In this Newsletter:<\/p>\n
1. Math tip - Hexadecimal Numbers
\n2. Encryption - and a challenge
\n3. Current IntMath Poll - Math Anxiety
\n4. From the math blog
\n5. Final thought<\/p>\n
First, so we can better understand hexadecimal numbers, let's talk about our normal number system<\/b>: 0, 1, 2, 3, 4, ...<\/p>\n
This system is called the Hindu-Arabic<\/b> number system because it was developed in India (by the Hindus) and was spread throughout the Middle East and Europe by Muslim scholars around 1000 years ago. Hindu-Arabic numerals finally replaced the Roman Numeral system (I, II, III, IV, V,...) in Europe around 1500.<\/p>\n
The biggest contribution of the new number system was that it was positional<\/b>. That is, if I have the number 439, the \"4\" really means \"4x100 = 400\", the \"3\" means \"3x10 = 30\" and the \"9\" means \"9x1 = 9 units\". The position<\/b> of the digit in the number determines its value<\/b>. <\/p>\n
Our number system uses base 10<\/b>. This means that after every 10 digits, we need to add \"1\" to the column on the left. Consider 36, 37, 38, 39. The next number is 40. We have added \"1\" to the column on the left, and the 9 \"ticks over\" to 0 once again.<\/p>\n
Using exponents, base 10 numbers are based on the following values in each column (right to left):<\/p>\n
100<\/sup> = 1 Now let's look at an alternative number system that has important applications in computers. <\/p>\n Here we learn that 100 is not always \"one hundred\"!<\/p>\n Hexadecimal Numbers<\/b> In hexadecimal numbers, the digits 0 to 9 are used and the next number is written using the letter \"A\" (to mean \"ten\"). Continuing, \"B\" means \"eleven\", \"C\" means \"twelve\", \"D\" means \"thirteen\", \"E\" means \"fourteen\" and \"F\" means \"fifteen\".<\/p>\n The next number is written \"10\" (which actually means \"sixteen\" in our normal system), and the following number is \"11\" (which is really \"seventeen\"). Next is \"12\" which means \"16+2=18\", and so on. <\/p>\n Like the Hindu-Arabic decimal system, the hexadecimal system is positional<\/b>. <\/p>\n For example, in the hexadecimal number \"34A\", <\/p>\n Altogether, the number \"34A\" has decimal value The hexadecimal system is based on powers of 16, as follows:<\/p>\n 160<\/sup> = 1 So in the hexadecimal system, the number \"100\" is actually equal to 256 in the ordinary decimal system.<\/p>\n Similar to the decimal system, we add \"1\" to the column on the left as we count up. In the hexadecimal system, \"E + 1 = 10\" (which means \"fifteen plus one equals sixteen\"). Or we could have the counting sequence 39, 3A, 3B, 3C, 3D, 3E, 3F, 40. That last number is obtained by adding one to the left column (giving \"4\") and the \"F\" ticks over back to \"0\".<\/p>\n For an example of how hexadecimal numbers are used in computing, the colors of the Web page<\/b> that you are looking at right now are coded using hexadecimal values. A white background will be written #FFFFFF, whereas the light blue used as the background for Interactive Mathematics<\/a> is #E5EDEF.<\/p>\n [This topic was requested by a reader. It involves ETAOIN, computers and Facebook.]<\/p>\n One of the hot areas in current mathematical research is data encryption<\/strong>. With so much sensitive data floating around the Internet, encryption has become vital to protect online banking, e-commerce and even our FaceBook profiles. \ud83d\ude42<\/p>\n I guess you’re all familiar with simple encryption, like the following:<\/p>\n I have simply used A=1, B=2, C=3,..., Z=26 to turn my text message \"knowledge is power\" into a set of numbers. Such encoding of a message is called a \"cipher\" (or \"cypher\").<\/p>\n However, this cipher is too simple and very easy to crack. One way to make it more secure is to use what is called a \"shift cipher<\/strong>\", where we add the same number to each of the digits in our code. Let's add 9. So the first number (representing \"K\") will become 11+9=20, and N will become 14+9=23. <\/p>\n When we get to \"W\", we have a little problem, since 23+9=32. There is no 32nd letter of the alphabet, so we just use modulo<\/b> arithmetic. (What is the remainder when we divide 32 by 26? Answer: 6. So we say \"32 is equivalent to 6, modulo 26\").<\/p>\n By applying our shift cipher and modulo arithmetic, our coded message is now:<\/p>\n
\n101<\/sup> = 10
\n102<\/sup> = 100
\n103<\/sup> = 1,000
\n104<\/sup> = 10,000<\/p>\n
\nHexadecimal numbers use base 16<\/b> (not base 10 like our normal number system). \"Hexa-\" means \"6\" and \"deci-\" is \"ten\". <\/p>\n\n
\n768+64+10 = 842.<\/p>\n
\n161<\/sup> = 16
\n162<\/sup> = 256
\n163<\/sup> = 4,096
\n164<\/sup> = 65,536<\/p>\n
\n2. Encryption<\/h2>\n
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\n K<\/td>\n N<\/td>\n O<\/td>\n W<\/td>\n L<\/td>\n E<\/td>\n D<\/td>\n G<\/td>\n E<\/td>\n I<\/td>\n S<\/td>\n P<\/td>\n O<\/td>\n W<\/td>\n E<\/td>\n R<\/td>\n<\/tr>\n \n 11<\/td>\n 14<\/td>\n 15<\/td>\n 23<\/td>\n 12<\/td>\n 5 <\/td>\n 4 <\/td>\n 7 <\/td>\n 5 <\/td>\n 9 <\/td>\n 19<\/td>\n 16<\/td>\n 15<\/td>\n 23<\/td>\n 5 <\/td>\n 18<\/td>\n<\/tr>\n<\/table>\n