The ancient Egyptians used a decimal system in which pictographs were used to represent powers of 10.

Transcription

1 Section 3.1 The ancient Egyptians used a decimal system in which pictographs were used to represent powers of 10. Value Symbol Represents 1 Staff 10 Heel bone 100 Coil of rope 1,000 Lotus flower 10,000 Pointing finger 100,000 Tadpole 1,000,000 Astonished person Example: Example: Write 2,041,365 in Egyptian Hieroglyphs.

2 Section 3.2 Roman numerals use capital letters for different values, which are added together to get the number being represented (similar to Egyptian numerals). Symbol I V X L C D M Value "If Victor's X-ray Looks Clear, Don't Medicate" Roman numerals appear in things like buildings and copyrights. Generally speaking, symbols will be written left to right in decreasing value. LX = CC = DCXII = MDCCCLXXXVIII = Unlike Egyptian numerals, Roman numerals can pair two symbols together to lessen the value of the second symbol. This is called an increase. Possible increase pairs are: IV = IX = XL = XC = CD = CM = Notice how the values of the symbols increase from left to right. the value of the first symbol from the value of the second symbol. Increases must be done first (kind of like grouping symbols), then add up the values. DCCXCVI = MCMXLIX =

3 For numbers greater than or equal to 4,000 a bar is written over the number: One bar = multiply the number by 1,000 XLII XC CDV Example: Write using Roman numerals

4 Section 3.3 Our number system is called a base-10 place value system. Ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Each "place" represents a power of 10 Powers increase from right to left The first power used is always Example: 27, However, other bases are or were used in certain applications or languages: Base 2 (binary): computers Base 4 (quaternary): Chumash Base 5 (quinary): Saraveca and Gumatj Base 8 (octal): computers Base 16 (hexadecimal): computers Base 60 (sexagesimal): Babylonian, time, angles When a number is written using another base, we will use the notation (number)base. Example: Write (2403)5 as a base 10 number. Five digits: Powers of! Change each number to base 10: (571)8 (1011)2 (1021)3

5 For hexadecimal (base 16), we need more than 10 digits, so letters are used in addition to numbers. Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F A = B = C = D = E = F = Change each number to base 10: (5E)16 (1A2)16 To convert from base 10 to another base requires repeated subtraction or division (you will want a calculator for this). Example: Change 238 to base 5 (by subtraction) Example: Change 135 to octal and to binary. Example: Change 3391 to hexadecimal. Five hundred begins it, five hundred ends it, Five in the middle is seen; First of all letters, the first of all figures, Take up their stations between. Join all together, and then you will bring Before you the name of an eminent king.

6 Section 3.4 In 1947 the transistor was invented. Transistors are tiny switches that can be triggered by electric signals. They enabled the invention of the modern computer. In a computer, information is stored in memory as a bit (short for binary digit) using transistors. A bit is a basic unit of information used in computing and digital communications. A bit can only have one of two values and may be physically represented with a two-state device. These state values are most commonly represented as either a or. Think of representing the state when the switch (transistor) is on, and when it is off. The byte is a unit of digital information that most commonly consists of eight bits, representing a binary number. Think of a byte as a set of 8 switches, whose state is indicated by a 0 or 1. ASCII, abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication that was introduced in ASCII codes represent in computers. Here is a table of some ASCII codes. We will only be interested in the alphabet characters. Upper case: Lower case: The computer will store an ASCII character as an 8-digit binary number (a byte). Example: Translate the binary code to English Example: Translate the word "Math" to ASCII (use base 10 codes).

7 Section 3.5 POSTNET stands for POSTal Numeric Encoding Technique. It is a barcode used by USPS to encode ZIP code information. How many bars for each digit? How many tall bars for each digit? How does it work? 1-12 in Binary Add 5 th bit Bars Renumber The number will help you find the value for each POSTNET code: The barcode starts and ends with a full bar (often called a guard rail) and has a check digit after the ZIP, ZIP+4, or delivery point.

8 To determine the check digit, first add up all of the digits in the Zip code. The check digit is chosen so that the sum of all digits in the barcode is a multiple of. Find the check digit for each ZIP code: Example: Write in POSTNET code. Check digit: Note: In 2009 the POSTNET barcode was replaced by Intelligent Mail barcode. The IM barcode is intended to provide greater information and functionality. It effectively incorporates the routing ZIP code and tracking information included in previously used postal barcode standards.

9 Section 3.6 In the Middle Ages, Königsberg, Prussia, became a very important city and trading center strategically positioned on the river. The healthy economy allowed the people of the city to build seven bridges across the river. The bridges were called Blacksmith s bridge, Connecting Bridge, Green Bridge, Merchant s Bridge, Wooden Bridge, High Bridge, and Honey Bridge. According to lore, the citizens of Königsberg used to spend Sunday afternoons walking, and they wondered if they could walk around the city crossing each of the seven bridges only once. None of the citizens of Königsberg could invent such a route, but they could not prove that it was impossible. Here is a map of Königsberg showing the river and seven bridges. Can you find a route? On August 26, 1735, Leonhard Euler presented a paper containing the solution to the Königsberg bridge problem. He addressed both this specific problem, as well as a general solution with any number of landmasses and any number of bridges. Euler accidentally sparked a new branch of mathematics called graph theory. A graph is a collection of and connecting them. We want to know if a graph is traversable, which means we can trace the graph following each edge just, without having to lift our pencil from the paper. Try it! It turns out, a graph is traversable if and only if it has or odd vertices. If odd vertices, start at one and you will end at the other. If it has odd vertices, it is called an Euler Circuit. You can start at any vertex, traverse the graph, and end back where you began.

10 Which graphs are traversable? Which are Euler circuits?

11 Section 3.7 We return to 18th century Königsberg. Could a citizen walk a path across all seven bridges without crossing a bridge twice? Euler proved it was. In 1875, the people of Königsberg decided to build a new bridge. Now is it traversable? Euler circuit? Is this an Euler circuit?

12 Is this an Euler circuit? What is the minimum number of bridges to burn so that it is? Modern applications of graph theory:

13 Section 3.8 Like bridge problems, we can try to find routes through buildings with multiple rooms and doors. For example, can you pass through all the rooms while going through each door only once? Try it! We can use graph theory to solve this! Represent each room and the outside as a vertex, and each doorway as an edge. You can also just count the number of doorways to each room (including the outside). Is the game of Clue traversable?