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Chapter 5 Data representation Data Rep

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**Learning outcomes By the end of this Chapter you will be able to:**

Explain how integers are represented in computers using: Unsigned, signed magnitude, excess, and two’s complement notations Explain how fractional numbers are represented in computers Floating point notation (IEEE 754 single format) Calculate the decimal value represented by a binary sequence in: Unsigned, signed notation, excess, two’s complement, and the IEEE 754 notations. Explain how characters are represented in computers E.g. using ASCII and Unicode Explain how colours, images, sound and movies are represented Data Rep

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**Additional Reading Essential Reading Further Reading**

Stalling (2003): Chapter 9 Further Reading Brookshear (2003): Chapter Burrell (2004): Chapter 2 Schneider and Gersting (2004): Chapter 4.2 Data Rep

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**Number representation**

Representing whole numbers Representing fractional numbers Data Rep

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**Integer Representations**

Unsigned notation Signed magnitude notion Excess notation Two’s complement notation. Data Rep

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**Unsigned Representation**

Represents positive integers. Unsigned representation of 157: Addition is simple: = position 7 6 5 4 3 2 1 Bit pattern 128 64 32 16 8 contribution 27 26 25 24 23 22 21 20 Data Rep

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**Advantages and disadvantages of unsigned notation**

One representation of zero Simple addition Disadvantages Negative numbers can not be represented. The need of different notation to represent negative numbers. Data Rep

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**Representation of negative numbers**

Is a representation of negative numbers possible? Unfortunately: you can not just stick a negative sign in front of a binary number. (it does not work like that) There are three methods used to represent negative numbers. Signed magnitude notation Excess notation notation Two’s complement notation Data Rep

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**Signed Magnitude Representation**

Unsigned: - and + are the same. In signed magnitude the left-most bit represents the sign of the integer. 0 for positive numbers. 1 for negative numbers. The remaining bits represent to magnitude of the numbers. Data Rep

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**Example - Suppose 10011101 is a signed magnitude representation.**

The sign bit is 1, then the number represented is negative The magnitude is with a value = 29 Then the number represented by is –29. position 7 6 5 4 3 2 1 Bit pattern contribution - 24 23 22 20 Data Rep

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Exercise 1 3710 has in signed magnitude notation. Find the signed magnitude of –3710 ? Using the signed magnitude notation find the 8-bit binary representation of the decimal value 2410 and Find the signed magnitude of –63 using 8-bit binary sequence? Data Rep

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**Disadvantage of Signed Magnitude**

Addition and subtractions are difficult. Signs and magnitude, both have to carry out the required operation. They are two representations of 0 = = To test if a number is 0 or not, the CPU will need to see whether it is or 0 is always performed in programs. Therefore, having two representations of 0 is inconvenient. Data Rep

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**Signed-Summary In signed magnitude notation, Advantages:**

The most significant bit is used to represent the sign. 1 represents negative numbers 0 represents positive numbers. The unsigned value of the remaining bits represent The magnitude. Advantages: Represents positive and negative numbers Disadvantages: two representations of zero, Arithmetic operations are difficult. Data Rep

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**Excess Notation In excess notation:**

The value represented is the unsigned value with a fixed value subtracted from it. For n-bit binary sequences the value subtracted fixed value is 2(n-1). Most significant bit: 0 for negative numbers 1 for positive numbers Data Rep

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**Excess Notation with n bits**

1000…0 represent 2n-1 is the decimal value in unsigned notation. Therefore, in excess notation: 1000…0 will represent 0 . Decimal value In unsigned notation Decimal value In excess notation - 2n-1 = Data Rep

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**Example (1) - excess to decimal**

Find the decimal number represented by in excess notation. Unsigned value = = = 15310 Excess value: excess value = 153 – 27 = 152 – 128 = 25. Data Rep

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**Example (2) - decimal to excess**

Represent the decimal value 24 in 8-bit excess notation. We first add, 28-1, the fixed value = = 152 then, find the unsigned value of 152 15210 = (unsigned notation). = (excess notation) Data Rep

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**example (3) Represent the decimal value -24 in 8-bit excess notation.**

We first add, 28-1, the fixed value = = 104 then, find the unsigned value of 104 10410 = (unsigned notation). -2410 = (excess notation) Data Rep

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**Example (4) (10101) Unsigned Sign Magnitude Excess notation**

= = 2110 The value represented in unsigned notation is 21 Sign Magnitude The sign bit is 1, so the sign is negative The magnitude is the unsigned value = 510 So the value represented in signed magnitude is -510 Excess notation As an unsigned binary integer = 2110 subtracting = = 16, we get = 510. So the value represented in excess notation is 510. Data Rep

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**Advantages of Excess Notation**

It can represent positive and negative integers. There is only one representation for 0. It is easy to compare two numbers. When comparing the bits can be treated as unsigned integers. Excess notation is not normally used to represent integers. It is mainly used in floating point representation for representing fractions (later floating point rep.). Data Rep

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**Exercise 2 Find 10011001 is an 8-bit binary sequence.**

Find the decimal value it represents if it was in unsigned and signed magnitude. Suppose this representation is excess notation, find the decimal value it represents? Using 8-bit binary sequence notation, find the unsigned, signed magnitude and excess notation of the decimal value ? Data Rep

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**Excess notation - Summary**

In excess notation, the value represented is the unsigned value with a fixed value subtracted from it. i.e. for n-bit binary sequences the value subtracted is 2(n-1). Most significant bit: 0 for negative numbers . 1 positive numbers. Advantages: Only one representation of zero. Easy for comparison. Data Rep

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**Two’s Complement Notation**

The most used representation for integers. All positive numbers begin with 0. All negative numbers begin with 1. One representation of zero i.e. 0 is represented as 0000 using 4-bit binary sequence. Data Rep

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**Two’s Complement Notation with 4-bits**

Binary pattern Value in 2’s complement. Data Rep

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**Properties of Two’s Complement Notation**

Positive numbers begin with 0 Negative numbers begin with 1 Only one representation of 0, i.e. 0000 Relationship between +n and –n. Data Rep

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**Advantages of Two’s Complement Notation**

It is easy to add two numbers. + + Subtraction can be easily performed. Multiplication is just a repeated addition. Division is just a repeated subtraction Two’s complement is widely used in ALU Data Rep

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**Evaluating numbers in two’s complement notation**

Sign bit = 0, the number is positive. The value is determined in the usual way. Sign bit = 1, the number is negative. three methods can be used: Method 1 decimal value of (n-1) bits, then subtract 2n-1 Method 2 - 2n-1 is the contribution of the sign bit. Method 3 Binary rep. of the corresponding positive number. Let V be its decimal value. - V is the required value. Data Rep

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**Example- 10101 in Two’s Complement**

The most significant bit is 1, hence it is a negative number. Method 1 0101 = (+5 – 25-1 = 5 – 24 = 5-16 = -11) Method 2 = Method 3 Corresponding + number is = = 11 the result is then –11. Data Rep

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**Two’s complement-summary**

In two’s complement the most significant for an n-bit number has a contribution of –2(n-1). One representation of zero All arithmetic operations can be performed by using addition and inversion. The most significant bit: 0 for positive and 1 for negative. Three methods can the decimal value of a negative number: Method 1 decimal value of (n-1) bits, then subtract 2n-1 Method 2 - 2n-1 is the contribution of the sign bit. Method 3 Binary rep. of the corresponding positive number. Let V be its decimal value. - V is the required value. Data Rep

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Exercise Determine the decimal value represented by in each of the following four systems. Unsigned notation? Signed magnitude notation? Excess notation? Tow’s complements? Data Rep

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**Fraction Representation**

To represent fraction we need other representations: Fixed point representation Floating point representation. Data Rep

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**Fixed-Point Representation**

old position New position Bit pattern Contribution =19.625 Radix-point Data Rep

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**Limitation of Fixed-Point Representation**

To represent large numbers or very small numbers we need a very long sequences of bits. This is because we have to give bits to both the integer part and the fraction part. Data Rep

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**Floating Point Representation**

In decimal notation we can get around this problem using scientific notation or floating point notation. Number Scientific notation Floating-point notation 1,245,000,000,000 Data Rep

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**Base Mantissa Exponent Sign**

Floating Point could be represented as Base Mantissa Exponent Sign - 159 * 1014 When we deal with very large and very small numbers we often resort to using scientific notation. Calculators, for example, often represent the results of large calculations this way because there is an insufficient number of digits to show the result normally 159 is called the mantissa 14 the exponent which is the number of places to move the decimal point. * 1015 * 1016 A calculator might display 159 E14 Data Rep

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**Floating point format M B E Sign mantissa or base exponent**

significand Sign Exponent Mantissa Data Rep

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**Floating Point Representation format**

The exponent is biased by a fixed value b, called the bias. The mantissa should be normalised, e.g. if the real mantissa if of the form 1.f then the normalised mantissa should be f, where f is a binary sequence. Sign Exponent Mantissa Data Rep

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**The number will occupy 32 bits**

IEEE 745 Single Precision The number will occupy 32 bits The first bit represents the sign of the number; 1= negative 0= positive. The next 8 bits will specify the exponent stored in biased 127 form. The remaining 23 bits will carry the mantissa normalised to be between 1 and 2. i.e. 1<= mantissa < 2 Data Rep

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**Representation in IEEE 754 single precision**

sign bit: 0 for positive and, 1 for negative numbers 8 biased exponent by 127 23 bit normalised mantissa Sign Exponent Mantissa Data Rep

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Basic Conversion Converting a decimal number to a floating point number. 1.Take the integer part of the number and generate the binary equivalent. 2.Take the fractional part and generate a binary fraction 3.Then place the two parts together and normalise. Data Rep

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**IEEE – Example 1 Convert 6.75 to 32 bit IEEE format. = 1102 = 0.112**

1. The Mantissa. The Integer first. 6 / 2 = 3 r 0 3 / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next. .75 * 2 = 1.5 .5 * 2 = 1.0 3. put the two parts together… Now normalise * 22 = 1102 = 0.112 Data Rep

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**IEEE – Example 1 Convert 6.75 to 32 bit IEEE format. = 1102 = 0.112**

1. The Mantissa. The Integer first. 6 / 2 = 3 r 0 3 / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next. .75 * 2 = 1.5 .5 * 2 = 1.0 3. put the two parts together… Now normalise * 22 = 1102 = 0.112 Data Rep

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**IEEE – Example 1 Convert 6.75 to 32 bit IEEE format. = 1102 = 0.112**

1. The Mantissa. The Integer first. 6 / 2 = 3 r 0 3 / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next. .75 * 2 = 1.5 .5 * 2 = 1.0 3. put the two parts together… Now normalise * 22 = 1102 = 0.112 Data Rep

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**IEEE Biased 127 Exponent To generate a biased 127 exponent**

Take the value of the signed exponent and add 127. Example. 216 then = and my value for the exponent would be 143 = So it is simply now an unsigned value .... Data Rep

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**Possible Representations of an Exponent**

Data Rep

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**Why Biased ? The smallest exponent 00000000**

Only one exponent zero The highest exponent is To increase the exponent by one simply add 1 to the present pattern. Data Rep

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**Back to the example Our original example revisited…. 1.1011 * 22**

Exponent is =129 or in binary. NOTE: Mantissa always ends up with a value of ‘1’ before the Dot. This is a waste of storage therefore it is implied but not actually stored is stored .1000 6.75 in 32 bit floating point IEEE representation:- sign(1) exponent(8) mantissa(23) Data Rep

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**Representation in IEEE 754 single precision**

sign bit: 0 for positive and, 1 for negative numbers 8 biased exponent by 127 23 bit normalised mantissa Sign Exponent Mantissa Data Rep

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Example (2) which number does the following IEEE single precision notation represent? The sign bit is 1, hence it is a negative number. The exponent is = 12810 It is biased by 127, hence the real exponent is 128 –127 = 1. The mantissa: It is normalised, hence the true mantissa is 1.01 = Finally, the number represented is: x 21 = -2.50 1 Data Rep

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**Single Precision Format**

The exponent is formatted using excess-127 notation, with an implied base of 2 Example: Exponent: Representation: 135 – 127 = 8 The stored values 0 and 255 of the exponent are used to indicate special values, the exponential range is restricted to 2-126 to 2127 The number 0.0 is defined by a mantissa of 0 together with the special exponential value 0 The standard allows also values +/-∞ (represented as mantissa +/-0 and exponent 255 Allows various other special conditions Data Rep

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In comparison The smallest and largest possible 32-bit integers in two’s complement are only -232 and 2017/3/28 PITT CS 1621 51 Data Rep 51

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**Numbers in 32-bit Formats**

Two’s complement integers Floating point numbers Ref: W. Stallings, Computer Organization and Architecture, Sixth Edition, Upper Saddle River, NJ: Prentice-Hall. Expressible numbers -231 231-1 Positive underflow Negative underflow Negative Overflow Expressible negative numbers Expressible positive numbers Positive Overflow -2-127 2-127 - (2 – 2-23)×2127 (2 – 2-23)×2127 Data Rep

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**Positive Zero in IEEE 754 + 1.0 × 2-127**

Biased exponent Fraction + 1.0 × 2-127 Smallest positive number in single-precision IEEE 754 standard. Interpreted as positive zero. True exponent less than -127 is positive underflow; can be regarded as zero. Data Rep

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**Negative Zero in IEEE 754 - 1.0 × 2-127**

Biased exponent Fraction - 1.0 × 2-127 Smallest negative number in single-precision IEEE 754 standard. Interpreted as negative zero. True exponent less than -127 is negative underflow; may be regarded as 0. Data Rep

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**Positive Infinity in IEEE 754**

Biased exponent Fraction + 1.0 × 2128 Largest positive number in single-precision IEEE 754 standard. Interpreted as + ∞ If true exponent = 128 and fraction ≠ 0, then the number is greater than ∞. It is called “not a number” or NaN and may be interpreted as ∞. Data Rep

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**Negative Infinity in IEEE 754**

Biased exponent Fraction -1.0 × 2128 Smallest negative number in single-precision IEEE 754 standard. Interpreted as - ∞ If true exponent = 128 and fraction ≠ 0, then the number is less than - ∞. It is called “not a number” or NaN and may be interpreted as - ∞. Data Rep

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**Normalized (positive range; negative is symmetric)**

Range of numbers Normalized (positive range; negative is symmetric) smallest +2-126× (1+0) = 2-126 largest +2127× (2-2-23) 2127(2-2-23) 2-126 Positive overflow Positive underflow 2017/3/28 PITT CS 1621 57 Data Rep 57

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**Representation in IEEE 754 double precision format**

It uses 64 bits 1 bit sign 11 bit biased exponent 52 bit mantissa Sign Exponent Mantissa Data Rep

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**IEEE 754 double precision Biased = 1023**

11-bit exponent with an excess of 1023. For example: If the exponent is -1 we then add 1023 to it = 1022 We then find the binary representation of 1022 Which is The exponent field will now hold This means that we just represent -1 with an excess of 1023. Data Rep

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**IEEE 754 Encoding Single Precision Double Precision Represented Object**

Exponent Fraction non-zero +/- denormalized number 1~254 anything 1~2046 +/- floating-point numbers 255 2047 +/- infinity NaN (Not a Number) 60 Data Rep 60

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**Floating Point Representation format (summary)**

the sign bit represents the sign 0 for positive numbers 1 for negative numbers The exponent is biased by a fixed value b, called the bias. The mantissa should be normalised, e.g. if the real mantissa if of the form 1.f then the normalised mantissa should be f, where f is a binary sequence. Sign Exponent Mantissa Data Rep

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**Character representation- ASCII**

ASCII (American Standard Code for Information Interchange) It is the scheme used to represent characters. Each character is represented using 7-bit binary code. If 8-bits are used, the first bit is always set to 0 See (table 5.1 p56, study guide) for character representation in ASCII. Data Rep

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**ASCII – example Symbol decimal Binary 7 55 00110111 8 56 00111000**

: ; < = > ? @ A B C Data Rep

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**Character strings How to represent character strings?**

A collection of adjacent “words” (bit-string units) can store a sequence of letters Notation: enclose strings in double quotes "Hello world" Representation convention: null character defines end of string Null is sometimes written as '\0' Its binary representation is the number 0 'H' 'e' 'l' 'l' o' ' ' 'W' 'o' 'r' 'l' 'd' '\0' Data Rep

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**Layered View of Representation**

Text string Information Data Sequence of characters Information Data Character Bit string Data Rep

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**Working With A Layered View of Representation**

Represent “SI” at the two layers shown on the previous slide. Representation schemes: Top layer - Character string to character sequence: Write each letter separately, enclosed in quotes. End string with ‘\0’. Bottom layer - Character to bit-string: Represent a character using the binary equivalent according to the ASCII table provided. NEED TO PROVIDE PRINTED ASCII TABLES Data Rep

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Solution SI ‘S’ ‘I’ ‘\0’ The colors are intended to help you read it; computers don’t care that all the bits run together. Data Rep

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exercise Use the ASCII table to write the ASCII code for the following: CIS110 6=2*3 Data Rep

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**Unicode - representation**

ASCII code can represent only 128 = 27 characters. It only represents the English Alphabet plus some control characters. Unicode is designed to represent the worldwide interchange. It uses 16 bits and can represents 32,768 characters. For compatibility, the first 128 Unicode are the same as the one of the ASCII. Data Rep

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**Colour representation**

Colours can represented using a sequence of bits. 256 colours – how many bits? Hint for calculating To figure out how many bits are needed to represent a range of values, figure out the smallest power of 2 that is equal to or bigger than the size of the range. That is, find x for 2 x => 256 24-bit colour – how many possible colors can be represented? Hints 16 million possible colours (why 16 millions?) Data Rep

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**24-bits -- the True colour**

24-bit color is often referred to as the true colour. Any real-life shade, detected by the naked eye, will be among the 16 million possible colours. Data Rep

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**Example: 2-bit per pixel**

= (white) 1 (dark grey) (light grey) (black) 4=22 choices 00 (off, off)=white 01 (off, on)=light grey 10 (on, off)=dark grey 11 (on, on)=black Data Rep

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Image representation An image can be divided into many tiny squares, called pixels. Each pixel has a particular colour. The quality of the picture depends on two factors: the density of pixels. The length of the word representing colours. The resolution of an image is the density of pixels. The higher the resolution the more information information the image contains. Data Rep

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Bitmap Images Each individual pixel (pi(x)cture element) in a graphic stored as a binary number Pixel: A small area with associated coordinate location Example: each point below is represented by a 4-bit code corresponding to 1 of 16 shades of gray Data Rep

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**Representing Sound Graphically**

X axis: time Y axis: pressure A: amplitude (volume) : wavelength (inverse of frequency = 1/) Data Rep

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**Sampling Sampling is a method used to digitise sound waves.**

A sample is the measurement of the amplitude at a point in time. The quality of the sound depends on: The sampling rate, the faster the better The size of the word used to represent a sample. Data Rep

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**Digitizing Sound Capture amplitude at these points**

Lose all variation between data points Zoomed Low Frequency Signal Data Rep

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**Summary Integer representation Fraction representation**

Unsigned, Signed, Excess notation, and Two’s complement. Fraction representation Floating point (IEEE 754 format ) Single and double precision Character representation Colour representation Sound representation Data Rep

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Exercise Represent +0.8 in the following floating-point representation: 1-bit sign 4-bit exponent 6-bit normalised mantissa (significand). Convert the value represented back to decimal. Calculate the relative error of the representation. Data Rep

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KS3 Mathematics N2 Negative numbers

KS3 Mathematics N2 Negative numbers

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