NUMBER SYSTEM

1.1 BASIC TYPES OF NUMBER SYSTEM

What is number system? A number system is a basic symbol to represent a set of quantities. There are many types of number system. But here, we are only focus on three number system which are decimal, binary and hexadecimal number.

Here, i put a video that i think it will help you to understand more what are the binary, decimal and hexadecimal number. This video is from Khan Academy, a world-class education on  youtube.

1.2 NUMBER SYSTEM BASE

What is a base for number system types?

There are of 4 types of number system types which are :
1. binary no. systems- which has its base as 2 
2. octal - base-8 
3. decimal- base-10 
4. hexadecimal - base 16

but here, i just focus on there of them which are binary(base 2), decimal(base 10) and hexadecimal(base 16).

• BINARY NUMBER (BASE 2)

* base 2

*The number consists only two digit : 1 and 0

The weight structure of binary number

*The least significant bit (LBS) and most significant bits (MBS) is depends on the size of binary number.

• DECIMAL NUMBER

*Base 10

*The value of the assigned weight is composed by 10 digits : from 0 to 9.

• HEXADECIMAL NUMBER

*Base 16

*The composed number start from 0 until F

*The hexadecimal system is useful because it can represent every byte (8 bits) as two consecutive hexadecimal digits.

Number System Conversion

PREPARED BY : SITI ROSNIEZA EILISA BINTI JAMAL (B031410230)

1.2: NUMBER SYSTEM CONVERSION

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. 

There are many methods or techniques which can be used to convert numbers from one base to another.

Binary Number System:

BINARY to DECIMAL

steps:

• Determine column( positional) value of each  unit
• Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.
• Sum the products calculated in Step 2. The total is the equivalent value in decimal.

Example 1:

 Binary Number : 11101₂

Calculating Decimal Equivalent:

10111_{2 =} 

Weight	24	23	22	21	20
Value represented	16	8	4	2	1
Binary(*)	1	0	1	1	1

 10111_{2  = 16+4+2+1}

                   _{=  23}

Binary Number : 10111₂ = Decimal Number : 23<sub>10

BINARY to HEXADECIMAL</sub>

steps:

•  Divide the binary digits into groups of four (starting from the right).
• Convert each group of four binary digits to one hexadecimal symbol.

Example 2:

0.1011_{2    =} 

                                     _{8+0+2+0=10(A)                                      8+4+0+1=13(D)}

                                              _{↑                                                                          ↑}

1110

1010

0101

1101

1001

.

011

     ↓                                                ↓                                              ↓                                     ↓      8+4+2+0+14(E)                    0+4+0+1=5                               8+0+0+1=9                           0+4+2+0=6

11101010010111011001.011_{2  =} EA5D9.6₁₆

Octal Number System

 The base of octal system is 8.

OCTAL to BINARY:

Example 1:

Octal:	0	1	2	3	4	5	6	7
Binary:	000	001	010	011	100	101	110	111

Octal  = 3 4 5
Binary =  011100101 binary

OCTAL to HEXADECIMAL:

Example 2 :

When converting from octal to hexadecimal, it is often easier to first convert the octal number into binary and then from binary into hexadecimal. 

For example, to convert 345 octal into hex:

Octal  =	3	4	5
Binary =	011	100	101	= 011100101 binary

Drop any leading zeros or pad with leading zeros to get groups of four binary digits (bits):
Binary 011100101 = 1110 0101

Then, look up the groups in a table to convert to hexadecimal digits.

Binary:	0000	0001	0010	0011	0100	0101	0110	0111
Hexadecimal:	0	1	2	3	4	5	6	7

Binary:	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal:	8	9	A	B	C	D	E	F

Binary =

1110

0101

Hexadecimal = E5 hex

Decimal Number System:

 DECIMAL to HEXADECIMAL:

Example 1:

DECIMAL  = 1792

Decimal:	0	1	2	3	4	5	6	7
Hexadecimal:	0	1	2	3	4	5	6	7

Decimal:	8	9	10	11	12	13	14	15
Hexadecimal:	8	9	A	B	C	D	E	F

             

DECIMAL to BINARY:

Example 2:

Example 1 - (Convert Decimal 44 to Binary)

Hexadecimal Number System:

Steps:

•  Convert each hexadecimal digit to a 4 digit binary number
•  Combine all the resulting binary groups (of 4 digits each) into a single binary number.

Example 1: 

Hexadecimal to Binary

Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.

Hexadecimal:	0	1	2	3	4	5	6	7
Binary:	0000	0001	0010	0011	0100	0101	0110	0111

Hexadecimal:	8	9	A	B	C	D	E	F
Binary:	1000	1001	1010	1011	1100	1101	1110	1111

Hexadecimal =A2DE

Binary =1010001011011110 binary

Example 2: 

Hexadecimal to Decimal:

This is quite simple.Multiply the second hex digit by 16 and then add the first digit to it

Hexadecimal = 2E

2 x 16 = 32

      E  =  14

14 + 32 = 46

                                          https://www.youtube.com/watch?v=Y0ybldaI5_M

• http://www.byte-notes.com/number-system-computer
• http://www.tutorialspoint.com/computer_fundamentals/computer_number_conversion.htm
• http://www.robotroom.com/NumberSystems3.html
• http://www.teach-ict.com/gcse_computing/ocr/214_representing_data/number/miniweb/pg4.htm
• http://www.is.wayne.edu/OLMT/BINARY/PAGE3.HTM

       PREPARED BY : WAN NORAQILAH BINTI A.RAZAK

2.4) COMPLEMENT NUMBER

Property

Two's complement representation allows the use of binary arithmetic operations on signed integers, yielding the correct 2's complement results.

Positive Numbers

Positive 2's complement numbers are represented as the simple binary.

Negative Numbers

Negative 2's complement numbers are represented as the binary number that when added to a positive number of the same magnitude equals zero.

Negative Number Conversion

• Complements are used to simplify the subtraction 

operation and for logical manipulations. 

• For each base (r), there are two complements: 

• (r-1)'s complement, also called diminished radix 

complement 

• r's complement, also called radix complement. 

• (r-1)'s complement Æ For any integer number N in base r 

with number of digits equal n we define: 

• (r-1)'s complement of N = r N n ( −1)− . 

• R's complement of N = r N n − = (r-1)'s complement + 1 

If the sign bit is zero,

then the number is greater than or equal to zero, or positive.

If the sign bit is one,

then the number is less than zero, or negative.

Calculation Of 2's Complement

ADDITON

5 + (-3)  =  2   0000 0101 = +5
                    + 1111 1101 = -3
                        0000 0010 = +2

SUBTRACTION

7 - 12  =  -5    0000 0111 =  +7
                     - 1111 0100 = -12
                       1111 1011 =  -5 ← answer from 2's complement must be convert to the binary number.
                                                      Hence, 1111 1011 to be 0000 0101.


MULTIPLICATION

(-4) × 4  =  -16    1111 1100 =  -4
                         x 0000 0100 =   4
                            1111 0000 = -16←  answer from 2's complement must be convert to the binary
                                                            number. Hence, 1111 0000 to be 0001 0000.

DIVISION

7 ÷ 3  =  2 remainder 1   0000 0111 =  +7        0000 0100 = +4
                                        1111 1101 =  -3         1111 1101 = -3
                                        0000 0100 =  +4         0000 0001 = +1 (remainder)   

• http://opencourseware.kfupm.edu.sa/colleges/ces/ee/ee200/files%5C3-Handouts_Lecture_3.pdf
• http://academic.evergreen.edu/projects/biophysics/technotes/program/2s_comp.htm
• https://www.youtube.com/watch?v=u01imtvqsrU

PREPARED BY : SITI NORHAYATI BINTI MASHUDI



                BINARY NUMBER OPERATION

  Binary Addition

Example:

Consider the addition of decimal numbers:

     23

    +48

    ___

~We begin by adding 3+8=11. Since 11 is greater than 10, a one is put into the 10's column (carried), and a 1 is recorded in the one's column of the sum.

~  Next, add {(2+4) +1} (the one is from the carry)=7, which is put in the 10's column of the sum. Thus, the answer is 71.

Binary addition works on the same principle, but the numerals are different. Begin with one-bit binary addition:

    0    0    1

   +0   +1   +0

  ___  ___  ___  

    0    1    1

~1+1 carries us into the next column. In decimal form, 1+1=2. In binary, any digit higher than 1 puts us a column to the left (as would 10 in decimal notation). The decimal number "2" is written in binary notation as "10" (1*2^1)+(0*2^0). 

~Record the 0 in the ones column, and carry the 1 to the twos column to get an answer of "10." In our vertical notation,

    1

   +1

  ___

   10

The process is the same for multiple-bit binary numbers:

         1010

        +1111

       ______

· Step one:
Column 2^0: 0+1=1.
Record the 1. 

Temporary Result: 1; Carry: 0

· Step two:
Column 2^1: 1+1=10. 
Record the 0, carry the 1.
Temporary Result: 01; Carry: 1

· Step three:
Column 2^2: 1+0=1 Add 1 from carry: 1+1=10. 
Record the 0, carry the 1.
Temporary Result: 001; Carry: 1

· Step four:
Column 2^3: 1+1=10. Add 1 from carry: 10+1=11.
Record the 11. 
Final result: 11001

Alternately:

    11   (carry)

    1010

   +1111

  ______

   11001

Always remember

· 0+0=0

· 1+0=1

· 1+1=10

Binary substraction

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)

Binary Multiplication

Multiplication in the binary system works the same way as in the decimal system:

must remember

· 1*1=1

· 1*0=0

· 0*1=0

   101                      

  * 11

  ____

   101

  1010

 _____

  1111

Note that multiplying by two is extremely easy. To multiply by two, just add a 0 on the end.

Binary Division

Follow the same rules as in decimal division. For the sake of simplicity, throw away the remainder.

For Example: 111011/11

      10011 r 10

    _______

  11)111011

    -11

     ______

       101

       -11

     ______

        101

         11

     ______

         10

https://www.youtube.com/watch?v=sFd5bnDdB3Q

http://academic.evergreen.edu/projects/biophysics/technotes/misc/bin_math.htm

                                  PREPARED BY:NOR FADILA BT MOHD YUNUS

                   HEXADECIMAL NUMBER OPERATION

   HEXADECIMAL ADDITION.

Use the following steps to perform hexadecimal addition:

1. Add one column at a time.

2. Convert to decimal and add the numbers.

3.(a) If the result of step two is 16 or larger subtract the result from 16 and carry 1 to the next column.

(b) If the result of step two is less than 16, convert the number to hexadecimal

Example 1 :

                      Add 16F₁₆ + 4A2₁₆.

                              1  1            ←  Carried over digits
                              1  6   F
                     +       4  A  2
                          ----------------
                              6   1  1
                          -----------------

Example 2 :

                          Add C4BDF16 + E2CA916.

             

  HEXADECIMAL SUBTRACTION.

Firstly,we have to convert the hexadecimal number to binary number.Then,take the 2's complement of the binary number and change it back to hexadecimal number.Add both numbers to get the result.

 To compute 1B₁₆ - 0D₁₆, we first find the "15's complement" of 0D₁₆:

F	F
-	0	D
	-	-
	F	2

add 1 to that to get F3₁₆, and add that to 1B₁₆ to get:

	c
		1	B
+		F	3
		-	-
		0	E

But borrowing is not the same problem in hexadecimal as it is in the binary number system: in the previous example, we can borrow a 1 from the 16's place:

	1	B
-	0	D
	-	-
		?

becomes

		(1016)
	0	B
-	0	D
	-	-
	0	E

 since 10₁₆ + B₁₆ - D₁₆ is E₁₆.

HEXADECIMAL MULTIPLICATION.

Prepared by: Masturah Binti Mohammad Liza