By the end of this unit, you should be able to

Binary Number System


Binary or base 2 number systems use Bit (binary digit): 2 digits, 0 and 1

Binary to decimal



1.1.3 Octal Number System


Positional Notation: Octal





From Base 8 to Base 10

72638	= 3,76310
Power	8^3	8^2		8^1		8^0
	512	64		8		1
	x 7	x 2		x 6		x 3
Sum for Base 10	3,584	128		48		3

1.1.4 Hexadecimal Number System

Hex is easier to read and write than binary. Modern computer operating systems and networks

present variety of troubleshooting data in hex format.

Hexadecimal to decimal





1.2 Converting from Base 10

Power Base	8	7	6	5	4	3	2	1	0
2	256	128	64	32	16	8	4	2	1
8				32,768	4,096	512	64	8	1
16					65,536	4,096	256	16	1

1.2.1 From Base 10 to Base 2



1.2.2 From Base 10 to Base 8
To convert a decimal (base 10) number to Base-8 (Octal) number we
1) Divide the given decimal number by 8.
2) Write the quotient and remainder and continue the division till quotient becomes 0.
The remainders are taken and written from bottom to top form the base-8 number that is starting from the most significant digit to the least significant digit.

Example :

Given: Base 10 number, .
To find the base 8 numbers:
8 | 6 8 9
8 | 8 6 - 1
8 | 1 0 - 6
8 | 1 - 2
8 | 0 - 1

Therefore, the required base 8 number,

.

3. 
   From Base 10 to Base 16
   

1.3 From Base 16 to Base 2

We can use the nibble approach

Base 16	1	F	6	7
Base 2	0001	1111	0110	0111

Convert Binary to Hex

Example: Convert 1001 1110 0111 0000 to Hex

After grouping follow the procedure as discussed in the previous section use the symbols of Hex

number system like 13=E

1001 1110 0111 0000

9 E 7 0

Example: Convert 10010100110000 to Hex

10 0101 0011 0000

0010 0101 0011 0000
2 5 3 0
Convert Octal to Binary

For each of the Octal digit write its binary equivalent.

Example: Convert 2570 (octal) to binary

010 101 111 000


1.4 Counting in Base 2

Binary Number	Equivalent				Decimal Number
	8’s (2^3)	4’s (2^2)	2’s (2^1)	1’s (2^0)
0				0 x 2^0	0
1				1 x 2^0	1
10			1 x 2^1	0 x 2^0	2
11			1 x 2^1	1 x 2^0	3
100		1 x 2^2			4
101		1 x 2^2		1 x 2^0	5
110		1 x 2^2	1 x 2^1		6
111		1 x 2^2	1 x 2^1	1 x 2^0	7
1000	1 x 2^3				8
1001	1 x 2^3			1 x 2^0	9
1010	1 x 2^3		1 x 2^1		10

1.4 Arithmetic (Addition and Subtraction)

Base	Problem	Largest Single Digit
Decimal	6 +3	9
Octal	6 +1	7
Hexadecimal	6 +9	F
Binary	1 +0	1

Base	Problem	Carry	Answer
Decimal	6 +4	Carry the 10	10
Octal	6 +2	Carry the 8	10
Hexadecimal	6 +A	Carry the 16	10
Binary	1 +1	Carry the 2	10

1.4.1 Binary Arithmetic

Rules of Binary Addition

• 
  0 + 0 = 0
  
• 
  0 + 1 = 1
  
• 
  1 + 0 = 1
  
• 
  1 + 1 = 0, and carry 1 to the next more significant bit
  

For example,

00011010 + 00001100 = 00100110		1 1		carries
		0 0 0 1 1 0 1 0	=	26(base 10)
		+ 0 0 0 0 1 1 0 0	=	12(base 10)
		0 0 1 0 0 1 1 0	=	38(base 10)
00010011 + 00111110 = 01010001		1 1 1 1 1		carries
		0 0 0 1 0 0 1 1	=	19(base 10)
		+ 0 0 1 1 1 1 1 0	=	62(base 10)
		0 1 0 1 0 0 0 1	=	81(base 10)

Note: The rules of binary addition (without carries) are the same as the truths of the XOR gate.

• 
  EXCLUSIVE-OR
  
  • 
    Output is “1” only if either input, but not both inputs, is a “1”
    

1	1	1	1	1
	1	1	0	1	1	0	1
+			1	0	1	1	0
1	0	0	0	0	0	1	1

Rules of Binary Subtraction

• 
  0 - 0 = 0
  
• 
  0 - 1 = 1, and borrow 1 from the next more significant bit
  
• 
  1 - 0 = 1
  
• 
  1 - 1 = 0
  

For example, 00100101 - 00010001 = 00010100		0		borrows
		0 0 1 10 0 1 0 1	=	37(base 10)
		- 0 0 0 1 0 0 0 1	=	17(base 10)
		0 0 0 1 0 1 0 0	=	20(base 10)
00110011 - 00010110 = 00011101		0 10 1		borrows
		0 0 1 1 0 10 1 1	=	51(base 10)
		- 0 0 0 1 0 1 1 0	=	22(base 10)
		0 0 0 1 1 1 0 1	=	29(base 10)

1.4.2 Octal Arithmetic
Addition of Octal Numbers
The addition of octal numbers is not difficult provided you remember that anytime the sum of two digits exceeds 7, a carry is produced. Compare the two examples shown below:


Table 1.1. - Octal Addition Table
If you use the concepts of addition you have already learned, you are ready to add octal numbers.
Work through the solutions to the following problems:

Subtraction of Octal Numbers
The subtraction of octal numbers follows the same rules as the subtraction of numbers in any other number system. The only variation is in the quantity of the borrow.
In the decimal system, you had to borrow a group of 10. In the binary system, you borrowed a group of 2. In the octal system you will borrow a group of 8.
Consider the subtraction of 1 from 10 in decimal, binary, and octal number systems:

In each example, you cannot subtract 1 from 0 and have a positive difference. You must use a borrow from the next column of numbers.

Let's examine these problems:

When you use the borrow, the column you borrow from is reduced by 1, and the amount of the borrow is added to the column of the minuend being subtracted.
The following examples show this procedure:

In the octal example 78 cannot be subtracted from 68, so you must borrow from the 4. Reduce the 4 by 1 and add 108 (the borrow) to the 68 in the minuend.
By subtracting 78 from 168, you get a difference of 78. Write this number in the difference line and bring down the 3.
You may need to refer to table 1-1, the octal addition table, until you are familiar with octal numbers. To use the table for subtraction, follow these directions. Locate the subtrahend in column Y. Now find where this line intersects with the minuend in area Z. The remainder, or difference, will be in row X directly above this point.


1.4.3 Hexadecimal Arithmetic
Addition of Hex Numbers
Study the hex addition table in table 1-2. Using the table, add 7 and 7. Locate the number 7 in both columns X and Y. The point in area Z where these two columns intersect is the sum; in this case 7 + 7 = E.
As long as the sum of two numbers is 15 or less, only one symbol is used for the sum. A carry will be produced when the sum of two numbers is 16 or greater, as in the following examples:


Table 1-2. - Hexadecimal Addition Table

Use the addition table and follow the solution of the following problems:

In this example each column is straight addition with no carry. Now add the addend (78416) and the sum (BDA16) of the previous problem:

Here the sum of 4 and A is E. Adding 8 and D is 1516; write down 5 and carry a 1. Add the first carry to the 7 in the next column and add the sum, 8, to B. The result is 1316; write down 3 and carry a 1. Since only the last carry is left to add, bring it down to complete the problem.

Subtraction of Hex Numbers
In the preceding sections you learned all the rules for subtraction. Now you need only to apply those rules to a new number system. The symbols may be different and the amount of the borrow is different, but the rules remain the same.
Use the hex addition table (table 1-2) to follow the solution of the following problems:

Working from left to right, first locate the subtrahend (2) in column Y. Follow this line across area Z until you reach C. The difference is located in column X directly above the C - in this case A. Use this same procedure to reach the solution:

Now examine the following solutions:

In the previous example, when F was subtracted from 1E, a borrow was used. Since you cannot subtract F from E and have a positive difference, a borrow of 1016 was taken from the next higher value column. The borrow was added to E, and the higher value column was reduced by 1.

Exercises:

1. Convert the following base 10, decimal numbers to base 2, binary numbers.

A) 33
B) 87
C) 312
D) 972

2. Convert the following base 2, binary numbers to base 10 decimal numbers.

1. 
   1100111₂
   

2. 
   1110010₂
   

3. 
   1001₂
   

3. Convert the following base 10, decimal numbers to base 8, octal numbers.

1. 
   118
   

2. 
   2569
   

3. 
   1000
   

4. 
   45
   

4. Convert the following base 8, octal numbers to base 10, decimal numbers.

1. 
   467₈
   

2. 
   2543₈
   

3. 
   777₈
   

4. 
   35₈
   

5. Convert the following base 10, decimal numbers to base 16, hexadecimal numbers.

1. 
   5589
   

2. 
   198
   

3. 
   69
   

6. Convert the following base 16, Hexadecimal numbers to base 10, decimal numbers.

1. 
   7903₁₆
   

2. 
   9C5₁₆
   

3. 
   ABCD₁₆
   

4. 
   1147₁₆
   

7. Convert the following octal numbers to their hexadecimal equivalent.

1. 
   406₈
   

2. 
   34526₈
   

3. 
   555₈
   

8. Convert the following Hexadecimal numbers to their Octal equivalent.

1. 
   CAD₁₆
   

2. 
   2899₁₆
   

3. 
   1ABC₁₆
   

9. Perform the following additions:

1. 
   11111₂ + 11111₂
   

2. 
   10010110₂ + 10110111₂
   

3. 
   467₈ + 366₈
   

4. 
   1275₈ + 774₈
   

5. 
   AAA₁₆ + 359₁₆
   

6. 
   183266₁₆ + 10B7F₁₆
   

10. Perform the following subtractions:

1. 
   10000₂ - 1111₂
   

2. 
   10010110₂ - 11111₂
   

3. 
   467₈ - 366₈
   

4. 
   1275₈ - 774₈
   

5. 
   AAA₁₆ - 359₁₆
   

6. 
   183266₁₆ - 10B7F₁₆
   

MDIS-DIT 1- Mathematics for computing