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Computer Math

Computer Math . CPS120: Data Representation. Representing Data. The computer knows the type of data stored in a particular location from the context in which the data are being used; i.e. individual bytes, a word, a longword, etc 01100011 01100101 01000100 01000000

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Computer Math

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  1. Computer Math CPS120: Data Representation

  2. Representing Data • The computer knows the type of data stored in a particular location from the context in which the data are being used; • i.e. individual bytes, a word, a longword, etc • 01100011 01100101 01000100 01000000 • Bytes: 99(10, 101 (10, 68 (10, 64(10 • Two byte words: 24,445 (10 and17,472 (10 • Longword: 1,667,580,992 (10

  3. Numbers Natural Numbers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative Numbers A value less than 0, with a – sign Examples: -24, -1, -45645, -32 2

  4. Numbers (Cont’d) Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 Rational Numbers An integer or the quotient of two integers Examples: -249, -1, 0, ¼ , - ½ 3

  5. Natural Numbers How many ones are there in 642? 600 + 40 + 2 ? Or is it 384 + 32 + 2 ? -- Octal Or maybe… 1536 + 64 + 2 ? -- Hexadecimal 4

  6. Natural Numbers 642 is 600 + 40 + 2 in BASE 10 The base of a number determines the number of digits and the value of digit positions 5

  7. Positional Notation Continuing with our example… 642 in base 10 positional notation is: 6 x 10² = 6 x 100 = 600 + 4 x 10¹ = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10 The power indicates the position of the number This number is in base 10 6

  8. Positional Notation R is the base of the number As a formula: dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1 n is the number of digits in the number d is the digit in the ith position in the number 642 is:  63 * 102 +  42 * 10 +21 7

  9. Positional Notation What if 642 has the base of 13? 642 in base 13 is equivalent to 1068 in base 10 + 6 x 13² = 6 x 169 = 1014 + 4 x 13¹ = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2 = 1068 in base 10 6 8

  10. Representing Real Numbers • Real numbers have a whole part and a fractional part. For example 104.32, 0.999999, 357.0, and 3.14159 the digits represent values according to their position, and those position values are relative to the base. • The positions to the right of the decimal point are the tenths position (10-1 or one tenth), the hundredths position (10-2 or one hundredth), etc.

  11. Representing Real Numbers (Cont’d) • In binary, the same rules apply but the base value is 2. Since we are not working in base 10, the decimal point is referred to as a radix point. • The positions to the right of the radix point in binary are the halves position (2-1 or one half), the quarters position (2-2 or one quarter), etc.

  12. Representing Real Numbers (Cont’d) • A real value in base 10 can be defined by the following formula: • The representation is called floating point because the number of digits is fixed but the radix point floats.

  13. Representing Real Numbers (Cont’d) • Likewise, a binary floating –point value is defined by the following formula: sign * mantissa * 2exp

  14. Representing Real Numbers (Cont’d) • Scientific notation is a term with which you may already be familiar, so we mention it here. Scientific notation is a form of floating-point representation in which the decimal point is kept to the right of the leftmost digit. • For example, 12001.32708 would be written as 1.200132708E+4 in scientific notation.

  15. Representing Text • To represent a text document in digital form, we simply need to be able to represent every possible character that may appear. • There are finite number of characters to represent. So the general approach for representing characters is to list them all and assign each a binary string. • A character set is simply a list of characters and the codes used to represent each one. By agreeing to use a particular character set, computer manufacturers have made the processing of text data easier.

  16. Alphanumeric Codes • American Standard Code for Information Interchange (ASCII) • 7-bit code • Since the unit of storage is a bit, all ASCII codes are represented by 8 bits, with a zero in the most significant digit • H e l l o W o r l d • 48 65 6C 6C 6F 20 57 6F 72 6C 64 • Extended Binary Coded Decimal Interchange Code (EBCDIC)

  17. The ASCII Character Set • ASCII stands for American Standard Code for Information Interchange. The ASCII character set originally used seven bits to represent each character, allowing for 128 unique characters. • Later ASCII evolved so that all eight bits were used which allows for 256 characters.

  18. The ASCII Character Set (Cont’d)

  19. The ASCII Character Set (Cont’d) • Note that the first 32 characters in the ASCII character chart do not have a simple character representation that you could print to the screen.

  20. The Unicode Character Set • The extended version of the ASCII character set is not enough for international use. • The Unicode character set uses 16 bits per character. Therefore, the Unicode character set can represent 216, or over 65 thousand, characters. • Unicode was designed to be a superset of ASCII. That is, the first 256 characters in the Unicode character set correspond exactly to the extended ASCII character set.

  21. The Unicode Character Set (Cont’d) A few characters in the Unicode character set

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