bytes. This is the nearest power of two to In fact 2 10 = 1024 A megabit is 2 20 bits. This is the nearest power of 2 to 1 million. In fact 2 20 = Slide

Other Bases Decimal and Binary are two different number bases used by the computer, but there are others An important one is Hexadecimal which has 16 separate characters: 0,1,2,3,4,5,6,7, 8,9,A,B,C,D,E,F Slide

Hexadecimal The extra letters are so that the numbers can be written using one character each. This means that A8BC is a number written in Hexadecimal. These numbers are written in base 16, so that a number like 9E means the 9 is 9 x 16 = 144 the E is 14 x 1 = 14 Altogether this would be 158 Dec Hex 0  0 1  1 2  2 3  3 4  4 5  5 6  6 7  7 8  8 9  9 10  A 11  B 12  C 13  D 14  E 15  F Slide

Summary A bit has two states, ON or OFF, which means that at the core of a computer we need to use binary coding of numbers (powers of two) Registers count and add using in binary code There are algorithms for converting decimal to binary and vice-versa Binary addition has only four possible addition pairs, and a “carrying rule” Slide