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# Numerals and Numbers

Numerals and Numbers. How characters and integers are represented inside a computer (and in assembly language). What is ASCII?. Humans can communicate with machines But need a language each understands

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## Numerals and Numbers

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1. Numerals and Numbers How characters and integers are represented inside a computer (and in assembly language)

2. What is ASCII? • Humans can communicate with machines • But need a language each understands • English language employs a finite set of character-symbols (‘alphabet’) of letters, plus digits and various punctuation-marks • Character-symbols are assigned ‘values’ • American Standard Code for Information Interchange (ASCII) is a standard scheme

3. ‘A’, ‘B’, ‘C’, … , ‘Z’ ‘a’, ‘b’, ‘c’, … , ‘z’ ‘0’, ‘1’, ‘2’, … , ‘9’ ‘,’ (comma) ‘-’ (hyphen) ‘.’ (period) 65, 66, 67, … , 90 97, 98, 99, … , 122 48, 49, 50, … 57 44 45 46 Some ASCII illustrations There are 128 different numbers defined by the ASCII code (Type the Linux command: ‘man ascii’ to see the complete table)

4. ASCII ‘control-codes’ • ASCII was devised for typewriter terminals • Some code-values denote ‘typing motions’ • The ‘back-space’ motion is 8 • The ‘tab’ motion is 9 • The ‘new-line’ motion is 10 • The ‘carriage-return’ motion is 13 • A code exists for the typewriter’s bell: 7

5. Numerals versus Numbers • A ‘numeral’ is a digit-character: ‘1’, ‘2’, … • The digit-character ‘0’ has ASCII value 48 • The digit-character ‘1’ has ASCII value 49 … • The digit-character ‘9’ has ASCII value 57 • So a ‘numeral’ isn’t what it appear to be! e.g., character ‘7’ isn’t equal to number 7

6. Communicating in ASCII • Humans talk to computers using ASCII • Computer says: Give me a number • Human replies: ok, I will say ‘6’ • Computer must ‘interpret’ this input • Must ‘convert’ the character’s ASCII-value into the number-value that was intended • It’s easy: computer can just do subtraction ‘6’ – ‘0’ = 54 – 48 = 6

7. Bigger numbers? • Computer says: Tell me a bigger number • Human thinks: ok, how about ninety-five? • Human types in two digits: ‘9’ and ‘5’ • Here the typing-order is important: because “95” means 9-times-10, plus 5 • Computer sees two ASCII values: 57, then 53 • It must convert 57 into 9, and 53 into 5, and then do a multiplication (by 10) and an addition step

8. Positional Notation • We use the Hindu-Arabic notation system • It uses ten digit-symbols: ‘0’, ‘1’, ‘2’. … , ‘9’ • For writing numbers bigger than nine, the digit-symbols are comined in a sequence: so three hundred sixty-five is written “365” • The “position” of each digit is significant, as it affects what that digit really means • Meanings determined by the number-base

9. Other number bases? • An analogy: base two versus base ten • Base-ten uses ten digits: 0, 1, 2, … , 0 • Base-two uses two digits: 0, 1 • 3-digit number using base ten: 365 • 3-digit number using base two: 101 • 365 (base 10) means 3x100 + 6x10 + 5x1 • 101 (base 2) means 1x4 + 0x2 + 1x1 • 101 (base 10) means 1x100 + 0x10 + 1x1

10. Conversions • How do we write twenty-five using base 2? • Algorithm: use repeated division by 2, but save the remainders as they’re generated • 25 divided by 2: quot=12 rem=1 • 12 divided by 2: quot=6 rem=0 • 6 divided by 2: quot=3 rem=0 • 3 divided by 2: quot=1 rem=1 • 1 divided by 2: quot=0 rem=1 • So twenty-five (base 2) can be written: 11001 • It means: 1x16 + 1x8 + 0x4 + 0x2 + 1x1

11. Exercise • Write forty-nine in base 8 notation? • 49 divided by 8: quot=6 rem=1 • 6 divided by 8: quot=0 rem=6 • We stop when quotient equals zero • Answer: 49 (base 8) is written as 61 • It means (positional notation): 6x8 + 1x1 • Note: remainders used in backward order!

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