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№1 слайд![Arithmetic fundamentals of](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img0.jpg)
Содержание слайда: Arithmetic fundamentals of number systems
Sabina Batyrkhanovna
PhD, assistant professor
№2 слайд![Outline Different number](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img1.jpg)
Содержание слайда: Outline
Different number systems
Why use different ones?
Binary / Octal / Hexadecimal
Conversions
Negative number representation
Binary arithmetic
Overflow / Underflow
№3 слайд![Number Systems Four number](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img2.jpg)
Содержание слайда: Number Systems
Four number systems:
Decimal (10)
Binary (2)
Octal (8)
Hexadecimal (16)
№4 слайд![Binary numbers Computers work](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img3.jpg)
Содержание слайда: 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 !
№5 слайд![Decimal numbers x x x x](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img4.jpg)
Содержание слайда: Decimal numbers
1439 = 1 x 103 + 4 x 102 + 3 x 101 + 9 x 100
Thousands Hundreds Tens Ones
Radix = 10
№6 слайд![Binary Decimal x x x x x x x](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img5.jpg)
Содержание слайда: 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, …
№7 слайд![Decimal Binary](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img6.jpg)
Содержание слайда: Decimal Binary
№8 слайд![Octal Decimal x x x x x x](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img7.jpg)
Содержание слайда: 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
№9 слайд![Decimal Octal](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img8.jpg)
Содержание слайда: Decimal Octal
№10 слайд![Hex Decimal BAD x x x x x x](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img9.jpg)
Содержание слайда: 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
№11 слайд![Decimal Hex](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img10.jpg)
Содержание слайда: Decimal Hex
№12 слайд![Why octal or hex? Ease of use](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img11.jpg)
Содержание 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
№13 слайд![Roman Numerals](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img12.jpg)
Содержание слайда: Roman Numerals
№14 слайд![A Brief History of Roman](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img13.jpg)
Содержание 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.
№15 слайд![Arabic numbers Roman numerals](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img14.jpg)
Содержание 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).
№16 слайд![As a general guide Roman](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img15.jpg)
Содержание 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
№17 слайд![Some more examples VIII XIX](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img16.jpg)
Содержание 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 =
№18 слайд![Check your answers.](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img17.jpg)
Содержание слайда: Check your answers.
№19 слайд![Can you convert these numbers](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img18.jpg)
Содержание слайда: Can you convert these numbers to Roman Numerals?
17 = 22 = 26 = 29 = 30 =
32 = 35 = 50 = 40 =
44 = 49 = 58 = 60 =
№20 слайд![Were you correct? XVII XXII](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img19.jpg)
Содержание 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
№21 слайд![Some more large numbers to try](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img20.jpg)
Содержание слайда: Some more large numbers to try:
600 = 700 = 800 =
1000 = 900 =
1600 = 1700 = 1900 =
2000 =
№22 слайд![Check your answers. DC DCC](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img21.jpg)
Содержание слайда: Check your answers.
600 = DC 700 = DCC 800 = DCCC
1000 = M 900 = CM
1600 =MDC 1700 =MDCC
1900 = MCM 2000 =MM
№23 слайд![The last one Can you convert](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img22.jpg)
Содержание 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!
№24 слайд![Binary Arithmetic Addition](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img23.jpg)
Содержание слайда: Binary Arithmetic
Addition
Subtraction
№25 слайд![Addition Like normal decimal](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img24.jpg)
Содержание слайда: Addition
Like normal decimal addition
B
A
The carry out of the MSB is neglected
№26 слайд![Subtraction Like normal](/documents_6/37a99e480109add6fc92fa83ac7f83f5/img25.jpg)
Содержание слайда: Subtraction
Like normal decimal subtraction
B
A
A borrow (shown in red) from the MSB implies a negative