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Using Alternative Numeral Systems in Teaching Mathematics

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Using Alternative Numeral Systems in Teaching Mathematics Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College) Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York Volume 2 Issue 1 Date September 2007 Using Alternative Numeral Systems in Teaching Mathematics Farida Kachapova Murray Black Ilias Kachapov School of Computing School of Computing 13 Fairlands Ave and Mathematical Sciences and Mathematical Sciences Waterview Auckland University of Auckland University of Auckland 1026, New Technology Technology Zealand [email protected] [email protected] [email protected] 1. Introduction The purpose of this article is to stimulate interest in number theory and history of mathematics. Teachers can include parts of this article as additional topics in their classroom teaching. On one hand, the origin of our decimal numbers and possible numeral systems can make an exciting lesson and lead to quite advanced topics in mathematics. On the other hand, this material does not require any specific knowledge and can be explained even to primary students. So it can be used by high school teachers to motivate students and expand their horizons. A numeral system is a language where numbers are represented by symbols – numerals. In the modern mathematics we use the decimal numeral system. It has base 10, which means that all numerals are made of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The probable reason for using base 10 is that humans have 10 fingers and they used them for counting. 10 is the base of the most common numeral system but it is not the only possible one. Some native Americans used spaces between their fingers for counting, so their numeral system had base 8. Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College) Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York Volume 2 Issue 1 Date September 2007 The binary system (with base 2) has originated in China, was studied by G.Leibniz in the 17th century and had useful applications in computer science in the modern time. The hexadecimal system (with base 16) also has applications in computer science because its base can be written as a power of two: 16 = 2 4. The numeral systems with base 12 and base 60 were also used in the past. We still have 12 hours in the clock and 12 months in the year. When we measure time we use 60 minutes in an hour and 60 seconds in a minute; we have 60 seconds in a degree as angular measure. In practice we mostly use the decimal numeral system. The alternative numeral systems can be used by mathematics teachers to stimulate students’ interest in mathematics and research. The students will realise that the decimal system is not the only possible one and even not the best one, so they will learn to think “outside the square”. The binary, senary and other alternative numeral systems can be explained in simple terms at different levels starting from primary school. At the same time they lead to some interesting topics in number theory and general algebra at the tertiary level, such as modular arithmetic and Mersenne primes. 2. Senary Numeral System Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College) Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York Volume 2 Issue 1 Date September 2007 Now we will take a closer look at the senary numeral system – the numeral system with base 6. Imagine aliens from another planet that have three fingers on each hand. They use 6 fingers for counting, so their numeral system has base 6 and all the numerals are constructed from 6 digits: 0, 1, 2, 3, 4, 5. Besides each alien has 6 limbs: two arms, two legs and two wings. The aliens have the same arithmetic as we do but their arithmetic has a simpler representation because it is expressed in the senary numeral system instead of our decimal numeral system. We will write numerals in the senary system with a subscript 6. So numbers 0, 1, 2, 3, 4, 5 are represented by numerals 06 , 16 , 26 , 36 , 46 , 56 respectively. Next number 6 is represented by 106 , 7 is represented by 116 , etc. Actually humans with 5 fingers on each hand can show two-digit senary numerals if they use each hand to show a digit from 0 (a fist with no fingers out) to 5 (all fingers out). 3. Arithmetic in the Senary System Addition table: 16 + 16 = 26 26 + 16 = 36 26 + 26 = 46 Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College) Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York Volume 2 Issue 1 Date September 2007 36 + 16 = 46 36 + 26 = 56 36 + 36 = 106 46 + 16 = 56 46 + 26 = 106 46 + 36 = 116 46 + 46 = 126 56 + 16 = 106 56 + 26 = 116 56 + 36 = 126 56 + 46 = 136 56 + 56 = 146 Multiplication table: 26 × 16 = 26 36 × 16 = 36 46 × 16 = 46 56 × 16 = 056 26 × 26 = 46 36 × 26 = 106 46 × 26 = 126 56 × 26 = 146 26 × 36 = 106 36 × 36 = 136 46 × 36 = 206 56 × 36 = 236 26 × 46 = 126 36 × 46 = 206 46 × 46 = 246 56 × 46 = 326 26 × 56 = 146 36 × 56 = 236 46 × 56 = 326 56 × 56 = 416 26 ×106 = 206 36 ×106 = 306 46 ×106 = 406 56 ×106 = 506 The senary addition table is shorter than the decimal one but otherwise is not very different. The senary multiplication table is much easier to learn for the aliens’ children than the decimal one for the humans’ children because it has several patterns in it: 1. Last digits 2, 4 and 0 make a cycle in the first and third columns. Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College) Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York Volume 2 Issue 1 Date September 2007 2. Last digits 3 and 0 make a cycle in the second column. 3. In the fourth column the first digits are 0, 1, 2, 3, 4, 5 and the last digits are the same in reverse order. Notation: for digits an, …, a1, a0, the symbol an … a1 a0 will denote the numeral in the senary system made of these digits (versus a product an ⋅ …⋅ a1 ⋅ a0). As with any other base, there are algorithms for transferring numbers from senary form to decimal form and vice versa. Senary to Decimal. For a numeral an an-1… a1 a0 in the senary system its decimal n n-1 1 0 form is given by the formula an ⋅6 + an-1 ⋅6 +…+ a1 ⋅ 6 + a0 ⋅ 6 . Examples: 3 2 12346 = 1×6 + 2×6 + 3×6 + 4 = 310, 4 3 2 543216 = 5×6 + 4×6 + 3×6 + 2×6 + 1 = 7465. Decimal to Senary. To transform a natural number in decimal form to senary form we keep dividing numbers by 6 and then write all remainders from right to left. Example: 1244 ÷ 6 = 207 with remainder 2 207 ÷ 6 = 34 with remainder 3 34 ÷ 6 = 5 with remainder 4 5 ÷ 6 = 0 with remainder 5 So 1244 = 54326. Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
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