🗊Презентация Arithmetic fundamentals of number systems

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Arithmetic fundamentals of number systems Sabina Batyrkhanovna PhD, assistant professor
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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
Описание слайда:
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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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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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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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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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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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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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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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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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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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
Описание слайда:
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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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.
Описание слайда:
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).  
Описание слайда:
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
Описание слайда:
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 =
Описание слайда:
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 =

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Check your answers.
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Check your answers.

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Can you convert these numbers to Roman Numerals? 17 = 22 = 26 = 29 = 30 = 32 = 35 = 50 = 40 = 44 = 49 = 58 = 60 =
Описание слайда:
Can you convert these numbers to Roman Numerals? 17 = 22 = 26 = 29 = 30 = 32 = 35 = 50 = 40 = 44 = 49 = 58 = 60 =

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Were you correct? 17=XVII 22=XXII 26=XXVI 29=XXIX 30=XXX 32=XXXII 35=XXXV 50=L 40=XL 44=XLIV 49=XLIX 58=LVIII 60=LX
Описание слайда:
Were you correct? 17=XVII 22=XXII 26=XXVI 29=XXIX 30=XXX 32=XXXII 35=XXXV 50=L 40=XL 44=XLIV 49=XLIX 58=LVIII 60=LX

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Some more large numbers to try: 600 = 700 = 800 = 1000 = 900 = 1600 = 1700 = 1900 = 2000 =
Описание слайда:
Some more large numbers to try: 600 = 700 = 800 = 1000 = 900 = 1600 = 1700 = 1900 = 2000 =

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Check your answers. 600 = DC 700 = DCC 800 = DCCC 1000 = M 900 = CM 1600 =MDC 1700 =MDCC 1900 = MCM 2000 =MM
Описание слайда:
Check your answers. 600 = DC 700 = DCC 800 = DCCC 1000 = M 900 = CM 1600 =MDC 1700 =MDCC 1900 = MCM 2000 =MM

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The last one Can you convert 2017? MMXVII Now try to write today’s date. Day / Month / Year Well done. You are a Roman Numeral Converter!
Описание слайда:
The last one Can you convert 2017? MMXVII Now try to write today’s date. Day / Month / Year Well done. You are a Roman Numeral Converter!

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Binary Arithmetic Addition Subtraction
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Binary Arithmetic Addition Subtraction

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Addition Like normal decimal addition B A The carry out of the MSB is neglected
Описание слайда:
Addition Like normal decimal addition B A The carry out of the MSB is neglected

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Subtraction Like normal decimal subtraction B A A borrow (shown in red) from the MSB implies a negative
Описание слайда:
Subtraction Like normal decimal subtraction B A A borrow (shown in red) from the MSB implies a negative

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