1. COMPUTING FUNDAMENTALS
Instructor: Romana Farhan Lecture # 02

2.  How do computers represent data?
 Number systems
 Binary number system
 Binary Arithmetic

3. the binary number system.
 CPU consists of several million tiny electronic switches called transistors, that can be turned on or off.
 Computers know two distinct physical states
Switch is on or off
 Since these switches only have two states,
it makes sense for a computer to perform its computations with a number system that only has two digits.
the binary number system.

4. Several different number systems exist Used by humans to count
A manner of counting
Several different number systems exist
 Decimal number system
Used by humans to count
Contains ten distinct digits (0-9)
Digits combine to make larger numbers

5. Used by computers to count
 Binary number system
Used by computers to count
Two distinct digits, 0 and 1 corresponding to off or on state of switch.
0 and 1 combine to make numbers
With only these two digits, a computer can perform all the arithmetic that we can with ten digits
Goal of our study is to know
Converting between binary and decimal,
Performing binary arithmetic,

6. of any number by system is of digits in the the number
 The base determined system.
of any number by
system is
of digits in the
the number
 For example
Binary is a base 2 number system
Decimal is a base 10 number system
Octal ?????
Hexadecimal ?????
 Distinguish between various number systems by putting a small subscript after the number to indicate the base.

7. understand the relationship between the digits of a given number,
 For example ◦ ◦
 we need to
understand the relationship
between the digits of a given number,
the position of those digits, and the base of the number system.
Let's take another look at our example number in
the decimal system.

8.  Generalized formula

9. by both the number and its position.
 In decimal system value of number is defined
by both the number and its position.
 Most significant bit/digit (MSB,MSD)
The digit in a number which can affect the number
Significantly
It is the non zero digit of a number that is in the
farthest left.
 The least significant bit/ digit (LSB, LSD)
It is the digit in a number that has the least effect on that number.

10. Any changes in the LSD will have minimal effect on
the value.
It is the non zero digit of a number that is farthest right.
For example

11.  To convert a decimal number to a binary number,
 The decimal number is successively divided by 2.
 Follow the following steps
Divide the decimal number by and note the remainder.
Divide the quotient you got from the previous division once again by 2 and note the remainder. For each step, the division results in a remainder of 1 or 0.

12. Repeat step 2 until the quotient becomes 1.
The binary number is obtained by ordering the 1s and 0s obtained as remainders from bottom to top.

13.  To convert the fractional part of the decimal
number to a binary number, the part is repeatedly multiplied by 2.
 Follow the following steps
fractional
The decimal fraction is multiplied by 2. the integer
part, which is noted i.e. either 0 or 1
The fractional part of the previous product is multiplied by 2.
Repeat the first step until the fraction repeats or
terminates.
The integer part which is obtained as a result forms a string of 0s and 1s which becomes the binary equivalent of the decimal fraction.

14. 4. to obtain the binary number of the equivalent fractional, you need to consider the integer part from top to bottom.

15.  167
 0.875 

16. number, perform the following steps.
 To convert
a binary
number
to a decimal
number, perform the following steps.
1. Start
with the
rightmost digit and
multiply that
digit with 20. Here 2 represents the base of the
number
position,
system and 0 represents the
which is the rightmost digit
digit
whose
position always starts with 0.
Continue working from right to left, and multiply
each digit with an increasing power of 2
Repeat step 2 until all the digits have been multiplied
Add the result of each worked power of 2 for a final answer

18. decimal number in the same manner but
 Fractional binary number can converted to
decimal number in the same manner but
 The exponent are powers of two with negative values for digits on the right side of the binary point.

19.  

20.  Binary Addition
 Binary Subtraction
 Binary Division

21. 1 + 1 10  Basic four rules of addition
1. First, we apply Rule 4 to find the sum of the first two 1s. 1
+ 1 10 

22. answer to the second column. 10 + 1 11 to find the
Next, we take the previous result of 102 and add the final 1 to it. Notice that we use Rule 2 ( = 1) to find the answer to the first column,
3. We use Rule 3 (1 + 0 = 1)
answer to the second column. 10
+ 1 11
to find the 

23. 4. Now we have derived another rule for binary arithmetic
Now we have derived another rule for binary arithmetic. The sum of three 1s in binary is 112. 1
+ 1 11 

24.  Add 1112, 1102, 11012, 1012, and 11102

25. how to add the binary numbers 110.012 and 1.0112
 Addition is same as addition of binary integers, we need to take care of radix point while adding fractional part.
 Example
how to add the binary numbers and
1. First, we align the two numbers so that the radix
point of each number is located column
in the same
2. Next, we fill in the blank spaces with 0s and add the two numbers together

26. The first column adds to 1. 1
The second column adds to 102, so we write
a 0 below it
and carry a 1 to the next 1
column 01

27. 5. All of the remaining columns add to 1, so
we write 1 below them. 1
4. This gives us a final answer of

28.  Add 

29. 2(10)
3 0
The basic rules for binary subtraction are
Consider the problem of subtracting 12 from
102.

30. borrow a 1 from the next column.
 To compute
the first
column,
we need to
borrow a 1 from the next column.
 Recall that two 1s generated a carry in addition.
 If we reverse this process, we can borrow a 1 from the second column.
Once we borrow from the second column,we cross out the 1 and write 0 above it to show this column is now empty. 10 -1

31.  Since we must borrow a 1 from the next
 After cleaning up our work, we can see that the first column of our answer is identical to Rule 4.
 Since we must borrow a 1 from the next
column, = 1. 10
- 1 1

32.  Subtract
 −
 − 10010
 −
 − 
 When the binary fraction is subtracted from the binary fraction what is the correct result?

33. we multiply by is either
 Binary multiplication uses the same technique
as decimal multiplication.
 Binary multiplication is much easier because
each digit
one.
 Example
we multiply by is either
1102 multiplicand
10 2multiplier
zero or
 multiplying by 102
1. First, we note that 1102 is our multiplicand and 102 is our multiplier.

34. next digit of our multiplier which is 1.
2. We begin by multiplying
1102
by the
rightmost digit of our multiplier which is 0. Any number times zero is zero, so we just write zeros below.
110 x
3. Now we multiply the multiplicand by the
next digit of our multiplier which is 1.
To perform this multiplication, we just need to copy the multiplicand and shift it one column to the left as we do in decimal multiplication.

35. we add our results together. The
110 x
110
we add our results together. The
4. Now
product of our multiplication is x
110
1100

36. Ignore radix point, do not align them
Multiply numbers as whole numbers
Starting from right of the product, separate as many places as there are in the two numbers together

37. 10.01*1.01 1001 *101 --------- 0000 ------------- 101101
Now put the radix point, together binary fractions have four decimal places. starting from right separate the four decimal places

38. remember some important rules: the remainder is less than the divisor,
 When doing binary division, we need to
remember some important rules:
 When the remainder is greater than or equal to the divisor, write a 1 in the quotient and subtract.
 When
the
remainder is less than the divisor,
write
a 0
in the quotient and add another
digit
from the dividend.

39. mark a radix point in the dividend and append a zero.
 If all the digits
of the
dividend
have been
considered and
there
is still a
remainder,
mark a radix point in the dividend and append a zero.
 Remember that some fractions do not have an exact representation in binary, so not all division problems will terminate.

40. 1. 10|11 1
2. 10|11 10
3. 1.
10|11.0

41. 1.
10|11.0 10
1.1

42.  Divisor must be a whole number, if it is not, then we will multiply it so that it becomes one.
 Multiply the dividend by the same power
 Example
 /1.01

43.  Divide by 1102
 /11.011

44. therefore data and numbers must be represented in a numeral format.
 Digital computers
respond to
numbers,
therefore data and numbers must be represented in a numeral format.
 The ON/OFF positions of electronic switches in a digital computers correspond to binary digits 0 or 1.
 Inter-conversion of numbers from one number system into other becomes important for computation and data processing in digital computers.

45. required to perform addition, division and multiplication.
 In digital
computers very
few rules are
required to perform addition, division and multiplication.
subtraction,

46. Convert to decimal i
ii.
iii. iv

47.  45 i. 69
ii.
1066
iii. 99
iv. 1939
v. 998

48. DIVIDE
/1011
/11001
/10110

49. Multiply
1101 * 1101
1011* 1001 * *
* 10110