Number system part 1

NUMBER SYSTEMPart 1

Objectives - To understand3.1 Analyses how numeric data are

represented in computers

I. Decimal representation of numbers (Signed and Unsigned)

II. Integers III. Fixed Point and Floating-Point numbers IV. Number systems used in computing

Binary, Octal, Hexa-decimal logic operationV. Conversions among number systems

Introduction

Introduction

A number system defines a set of values used to represent a quantity.

The study of numbers is not only related to computers. We apply numbers everyday, and knowing how numbers work, will give us an insight of how computers manipulate and store numbers.

A number is a mathematical object used in counting and measuring. It is used in counting and measuring. Numerals are often used for labels, for ordering serial numbers, and for codes like ISBNs. In mathematics, the definition of number has been extended over the years to include such numbers as zero, Positive numbers , Negative numbers , Integers numbers , Rational numbers, Irrational numbers, Whole number and complex numbers.

Certain procedures which take one or more numbers as input and produce a number as output are called numerical operation.

Complex number:- The complex numbers consist of all numbers of the form (a+bi) ; Where a and b are real numbers and i=0.1

Ex :- 2+5i , 3+3i , … . .. . . . , 14+20i

Real Numbers (R) :- The real numbers include all of the measuring numbers

Ex :- -69 , -2 , -1.5 , +5 …+ 7.5 . .. . . . , +14

Rational Numbers (Q) :- A rational number is a number that can be expressed as a fraction with an integer and a non-zero natural number denominator Ex :- -25 , -15/10 , 0 , 0.05 , 0.2 , 7/3 , 25 ,36

Irrational Numbers:- A decimal that can be written as a fraction either ends (terminates)or forever repeats about which we will see in detail further.

Ex – π =3.14159365358979

Natural numbers (N) :-All positive numbers (Counting numbers) started with 1.However in the 19th century, mathematicians started including 0 in the set of natural

numbers.

Ex :- 0,1,2,3,4,……,55 ,105 . .. . . .

Types of numbers

Types of numbers

Integers (Z):- Integers are the number which includes positive and negative numbers. Negative numbers are usually written with a negative sign (also called a minus sign)in front of the number they are opposite of .When the set of negative numbers is combined with the natural numbers zero, the result is the set of integer numbers

Ex :- -25 , -15 , +0 , +25 , +36

Fractions :- This is a type of a rational number. Fractions are written as two numbers (m/n), the numerator (m) and the denominator(n)

1. Fixed point Number2.Floating point number

Fixed point number :-Involving or being a mathematical notation(as in a decimal

system) in which the point separating whole numbers and fractions is

fixed

A fixed point number has a specific number of bits (or digits) reserved

for the integer part (the part to the left of the decimal point) and a

specific number of bits reserved for the fractional part (the part tothe

right of the decimal point). No matter how large or small your number is, it will always use the same number of bits for each portion.

For example:- if your fixed point format was in decimal IIIII.FFFFF then the largest number you could represent would be 99999.99999 and the smallest would be 00000.00001. Every bit of code that processes such numbers has to have built-in knowledge of where the decimal point is.

123.5625.238955.30.50.0098

A fixed point number just means that there are a fixed number of digits after the decimal point.

Floating point number :-A floating point number does not reserve a specific number of bits for the

integer part or the fractional part. Instead it reserves a certain number of bits for the number

(called the mantissa or significant) and a certain number of bits to say where within that number the decimal place sits (called the exponent).

For example:-So a floating point number that took up 10 digits with 2 digits reserved for the exponent might represent a largest value of

9.9999999e+50 and a smallest value of 0.0000001e-49.

1.2 × 102

35.02 × 1080

0.258 × 10-14 / 2.58 × 10-15 / 25.8 × 10-16

A floating point number allows for a varying number of digits after the decimal point.

Types of numbers

Precision({Í ý<) and Accuracy (ùÚØ<÷À]õ$<)

•Precision is the measure of how closely individual measurements agree with one another.

• Accuracy is how closely individual measurements with the correct value.

Significant Digits(µ<µ&&Ú &‘ç]$‘æ×) "Significant Digits", also called "Significant Digits of Precision", are the digits in a number starting with the first non-zero digit and ending with the last non-zero digit. Those digits can be anywhere relative to the decimal.

Non-zero digits are always significant.

22 has two significant digits,22.3 has three significant digits.

With zeroes, the situation is more complicated

a. Zeroes placed before other digits are not significant; 0.046 has two significant digits.

b. Zeroes placed between other digits are always significant; 4009 kg has four significant digits.

c. Zeroes placed after other digits but behind a decimal point are significant;

7.90 has three significant digits. 85.00 g has four significant digits9.000 000 000 mm has 10 significant digits

Significant Digits

d. Zeros at the end of a number but to the left of a decimal may or may not be significant. If such a zero has been measured, or is the first estimated digit, it is significant. On the other hand, if the zero has not been measured or estimated but is just a placeholder, it is not significant. A decimal placed after the zeros indicates that they are significant

1. 2000 m may contain from one to four significant digits, depending on how many zeros are placeholders. For measurements given in this text , assume that 2000 has one significant digits.

2. 2000. m contains four significant digits, indicated by the presence of the decimal point

3.1000.0 has five significant digits (the ".0" tells us something interesting about the presumed accuracy of the measurement being made: that the measurement is accurate to the tenths place, but that there happen to be zero tenths)

For example:- in the number 8200, it is not clear if the zeroes are significant or not. The number of significant digits in 8200 is at least two, but could be three or four. To avoid uncertainty, use scientific notation to place significant zeroes behind a decimal point:i.8.200 *103 has four significant digits ii.8.20 *103 has three significant digits iii.8.2 *103 has two significant digits

Significant Digits

Scientific notation - All digits expressed before the exponential term are significant.

5.060 x 10-3 m has four sig figs.

9.00 x 102 g has three sig figs.

Significant DigitMost Significant Digit

•The MSD in a number is the digit that has the greatest effect on that number. Highest power of base weightingThe digits on the left hand side are called the high-order digits (higher powers of 10)

Least Significant Digit (MSD and LSD)• The LSD in a number is the digit that has the least effect on that number.• Lowest power of base weighting• Digits on the right hand side are called the low-order digits (lower powers of 10).

Significant Digit

You can easily see that a change in the MSD will increase or decrease the value of the number the greatest amount. Changes in the LSD will have the smallest effect on the value. The nonzero digit of a number that is the farthest LEFT is the MSD, and the nonzero digit farthest RIGHT is the LSD, as in the following example

Key principle of numbering systems

Radix point (´ÕÙæ ÙæßÂ]× ) Divides fractional portion from the whole portion of a number

Weighting Factor (ýØ &$øæ× ) A multiplier value is used in each column position of a number. It represents the weight factor. Its value determines how many times the Base value is multiplied by itself thus giving the placeholders seen below from right to left labelled as "Ones", "Tens", "Hundreds", "Thousands", "Ten Thousands" and so on. . .

POSITIONAL (&ßö$ùÛ×)NUMBER SYSTEMS

In a positional number system, the position a symbol occupies in the number determines the value it represents. In this system, a number represented as:

In our decimal number system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right.A number in standard form is separated into groups of three digits using commas. Each of these groups is called a period.

http://www.math.com/school/subject1/lessons/S1U1L1GL.html

http://www.math.com/school/subject1/lessons/S1U1L1GL.html


In a positional number system, the position a symbol occupies in the number determines the value it represents. In this system, a number represented as:

has the value of:

in which S is the set of symbols, b is the base (or radix).

Key principle of numbering systems

Positional value significanceGives weight each digit contributes to the number’s overall value

Ex:-1.Steps for base 10Determine positional value of each digit by raising 10 to

position within number2. Steps for base 2 Determine positional value of each digit by raising 2 to position

within number

General procedure (any base) Calculate position value of the number by raising the base value to

the power of the position Multiply positional value by digit in that position Add each calculated value together

Calculate Position Value and Weighed

Number RepresentationsInteger numbers are those numbers which do not have fractional parts. Integer numbers include both positive numbers and negative numbers. They can be handled using any of the following representations

Unsigned notationSigned magnitude notion

Unsigned notation

An unsigned integer is an integer that can never be negative and can take only 0 or positive values. Its range is between 0 and positive infinity. All unsigned numbers are Positive

0 , 1 , 2 , 3 , 4 , . . . . , 1598 , . . . .

Range = 0 to (2n – 1) ; n is the number of bits used to store the unsigned integer.

Numbers with values GREATER than (2n – 1) would require more bits. If you try to store too large a value without using more bits, OVERFLOW will occur.

Example: On a system that stores unsigned integers in 16-bit words: Range = 0 to (216 – 1) = 0 to 65535

Therefore, you cannot store numbers larger than 65535 in 16 bits.

Advantages: 1.One representation of zero

2.Simple addition

Disadvantages : 1.Negative numbers can not be represented.

2.The need of different notation to represent negative numbers.

Unsigned representation

Signed notation Until now we've been concentrating on unsigned numbers. In real life we

also need to be able represent signed numbers (like : -12, -45, +78).

A signed number MUST have a sign (+/-). A method is needed to represent the sign as part of the binary representation. All signed numbers are negative and positive

-1.5 103 , - 25 , -9.610-4 , +0 , +5.710-4 , +0.5 , +98 , +4.6 103

Range =-(2(n-1) – 1) to +(2(n-1) – 1) ; n is the number of bits used to store the sign/magnitude integer

Numbers with values GREATER than +(2(n-1) – 1) and values LESS than -(2(n-1) – 1) would require more bits. If you try to store too large/too small a value without using more bits, OVERFLOW will occur.

Example: On a system that stores unsigned integers in 16-bit words: Range = -(2(16-1) – 1) to +(2(16-1) – 1) = -(2(15) – 1) to +(2(15) – 1)

= -32767 to +32767 Therefore, you cannot store numbers larger than 32767 or smaller than -32767 in 16 bits.

Advantages:Represents positive and negative numbers

Disadvantage : Arithmetic operations are difficult.

Addition and subtractions are difficult.Signs and magnitude, both have to carry out the required operation.They are two representations of 0 (Binary number system)

00000000 = + 010

10000000 = - 010

To test if a number is 0 or not, the CPU will need to see whether it is 00000000 or 10000000.0 is always performed in programs.

Therefore, having two representations of 0 is inconvenient.

Sign Representation

Summary No. of bits

Binary

Unsigned Sign-magnitude

Min Max Min Max

1 0 1

2 0 3 -1 1

3 0 7 -3 3

4 0 15 -7 7

5 0 31 -15 15

6 0 63 -31 31

Etc.

No. of bits

Binary

Unsigned Sign-magnitude

Min Max Min Max

n 0 2n - 1 -(2

n-1 – 1) 2n-1 - 1

Numeric

Representations

Numbering Systems

Integer ……………. Z Decimal …………… Base 10 (N10) Binary …………….. Base 2 (N2) Octal ………………. Base 8 (N8) Hexadecimal ……. Base 16 (N16)

Integer

Integer

A number with no fractional part Includes the counting numbers {1, 2, 3, ...}, zero {0}, and the negative of the counting numbers {-1, -2, -3, ...}

You can write them down like this: {..., -3, -2, -1, 0, 1, 2, 3, ...}

Examples of integers: -16, -3, 0, 1, 198

Decimal

Decimal Number System The decimal system is a base-10 system. There are 10 distinct digits (0 to 9) to represent any

quantity.

10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

For an n-digit number, the value that each digit represents depends on its weight or position.

Digit Position

Integer & fraction 123.456

The weights are based on powers of 10.

Weight = (Base) Position

1024 = 1*103 + 0*102 + 2*101 + 4*100 = 1000 + 20 + 4

Positional value example: 436.95To the left of the radix point : 4 in position two / 3 in position one / 6 in position zeroTo the right of the radix point: 9 in position negative one / 5 in position negative two

Decimal Number System

Base (also called radix) = 10 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Digit Position Integer & fraction

Digit Weight Weight = (Base) Position

Magnitude Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

5 1 2 7 4

10 1 0.1100 0.01

500 10 2 0.7 0.04

d2*B2+d1*B

1+d0*B0+d-1*B

-1+d-2*B-2

(512.74)10

Example

For D = 52.946

Uses the ten numbers from 0 to 9Each column represents a power of 10

Fixed-point representation

Uses the two numbers from 0 to 9Every column represents a power of 10

Standard decimal representation

Positional values for a base 10 number

Positional Value (cont’d.)

Whole Numbers (Radix = 10):

123410 = 1 103 + 2 102 + 3 101 + 4 100

With Fractional Part (Radix = 10):

36.7210 = 3 101 + 6 100 + 7 10-1 + 2 10-2

General Case (Radix = R):

(S1S0.S-1S-2)R =S1 R1 + S0 R0 + S-1 R -1 + S-2 R-2

Reals

Practice (base 10) (Scientific notation )

1) 258 = 2.58 x 102

Mantissa = 258Radix = 10Exponent = 2

3) One billion=1,000,000,000 =1 x 109

significand or mantissa: 1 base or radix: 10 exponent: 9

2) The following shows the place values for the integer +224 in the The following shows the place values for the integer +224 in the decimal system.decimal system.

Note that the digit 2 in position 1 has the value 20, but the same digit in position 2 has the value 200. Also note that we normally drop the plus sign, but it is implicit.

4) 1999 =1.999 x 103

significand or mantissa: 1999 base or radix: 10exponent: 3

=19.99 x 102

=199.9 x 1012) 24.25 = 2.425 x 101

=Mantissa = 2425=Radix = 10=Exponent = 1

The following shows the place values for the decimal number −7508. We have used 1, 10, 100, and 1000 instead of powers of 10.

Example

( ) Values

Note that the digit 2 in position 1 has the value 20, but the same digit in position 2 has the value 200. Also note that we normally drop the plus sign, but it is implicit.

ExampleThe following shows the place values for the real number +24.13.

Binary

Binary Number Systems The binary system is a base-2 system. There are 2 distinct digits (0 and 1) to represent any quantity.

2 digits { 0, 1 }, called binary digits or “bits”

For an n-digit number, the value of a digit in each column depends on its position.

The weights are based on powers of 2. Weight = (Base) Position

10112 = 1*23 + 0*22 + 1*21 + 1*20 =8+2+1 =1110

• 0 = represents no value

• 1 = represents a unit value

Binary Number System Weights


MagnitudeSum of “Bit x Weight”

Formal Notation Groups of bits 4 bits = Nibble

8 bits = Byte

1 0 -12 -2

2 1 1/24 1/4

1 0 1 0 1

=1 *22+0 *21+1 *20+0 *2-1+1 *2-2

=(5.25)10

=(101.01)2

1 0 1 1

1 1 0 0 0 1 0 1

Binary Number SystemBinary number Value encoded by the binary number

0 0×20 = 0

1 1×20 = 1

10 1×21 + 0 ×20 = 2

11 1×21 + 1 ×20 = 3

1010 1×23 + 0×22 + 1×21 + 0×20 = 8 + 2 = 10

Base-2 scientific notation1) 2.25ten = 10.01two = 10.01two x 20 = 1.001two x 21 normalized Numbers are usually normalized which means that the leading bit is always a 1.

Digit 1 1 0 0 . 1 0 1

Weight

23 22 21 20 Binary decimal Point

2-

1 2-2 2-3

2) 1100.101two

Standard binary representationUses the two numbers from 0 to 1Every column represents a power of 2

Fixed-point representation

Uses the two numbers from 0 to 1Every column represents a power of 2

Positional Value1)Positional value of the binary number 10112

Rightmost position: positional value of the base (2) raised to the 0 power

First Positional value: 1 (1 * 20) Next position: value of 2 raised to the power of 1 (1 * 21) Next position: 2 squared (0 * 22) Next position: 2 to the third (1 * 23)

(1 * 20) + (1 * 21) + (0 * 22) + (1 * 23)2)Positional value of the binary number 1011.0112 as follows

Whole Numbers (Radix = 2):

10112 = 1 23 + 0 22 + 1 21 + 1 20

With Fractional Part (Radix = 2):

11.012 = 1 21 + 1 20 + 0 2-1 + 1 2-2

General Case (Radix = R):

(S1S0.S-1S-2)R =S1 R1 + S0 R0 + S-1 R -1 + S-2 R-2

Real's

Example

The following shows that the number (11001)2 in binary is the same as 25 in decimal. The subscript 2 shows that the base is 2.

The equivalent decimal number is N = 16 + 8 + 0 + 0 + 1 = 25.

Example The following shows that the number (101.11)2 in binary is equal to the number 5.75 in decimal.

Octal Number Systems Octal and hexadecimal systems provide a shorthand way to deal

with the long strings of 1’s and 0’s in binary.

Octal is base-8 system using the digits 0 to 7.

8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }

To convert to decimal, you can again use a column weighted system


75128 = 7*83 + 5*82 + 1*81 + 2*80 = 391410

An octal number can easily be converted to binary by replacing each octal digit with the corresponding group of 3 binary digits

75128 = 1111010010102

Octal Number System

WeightsWeight = (Base) Position

MagnitudeSum of “Digit x Weight”

Formal Notation

1 0 -12 -2

8 1 1/864 1/64

5 1 2 7 4

5 *82+1 *81+2 *80+7 *8-1+4 *8-2

=(330.9375)10

=(512.74)8

Place values for an integer in the octal system

Real's

Binary Decimal Octal

0 0 0

1 1 1

10 2 2

11 3 3

100 4 4

101 5 5

110 6 6

111 7 7

Octal Number System

Octal is used as a shorthand for representing file permissions on UNIX systems. For example, file mode rwxr-xr-x would be 0755.

Example

The following shows that the number (1256)8 in octal is the same as 686 in decimal.

Note that the decimal number is N = 512 + 128 + 40 + 6 = 686.

The weight associated with each digit in the given octal number can be determined by raising 8 to a power equivalent to the position of the digit in the number.

Example

Hexadecimal

Number Systems - Hexadecimal Hexadecimal is a base-16 system.

It contains the digits 0 to 9 and the letters A to F (16 digit values).

16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }

Note :- symbols A, B, C, D, E, F are equal to 10, 11, 12, 13, 14, 15 respectively. The symbols in this system are often referred to as hexadecimal digits

The letters A to F represent the unit values 10 to 15.

This system is often used in programming as a condensed form for binary numbers (0x00FF, 00FF)

To convert to decimal, use a weighted system with powers of 16. Conversion to binary is done the same way as octal to

binary conversions. This time though the binary digits are organized into groups

of 4. Conversion from binary to hexadecimal involves breaking

the bits into groups of 4 and replacing them with the hexadecimal equivalent.

Hexadecimal Number System

Weights Weight = (Base) Position

Magnitude Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

16 1 1/16256 1/256

1 E 5 7 A

1 *162+14 *161+5 *160+7 *16-1+10 *16-2

=(485.4765625)10

=(1E5.7A)16

Integers

Place values for an integer in the hexadecimal system

The Hexadecimal Number SystemBinary Decimal Hexadecimal Binary Decimal Hexadecimal 0 0 0 1000 8 8

1 1 1 1001 9 9 10 2 2 1010 10 A 11 3 3 1011 11 B 100 4 4 1100 12 C 101 5 5 1101 13 D 110 6 6 1110 14 E 111 7 7 1111 15 F

Example of a hexadecimal number and the values of the positions 3 C 8 B 0 5 1 166 165 164 163 162 161 160

Hexadecimal SystemThe weight associated with each symbol in the given hexadecimal

number canbe determined by raising 16 to a power equivalent to the position of

the digit inthe number.

Digit 4 A 9 0 . 2 B C

Weight

163 162 161 160 Hexadecimal Point 16-1 16-2 16-3

4A90.2BCExample

The following shows that the number (2AE)16 in hexadecimal is equivalent to 686 in decimal.

The equivalent decimal number is N = 512 + 160 + 14 = 686.

Example

Value of 2001 in Binary, Octal and HexadecimalHexadecimal System

Hexadecimal System

Summary of the four positional systems

1).Why do we use the decimal system for everyday mathematics?

Answer: Fingers and Thumbs

2).Why do we use the binary system for computer mathematics?

Answer: Computers use voltage levels to perform mathematics.

0-Volts and 5-Volts correspond to 0’s and 1’s

Common PowersBase 10

Power Preface Symbol

10-12 pico p

10-9 nano n

10-6 micro

10-3 milli m

103 kilo k

106 mega M

109 giga G

1012 tera T

Value

.000000000001

.000000001

.000001

.001

1000

1000000

1000000000

1000000000000

Common Powers Base 2


210 kilo k

220 mega M

230 Giga G

Value

1024

1048576

1073741824

• What is the value of “k”, “M”, and “G”?• In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

Converting Numbers Between

Bases

Conversion Among Bases

The possibilities

Hexadecimal

Decimal Octal

Binary

Converting Numbers Between Bases Can represent any quantity in any base

Counting process similar for all bases Count until highest digit for base reached Add 1 to next higher position to left Return to 0 in current position

Conversion: map from one base to another Identities easily calculated Identities obtained by table lookup

Decimal to Decimal

Hexadecimal

Decimal Octal

Binary

12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100

125

Base

Weight

Example

Decimal to Binary

Hexadecimal

Decimal Octal

Binary

Converting From Decimal to Binary

• Make a list of the binary place values up to the number being converted.

• Perform successive divisions by 2, placing the remainder of 0 or 1 in each of the positions from right to left.

• Continue until the quotient is zero.

• Example: 4210

25 24 23 22 21 20

32 16 8 4 2 1 1 0 1 0 1 0

12510 = ?22 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1

12510 = 11111012

Example

Decimal to Binary 149210 (decimal) = ???2 (binary)

Repeated Divide by 2

Example

ExampleThe following shows how to convert 35 in decimal to binary. We start with the number in decimal, we move to the left while continuously finding the quotients and the remainder of division by 2. The result is 35 = (100011)2.

Example

Convert the decimal number 0.625 to binary.

Since the number 0.625 = (0.101)2 has no integral part, the example shows how the fractional part is calculated.

Decimal (Fraction) to Binary Conversion

Multiply the number by the ‘Base’ (=2) Take the integer (either 0 or 1) as a coefficient Take the resultant fraction and repeat the division

Example: (0.625)10

Integer Fraction Coefficient

Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2

MSB LSBMSB LSB

0.625 * 2 = 1 . 25 a-1 = 10.25 * 2 = 0 . 5 a-2 = 00.5 * 2 = 1 . 0 a-3 = 1

Example

A similar method can be used to convert a decimal fraction to binary when the denominator is a power of two:

The answer is then (0.011011)The answer is then (0.011011)22

ExampleAn alternative method for converting a small decimal integer (usually less than 256) to binary is to break the number as the sum of numbers that are equivalent to the binary place values shown:

Binary to Decimal

Hexadecimal

Decimal Octal

Binary

Binary to Decimal Technique

Multiply each bit by 2n, where n is the “weight” of the bit

The weight is the position of the bit, starting from 0 on the right

Add the results

Converting from Binary to Decimal

Example

Example

1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32

4310

Bit “0”

Example

Example

The following shows how to convert the binary The following shows how to convert the binary number (110.11)number (110.11)22 to decimal: (110.11) to decimal: (110.11)22 = 6.75. = 6.75.

Decimal to Octal

Hexadecimal

Decimal Octal

Binary

123410 = ?8

8 1234 154 28 19 28 2 38 0 2

123410 = 23228

Example

Decimal to Octal Conversion

(175)10

Quotient Remainder Coefficient

Answer: (175)10 = (a2 a1 a0)8 = (257)8

175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2

(0.3125)10Integer Fraction Coefficient

Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8

0.3125 * 8 = 2 . 50.5 * 8 = 4 . 0 a-2 = 4

a-1 = 2

Example

Example

ExampleThe following shows how to convert 126 in decimal to its equivalent in the octal system. We move to the right while continuously finding the quotients and the remainder of division by 8. The result is 126 = (176)8.

Example

The following shows how to convert 0.634 to octal using a maximum of four digits. The result is 0.634 = (0.5044)8. Note that we multiple by 8 (base octal).

Octal to Decimal

Hexadecimal

Decimal Octal

Binary

Octal to Decimal Technique



Add the results

86 85 84 83 82 81 80

26214410 3276810 409610 51210 6410 810 110

Example

The following shows how to convert (724)8 to decimal.

The following shows how to convert (23.17)8 to decimal.

This means that (23.17)8 ≈ 19.234 in decimal. Again, we have rounded up 7 × 8−2 = 0.109375.

Example

Decimal to Hexadecimal

Hexadecimal

Decimal Octal

Binary

123410 = ?16

123410 = 4D216

16 1234 77 216 4 13 = D16 0 4

Example

ExampleThe following shows how we convert 126 in decimal to its equivalent in the hexadecimal system. We move to the right while continuously finding the quotients and the remainder of division by 16. The result is 126 = (7E)16

The following shows how to convert 178.6 in decimal to hexadecimal using only one digit to the right of the decimal point. The result is 178.6 = (B2.9)16 Note that we divide or multiple by 16 (base hexadecimal).

Example

Hexadecimal to Decimal

Hexadecimal

Decimal Octal

Binary

Hexadecimal to Decimal Technique



Add the results

165 164 163 162 161 160

104857610 6553610 409610 25610 1610 110

ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560

274810

Example

The following shows how to convert the hexadecimal number (ABC)16 to decimal.

ExampleThe following shows how to convert the hexadecimal number (1A.23)16 to decimal.

Note :- The result in the decimal notation is not exact, because Note :- The result in the decimal notation is not exact, because 3 × 16 3 × 16−2−2 = 0.01171875. We have rounded this value to = 0.01171875. We have rounded this value to threethree digits (0.012). digits (0.012).

Base N to Decimal conversion

Decimal to Base N Conversions To convert from decimal to a different number base such as Octal, Binary or Hexadecimal involves repeated division by that number base

Keep dividing until the quotient is zero Use the remainders in reverse order as the digits of the

converted number

Binary to Octal

Hexadecimal

Decimal Octal

Binary

Binary-octal conversion8 = 23

Each group of 3 bits represents an octal digit

Group bits in threes, starting on right Convert to octal digits

Binary − Octal Conversion

Octal Binary

0 0 0 0

1 0 0 1

2 0 1 0

3 0 1 1

4 1 0 0

5 1 0 1

6 1 1 0

7 1 1 1

( 1 0 1 1 0 . 0 1 )2

( 2 6 . 2 )8

Assume Zeros

Works both ways (Binary to Octal & Octal to Binary)

10110101112 = ?81 011 010 111

1 3 2 7

10110101112 = 13278

Example

Example

Example

Show the octal equivalent of the binary number (101110010)2.

SolutionEach group of three bits is translated into one octal digit. The equivalent of each 3-bit group is shown in Table on above page

(101110010)2 = (562)8.

101 110 010

Example

What is the binary equivalent of for (24)8?

SolutionWrite each octal digit as its equivalent bit pattern to get

2 → 010 and 4 → 100

The result is (010100)2.

Octal to Binary

Hexadecimal

Decimal Octal

Binary

Octal to Binary• Technique

• Convert each octal digit to a 3-bit equivalent binary representation

• Useful to represent binary numbers indirectlyOctal and binary are nicely related; i.e. 8 = 23

Each octal digit represent 3 binary digits (bits)

7058 = ?2

7 0 5

111 000 101

7058 = 1110001012

Example

Binary to Hexadecimal

Hexadecimal

Decimal Octal

Binary

Binary-hexadecimal conversion

Technique

Group bits in fours, starting on rightConvert to hexadecimal digits

Binary − Hexadecimal Conversion

16 = 24

Each group of 4 bits represents a hexadecimal digit

Example:

( 1 0 1 1 0 . 0 1 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Binary to Hex & Hex to Binary)

Example

Show the hexadecimal equivalent of the binary number (110011100010)2.

SolutionSolutionWe first arrange the binary number in 4-bit patterns:We first arrange the binary number in 4-bit patterns:

1100 1110 00101100 1110 0010Note that the leftmost pattern can have one to four bits. We then use the equivalent of each pattern shown in Table 2.2 on page 25 to change the number to hexadecimal: (4E2)16.Example

Hexadecimal to Binary

Hexadecimal

Decimal Octal

Binary

Hexadecimal to Binary Technique

Convert each hexadecimal digit to a 4-bit equivalent binary representation

Useful to represent binary numbers indirectly Hex and binary are nicely related; i.e. 16 =

24

Each hex digit represent 4 binary digits (bits)

10AF16 = ?2

1 0 A F

0001 0000 1010 1111

10AF16 = 00010000101011112

Example

Example

What is the binary equivalent of (24C)16?Each hexadecimal digit is converted to 4-bit patterns:

2 → 0010, 4 → 0100, and C → 1100

The result is (001001001100)2

Octal to Hexadecimal

Hexadecimal

Decimal Octal

Binary

Octal-hexadecimal conversion

Octal−Hexadecimal Conversion

Convert to Binary as an intermediate step

( 0 0 1 0 1 1 0 . 0 1 0 0 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Octal to Hex & Hex to Octal)

( 2 6 . 2 )8

Assume Zeros

Example

10768 = ?16

1 0 7 6

001 000 111 110

2 3 E

10768 = 23E16

Example

Decimal, Binary, Octal and Hexadecimal

Decimal Binary Octal Hex00 0000 00 0

01 0001 01 1

02 0010 02 2

03 0011 03 3

04 0100 04 4

05 0101 05 5

06 0110 06 6

07 0111 07 7

08 1000 10 8

09 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F

Example

Find the minimum number of binary digits required to store decimal integers with a maximum of six digits.

k = 6, b1 = 10, and b2 = 2. Thenx = [x = [k × (logbk × (logb11 / logb / logb22) ] = [6 × (1 / 0.30103)]= 20.) ] = [6 × (1 / 0.30103)]= 20.

The largest six-digit decimal number is 999,999 and the largest 20-bit binary number is 1,048,575. Note that the largest number that can be represented by a 19-bit number is 524287, which is smaller than 999,999. We definitely need twenty bits.

Hexadecimal to Octal

Hexadecimal

Decimal Octal

Binary

Hexadecimal- Octal Conversion

Convert to Binary as an intermediate step

( 0 0 0 1 0 1 1 0 . 01 0 0 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Octal to Hex & Hex to Octal)

( 2 6 . 2 )8

Assume Zeros

Example

1F0C16 = ?8

1 F 0 C

0001 1111 0000 1100

1 7 4 1 4

1F0C16 = 174148

Example

Exercise – Convert ...Decimal Binary Octal Hexadecimal

33

1110101

703

1AF

Decimal Binary Octal Hexadecimal

33 100001 41 21

117 1110101 165 75

451 111000011 703 1C3

431 110101111 657 1AF

Common Powers (1 of 2)

Base 10


10-12 pico p

10-9 nano n

10-6 micro

10-3 milli m

103 kilo k

106 mega M

109 giga G

1012 tera T

Value

.000000000001

.000000001

.000001

.001

1000

1000000

1000000000

1000000000000


Base 2


210 kilo k

220 mega M

230 Giga G

Value

1024

1048576

1073741824

• What is the value of “k”, “M”, and “G”?• In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

Bits, Bytes, and Words A bit is a single binary digit (a 1 or 0). A byte is 8 bits A word is 32 bits or 4 bytes Long word = 8 bytes = 64 bits Quad word = 16 bytes = 128 bits Programming languages use these standard

number of bits when organizing data storage and access.

What do you call 4 bits?(hint: it is a small byte)

AddingSubtractionMultiplyingDivision

Binary Calculation

Decimal Addition

5 5

55+

011

= Ten ≥ Base Subtract a Base

11 Carry

Binary Addition

Column Addition

1 0 1111

1111 0+

0000 1 11

≥ (2)10

111111

= 61

= 23

= 84

Carry

Binary Addition

Octal Addition

c. 7338 + 748 = ?

Example

a. 1238 + 3218 = ?

1238

+3218

4448

b. 4578 + 2458 = ?

4578

+ 2458

7248

Hexadecimal Addition

c. DF16 + AB16 = ?

d. ADC16 + DEF16 = ?

Example

a. 3316 + 4716 = ?

3316

+ 4716

7A16

b. 20D316 + 12BC16 = ?

20D316

+ 12BC16

338F16

Note:- 2410 =1A16

Question: What is the result of adding 1 to the largest digit of some number system?

• (9)10 + 1 = (10)10

• (7)8 + 1 = (10)8

• (1)2 + 1 = (10)2

• (F)16 + 1 = (10)16

Conclusion: Adding 1 to the largest digit in any number system always has a result of (10) in that number system.

Addition

Binary SubtractionBorrow a “Base” when needed

0 0 1110

1111 0−

0101 1 10

Borrow 2 from next line (2= 10)2

2

22 2

1

000

1

= 77

= 23

= 54

A B A-B Borrow

0 0 0 0

0 1 1 1

1 0 1 0

1 1 0 0

Perform the binary subtraction of the following numbers: 10101 and 01110

Binary Subtraction

Octal Subtraction

a. 5248 - 1678 = ?

5 2 4 8 1 (8+4)

4 (1+8) (12)

(4) (9) (12)

- 1 6 7 8 3 3 5 8

5248 - 1678 = 3358

Octal Subtractiona. 5248 - 1678 = ?

5 2 48 1 (8+4)

4 (1+8) (12) (4) (9) (12) - 1 6 78

3 3 58

5248 - 1678 = 3358

b. 1678 - 248 = ?

1678

- 248

1438

c. 15238 - 3648 = ?

Hexadecimal Subtraction

a . 4416 - 1716 =?

4 416

3 (16 + 4)16

(3) (20)16

- 1 7 16

2 (13) 16

2 D 16

4416 - 1716 = 2D16

Hexadecimal Subtraction

a. 4416 - 1716 =?

4 416 3 (16 + 4)

(3) (20) - 1 716 2 (13)

2 D16

4416 - 1716 = 2D16

b. 20D316 - 12BC16 = ?

20D316

- 12BC16

0E1716

c. DF16 - AB16 = ?

Binary MultiplicationBit by bit

01 1 1 1

01 1 0

00 0 0 0

01 1 1 1

01 1 1 1

0 0 000

0110111 0

x

Example

Example : - Perform the binary multiplication of the decimal numbers 12 and 10.

The equivalent binary representation of the decimal number 12 is 1100.The equivalent binary representation of the decimal number 10 is 1010

Binary Multiplication

Octal Multiplication258 * 168 = ?

258 * 168 =4468

Method 1

Example

Octal MultiplicationMethod 2

Hexadecimal MultiplicationExampl

eEF * D = ?

EF * D = C23

2F * 1A =?

Hexadecimal Multiplication

Hexadecimal mortification table

Binary Division Binary division is also performed in the same way as we

perform decimal division. Like decimal division, we also need to follow the binary subtraction rules while performing the binary division. The dividend involved in binary division should be greater than the divisor. The following are the two important points, which need to be remembered while performing the binary division.

• If the remainder obtained by the division process is greater than or equal to the divisor, put 1 in the quotient and perform the binary subtraction.

• If the remainder obtained by the division process is less than the divisor, put 0 in the quotient and append the next most significant digit from the dividend to the remainder.

Binary DivisionThe equivalent binary representation of the decimal number 18 is 10010.The equivalent binary representation of the decimal number 8 is 1000.

Example

Find the quotient and remainder when 1111101 is divided by 1101 in modulo 2 arithmetic.

We find the quotient is 1011, and the remainder is 0010.

This procedure is very useful to us in calculating CRC syndromes

Binary DivisionExample

Example

Binary Division

Example

Octal Division

738 ÷ 48 = ?

738 ÷ 48 = 168 remain 3

Example

Octal Division

175348 ÷ 648 = ?648 × 2 = 1508

648 × 3 = 2348

Example

Hexadecimal Division

AB16 ÷ 316 = ?

AB16 ÷ 316 = 3916

AB16 ÷ C16 = ?

AB16 ÷ C16 = E.2516

Example

Hexadecimal DivisionDEF16 ÷ AB16 = ?

Note :- Short method

AB16 =10*16 + 11160+11171

DE16 = 13*16 + 14 208 + 14 222

51*16+F=816 + 15 = 831171*4 = 5C4171*E = 142E171*2 = 2E2

Summary of the Division

Bitwise logic

operations

What is an Operator? Operator is an operation performed over data at runtime

Takes one or more arguments (operands) Produces a new value

Operators have precedence Precedence defines which will be evaluated first

Expressions are sequences of operators and operands that are evaluated to a single value

Operators can be classified according to the type of their operands and of their output

Arithmetic Relational Logical Bitwise

the number of their operands Unary (one operand) Binary (two operands)

Categories of Operators in C#

Category Operators

Arithmetic + - * / % ++ --

Relational < <= > >= == !=

Logical && || ^ !

Binary & | ^ ~ << >>

Comparison == != < > <= >=

Assignment = += -= *= /= %= &= |= ^= <<= >>=

String concatenation +

Type conversion is as typeof

Other . [] () ?: new

Binary expression

Unary Expression

Relational operators These perform comparisons and the result is what is

called a boolean: a value TRUE or FALSE FALSE is represented by 0; anything else is TRUE The relational operators are:

< (less than) <= (less than or equal to) > (greater than) >= (greater than or equal to) == (equal to) != (not equal to)

Logical Operators (also called Boolean operators) These have Boolean operands and the result is also a Boolean. The basic Boolean operators are:

Operator ! turns true to false and false to true Behavior of the operators &&, || and ^ (1 == true, 0 == false)

Example :-

bool a = true;

bool b = false;

(a && b) False

(a || b) True

(a ^ b) True

(!b) True

(b || true) True

(b && true) False

(a || true) True

(a && true) True

(!a) False

((5>7) ^ (a==b))

False

&& (logical AND) || (logical OR) ! (logical NOT) -- unary

Logical Operators (also called Boolean operators)

Logical Operators (also called Boolean operators)

Operation ^ ^ ^ ^

Operand1 0 0 1 1

Operand2 0 1 0 1

Result 0 1 1 0

Operation || || || ||

Operand1 0 0 1 1

Operand2 0 1 0 1

Result 0 1 1 1

Operation && && && &&

Operand1 0 0 1 1

Operand2 0 1 0 1

Result 0 0 0 1

BITWISE OPERATORSBitwise operators operate on individual bits of integer (int and long) values.

If an operand is shorter than an int, it is promoted to int before doing the operations.

Negative integers are store in two's complement form. For example, -4 is 1111 1111 1111 1111 1111 1111 1111 1100.

Bitwise operator ~ turns all 0 to 1 and all 1 to 0 Like ! for boolean expressions but bit by bit

The operators |, & and ^ behave like ||, && and ^ for Boolean expressions but bit by bit

The << and >> move the bits (left or right)

Bitwise operators are used on integer numbers (byte, sbyte, int, uint, long, ulong)

Bitwise operators are applied bit by bit

Bitwise OperatorOperator Value

~ Bitwise unary NOT

& Bitwise AND

| Bitwise OR

^ Bitwise XOR

>> Shift Right

>>> Shift Right zero fill

<< Shift left

& = Bitwise AND Assignment

|= Bitwise OR Assignment

^= Bitwise XOR Assignment

>>= Shift Right Assignment

>>>= Shift Right zero fill Assignment

<<= Shift Left Assignment

A B AB

0 0 0

0 1 0

1 0 0

1 1 1

Bitwise Operator

{ A AND B } as A AND B=A ×

BA & B=A ×

B

A A

0 1

1 0

{ NOT A } as

NOT A = A

A B A+B

0 0 0

0 1 1

1 0 1

1 1 1

A B C A+B+C

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1

Bitwise Operator { A OR B } as follows

A OR B=A+B A | B=A+B

{ A OR B OR C} as follows

A OR B OR C =A+B+CA | B | C= A+B+C

Bitwise Operator

A B AB

0 0 0

0 1 1

1 0 1

1 1 0

{ A XOR B } as A XOR B = (A)B +

A(B)A ^ B = (A)B +

A(B)

Summary

Operation | | | | & & & & ^ ^ ^ ^

Operand1 0 0 1 1 0 0 1 1 0 0 1 1

Operand2 0 1 0 1 0 1 0 1 0 1 0 1

Result 0 1 1 1 0 0 0 1 0 1 1 0

BITWISE OPERATORS

Examples:

a = 3; is 00000011 b = 5; is 00000101

( a | b); // 00000111( a & b); // 00000001( a ^ b); // 00000110(~a & b); // 00000100( a<<1 ); // 00000110( a>>1 ); // 00000001

Bitwise Operator

<-------Bitwise Logical Operators------->

Example :- The binary value of a = 0010 and b = 01111.The Bitwise OR : a | b = 72.The Bitwise AND : a & b = 23.The Bitwise XOR(exclusive OR) : a ^ b = 54.The Bitwise unary NOT : ~a & a = 05.~a&b|a&~b = 5

<-------Bitewise Shift Operators------->

Example :- The original binary value of a = 0010 and Decimal value of a = 21.The Left shift : a = 82. The Right shift : b = b >> 2 = 13.The original decimal value of u = -14. The Unsigned Right shift : u = u >>> 30 means u = 11111111 11111111 11111111 11111111 >>> 30 hence u = 0011 and Decimal value of u = 3

<-------Bitewise Assignment Operators------->

Example :- The original binary value of p = 0101 and Decimal value of p = 5

1.The Bitewise Shift Right Assignment Operators : p >>= 2 means p = p >> hence p =0101 >> 2 so p = 0001 and Decimal value of p = 1

Bitwise OperatorBitwise operators are used in1.Communication stacks where the individual bits in the header attached to the data signify important information

2.Embedded software for controlling different functions in the chip and indicating the status of hardware by manipulating the individual bits of hardware registers of embedded microcontrollers

3.Low-level programming for applications such as device drivers, cryptographic software, video decoding software, memory allocators, compression software and graphics

4.Maintaining large sets of integers efficiently in search and optimization problems

5.Bitwise operations performed on bit flags, which can enable an instance of enumeration type to store any combination of values defined in an enumerator list

Low level languages are:

1 - Machine Language2 - Assembly Language3 - C (C is not the 100% Low-level language)

High level languages are:

1 - Visual Basic2 - Pascal3 - Java4 - C++and many more.

Number system part 1

Education

Transcript of Number system part 1