• 1900BC The Moscow papyrus is written. It gives details of Egyptian geometry. • 30000BC Palaeolithic peoples in central Europe and France record numbers • 1850BC on bones. Babylonians know Pythagoras's Theorem. • 5000BC • 1800BC A decimal number system is in use in Egypt. Babylonians use multiplication tables. • 4000BC Babylonian and Egyptian calendars in use. • 1750BC The Babylonians solve linear and quadratic algebraic equations, compile • 3400BC tables of square and cube roots. They use Pythagoras's theorem and use The first symbols for numbers, simple straight lines, are used in Egypt. mathematics to extend knowledge of astronomy. • 3000BC • 1700BC The abacus is developed in the Middle East and in areas around The Rhind papyrus (sometimes called the Ahmes papyrus) is written. It the Mediterranean. A somewhat different type of abacus is used in shows that Egyptian mathematics has developed many techniques to China. solve problems. Multiplication is based on repeated doubling, and division • 3000BC uses successive halving. Hieroglyphic numerals in use in Egypt. • 1360BC • 3000BC A decimal number system with no zero starts to be used in China. Babylonians begin to use a sexagesimal number system for recording financial transactions. It is a place-value system without a • 1000BC zero place value. Chinese use counting boards for calculation. • 2000BC • 540BC Harappans adopt a uniform decimal system of weights and Counting rods used in China. measures. • 500BC • 1950BC The Babylonian sexagesimal number system is used to record and predict Babylonians solve quadratic equations. the positions of the Sun, Moon and planets. Egyptian Numerals ��������� �������������� ������������ Babylonian numerals Mesopotamia Civilization ������������������������������ ������ ���������������������������������������� �������������� �����!������"���������#���������� $��������
Chinese numerals Indian numerals 4359 Greek number systems Roman Numerals � % (� � *� �� & (�� + / %�� ��� ' (��� , 0 *�� �( ) �- . 1 %��� ( * - %� 1110/// --(��� ',+, Mayan mathematics The numerals from al-Sizji's 250 AD to 900 AD, this period was built on top of a civilization which had lived in the region from about 2000 BC. treatise of 969 [8;14;3;1;12] represents 12 + 1 x 20 + 3 x 18 x 20 + 14 x 18 x 20 2 + 8 x 18 x 20 3 = 1253912.
Abaci Logarithm and Slide Rule 8����9��������� $"������� �������������� "��"������� ����������������� �0�%������ /����������"�� $���������������� ������������������� ����������#��� ���������������� �����%+����� ����� � 5 � 6������ � 5���� � � ����� � � ��5������ � ��7����� � � �������� �2����3�����"���� 4�����������"�������������� Numbers: positional notation Decimal Numbers: Base 10 • Number Base B � � � � B symbols per digit: Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 – Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base 2 (Binary): 0, 1 Example: • Number representation: – d 31 d 30 ... d 1 d 0 is a 32 digit number 3271 = – value = d 31 � � � B 31 + d 30 � � � � B 30 + ... + d 1 � � � � � B 1 + d 0 � � B 0 � � (3x10 3 ) + (2x10 2 ) + (7x10 1 ) + (1x10 0 ) • Binary: 0,1 (In binary digits called “bits”) = 1 � � � � 2 4 + 1 � � � � 2 3 + 0 � � � � 2 2 + 1 � � 2 1 + 0 � � � � 2 0 � � – 0b11010 = 16 + 8 + 2 = 26 – Here 5 digit binary # turns into a 2 digit decimal # – Can we find a base that converts to binary easily? #s often written 0b… Hexadecimal Numbers: Base 16 Decimal vs. Hexadecimal vs. Binary Examples: Examples: 00 0 0000 • Hexadecimal: 01 1 0001 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F 1010 1100 0011 (binary) 1010 1100 0011 (binary) 02 2 0010 – Normal digits + 6 more from the alphabet = 0xAC3 = 0xAC3 03 3 0011 – In C, written as 0x… (e.g., 0xFAB5) 04 4 0100 10111 (binary) 10111 (binary) • Conversion: Binary � Hex 05 5 0101 = 0001 0111 (binary) = 0001 0111 (binary) 06 6 0110 – 1 hex digit represents 16 decimal values = 0x17 = 0x17 07 7 0111 – 4 binary digits represent 16 decimal values 08 8 1000 � 1 hex digit replaces 4 binary digits 0x3F9 0x3F9 09 9 1001 • One hex digit is a “nibble”. Two is a “byte” = 11 1111 1001 (binary) = 11 1111 1001 (binary) 10 A 1010 How do we convert between hex and How do we convert between hex and • Example: 11 B 1011 Decimal? Decimal? 12 C 1100 – 1010 1100 0011 (binary) = 0x_____ ? 13 D 1101 14 E 1110 15 F 1111
What to do with representations of numbers? Which base do we use? • Just what we do with numbers! • Decimal: great for humans, especially when doing – Add them arithmetic – Subtract them • Hex: if human looking at long strings of binary 1 1 numbers, its much easier to convert to hex and look 4 – Multiply them bits/symbol – Divide them 1 0 1 0 – Terrible for arithmetic on paper – Compare them • Binary: what computers use; • Example: 10 + 7 = 17 you will learn how computers do +, -, *, / + 0 1 1 1 – …so simple to add in binary that we can – To a computer, numbers always binary build circuits to do it! – Regardless of how number is written: ------------------------- – subtraction just as you would in decimal 32 ten == 32 10 == 0x20 == 100000 2 == 0b100000 – Use subscripts “ten”, “hex”, “two” in book, slides when might – Comparison: How do you tell if X > Y ? be confusing 1 0 0 0 1 BIG IDEA: Bits can represent How to Represent Negative anything!! Numbers? • Characters? • So far, unsigned numbers – 26 letters � � 5 bits (2 5 = 32) � � • Obvious solution: define leftmost bit to be sign! – upper/lower case + punctuation – 0 � � � +, 1 � � � - � � � � � 7 bits (in 8) (“ASCII”) � – standard code to cover all the world’s languages � � 8,16,32 bits � � – Rest of bits can be numerical value of number (“Unicode”) • Representation called sign and magnitude www.unicode.com • Logical values? • MIPS uses 32-bit integers. +1 ten would be: – 0 � � False, 1 � � � � True � � • colors ? Ex: 0000 0000 0000 0000 0000 0000 0000 0001 • locations / addresses? commands? • And –1 ten in sign and magnitude would be: � at most 2 N things • MEMORIZE: N bits � � � 1000 0000 0000 0000 0000 0000 0000 0001 Red (00) Green (01) Blue (11) Shortcomings of sign and Another try: complement the bits magnitude? • Arithmetic circuit complicated • Example: 7 10 = 00111 2 -7 10 = 11000 2 – Special steps depending whether signs are the same or not • Called One’s Complement • Also, two zeros • Note: positive numbers have leading 0s, negative – 0x00000000 = +0 ten numbers have leadings 1s. – 0x80000000 = -0 ten – What would two 0s mean for programming? 00000 00001 ... 01111 • Therefore sign and magnitude abandoned 11111 10000 ... 11110 • What is -00000 ? Answer: 11111 • How many positive numbers in N bits? • How many negative ones?
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