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Arithmetic fundamentals of number systems Sabina Batyrkhanovna PhD, assistant professor

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Outline Different number systems Why use different ones? Binary / Octal / Hexadecimal Conversions Negative number representation Binary arithmetic Overflow / Underflow

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Number Systems Four number systems: Decimal (10) Binary (2) Octal (8) Hexadecimal (16)

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Binary numbers Computers work only on two states On Off Basic memory elements hold only two states Zero / One Thus a number system with two elements {0,1} A binary digit – bit !

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Decimal numbers 1439 = 1 x 103 + 4 x 102 + 3 x 101 + 9 x 100 Thousands Hundreds Tens Ones Radix = 10

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Binary Decimal 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 = 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1 = 8 + 4 + 0 + 1 (1101)2 = (13)10 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, …

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Decimal Binary

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Octal Decimal 137 = 1 x 82 + 3 x 81 + 7 x 80 = 1 x 64 + 3 x 8 + 7 x 1 = 64 + 24 + 7 (137)8 = (95)10 Digits used in Octal number system – 0 to 7

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Decimal Octal

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Hex Decimal BAD = 11 x 162 + 10 x 161 + 13 x 160 = 11 x 256 + 10 x 16 + 13 x 1 = 2816 + 160 + 13 (BAD)16 = (2989)10 A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

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Decimal Hex

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Why octal or hex? Ease of use and conversion Three bits make one octal digit 111 010 110 101 7 2 6 5 => 7265 in octal Four bits make one hexadecimal digit 1110 1011 0101 E B 5 => EB5 in hex

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Roman Numerals

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A Brief History of Roman Numerals Roman numerals originated in ancient Rome. This ancient counting system is believed to have started with the ancient Etruscans. The symbol for one in the roman numeral system probably represented a single tally mark which people would notch into wood or dirt to keep track of items or events they were counting. It would also be easy to write on a wax tablet.

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Arabic numbers → Roman numerals conversion Roman numerals are written as combinations of seven letters. I = 1 V = 5 X = 10 L = 50 C = 100 D =500 M = 1000 The letters can be written as capital (XVI) or lower-case letters (xvi).

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As a general guide Roman Numerals are made up by adding or subtracting numbers like this:- 11=10+1 = XI 9 = 10 – 1 = IX 40 = 50 –10 = XL If you want to say 1,100 in Roman Numerals, you would say M for 1000 and then put a C after it for 100; 1,100 = MC 900 =1000 – 100 so the C comes before M = CM

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Some more examples: VIII = 5+3 = 8 XIX = 10+ 9 = 19 (Remember 9 is always = IX (1 less than 10) XL = 50-10 = 40 XC = 100-10 = 90 Try these on whiteboards: 7 = 12 = 15 = 20 =