About Binary Numbers Arithmetic in bases other than ten is often used as an enrichment activity to deepen understanding of base ten and of place value systems in general. Binary numbers not only serve as an example of a place value system in another base but also are the fundamental number system of the computer. Several activities are associated with this module: the magic card trick, weighing objects to change base ten to binary, and doing some ordinary counting and calculation exercises in binary. The Magic Card Trick This trick piques the curiosity of kids, who in turn love to try it on others. It shows up in the Mathematics for Elementary Education course taught at Iowa State, and I have seen it in the Highlights Mathmania children's puzzle books. To play the trick, you ask your "victim" to think of a number between 1 and 31. Then you ask your "victim" which cards that number appears on. You rapidly produce an answer, leaving your "victim" wondering how you could possibly read their mind. The secret is that you add up the numbers on the top left corner of each card the "victim" identifies. The numbers on card A are all the numbers with a binary representation having a 1 in the ones place. The numbers on card B are all those with a binary representation having a 1 in the twos place. Card C: a 1 in the fours place. Card D: a 1 in the eights place. Card E: a 1 in the sixteens place. The trick is a great lead-in to explaining the binary system--once you've played it on your "victim" once or twice he or she will want to know how it works. The Binary Adding Board The 16th century mathematician John Napier was probably the first person to use a binary system as a means of calculation. His life work in mathematics was the simplification of computation, and he invented several mathematical devices to make arithmetic easier, including "Napier's bones", logarithms, and a system for computation in binary using checkers on a chessboard. His chessboard arithmetic is described in his book Rabdologia. While the invention of logarithms was revolutionary and the "bones" became wildly popular, the chessboard arithmetic system remained obscure. The binary adding board in this display is a simple version of his chessboard arithmetic. The materials needed for adding with the addition board are three or four copies of the adding board, instructions, and addition problems (Printable Materials) and about 30 counters (checkers, glass tokens, or poker chips). This will allow three children at a time to experiment with adding. The Binary Weights This is a physical way to discover the connection between binary numbers and decimal numbers. A group of elementary school students in the Ames area which I work with regularly did the weighing without any mention of binary numbers and quickly figured out that what they were doing was producing binary numbers. (They had worked with binary numbers a month previous to this activity). The activity was inspired by an article in the December, 2001 issue of American Scientist: "Third Base" by Brian Hayes. The article was on a base three number system and mentioned an old puzzle: "What is the smallest set of weights you need in order to weigh something which weighs between 1 and 31 grams (or any other units)?" A rocker balance is fairly commonplace elementary school equipment and can be easily borrowed for this activity. A weight set with units of 1, 2, 4, 8, and 16 is not very common, so that probably needs to be made. I chose to build my weight set from pennies, gluing together 2 pennies, then a stack of 4, and a stack of 8, and a stack of 16. It's a good idea to check for uniformity in the weight set--it turned out that while most of the pennies were uniform, there were some outliers which were heavier or lighter than the majority. (Photo). The objects to weigh were simply ordinary things from around the house: a spring link clip, a Hot Wheels car, a marble, a plastic Pokemon figure, etc. The math behind binary numbers Conversion from binary to decimal and the base 2 place value system are explained on the display board. The trick to converting a number from decimal to binary is to figure out exactly what sum of powers of 2 will give the desired number (there is exactly one of these sums). An algorithm for this is to find the largest power of 2 which is less than the number. Write a 1 in that place and subtract that power from the number to be converted. Find the largest power of 2 which is less than the result of your first step. Put a one in that place and zeroes in all the places in between that one and the first place. Continue until you reach the ones place. Example: write 149 in binary: Largest power of 2: 128. Put a 1 in the 128's place 1 149 - 128 = 21 Largest power of 2 less than 21: 16. Put a 1 in the 16's place. Put 0's in the 32's and 64's places. 1001 21 - 16 = 5 Largest power of 2 less than 5: 4. Put a 1 in the 4's place. Put a 0 in the 8's place. 100101 5 - 4 = 1 Put a 0 in the 2's place and a 1 in the ones place. 10010101 in binary is the same as 149 in decimal. Binary Bars by a Bassoonist I have added an optional add-on panel for the display. The motivation was a music tie-in for this module because the elementary school for which I wrote this originally was having a Math and Music Night. The inspiration was an article I read in the January 2001 issue (vol 24 no. 1 p. 103) of The Double Reed magazine, "Binary Bar Count" by Mitchell Clarke. In his article, he claims, "I do not share the fabled musical person's flare for mathematics." He finds this an easy and reliable way to count rests, though. I've included a brief summary of his article for the "gee whiz" value--an unusual application of binary numbers and a perfect fit for Math and Music Night. The panel can be found in Printable Materials. Written by Janet A. Dixon Jan 9, 2002 Modified Mar 18, 2002 Modified Jan 13, 2006