Using Alternative Numeral Systems in Teaching Mathematics

Home , Numeral system, Senary

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

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.

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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

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

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.

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.

Since the senary multiplication table has several patterns, divisibility criteria are easier in the senary system. To check divisibility of a number by 2, 3 or 6 we use only its last digit: if the last digit is a, then the number can be written as 6b + a for some integer b.

Suppose a natural number n ends on a digit a in the senary system.

Divisibility by 2 (3, 6). n is divisible by 2 (3, 6) if and only if a is divisible by 2 (3,

6).

We can write this in different terms.

Divisibility by 2. n is divisible by 2 if and only if a equals 0, 2 or 4.

Divisibility by 3. n is divisible by 3 if and only if a equals 0 or 3.

Divisibility by 6 = 10 6. n is divisible by 6 = 106 if and only if a equals 0.

Divisibility by 4. Suppose a natural number n has digits a and b as its last two digits in the senary system. Then n is divisible by 4 if and only if the number ab is divisible by 4.

Proof. The numeral ab denotes the number 6a + b. For some integer c, n = 36c + 6a + b = 4 ⋅ 9 + 6a + b.

So n is divisible by 4 if and only if 6a + b is divisible by 4, which is if and only if

ab is divisible by 4.

Next divisibility criterion is proven similarly.

Divisibility by 9 = 13 6. Suppose a natural number n has digits a and b as its last two digits in the senary system. Then n is divisible by 136 if and only if the number ab is divisible by 136.

Divisibility by 5. A natural number n is divisible by 5 if and only if a sum of all its digits in the senary system is divisible by 5.

Proof. The proof is easy if we use congruency modulo m. Two integers x and y are said to be congruent modulo m if (x − y) is divisible by m; this is denoted by x ≡ y (mod m).

k Suppose n = ak … a1 a0. Then n = ak ⋅6 +…+ a1 ⋅ 6 + a0 .

Apparently 6 ≡ 1 (mod 5). For i = 1, 2,…, k by properties of the congruency,

i i 6 ≡ 1 (mod 5), ai 6 ≡ ai (mod 5) and n ≡ ak +… + a1 + a0 (mod 5). Hence n is

divisible by 5 if and only if a0 + a1 +…+ ak is divisible by 5.

Next divisibility criterion is proven similarly.

Divisibility by 7 = 11 6. A natural number n = ak … a1 a0 is divisible by 116 if and only if a sum of its signed digits a0 − a1 + a2 −… ak is divisible by 116.

4. Fractions in the Senary System

A fraction 0.c1c2 c3 … in the senary system is transformed to the following

c c c decimal numeral: 1 + 2 + 3 + .... 6 62 63

3 4 11 Examples: 0.346 = + = = 0.6111… ; 6 62 18

1 1 1 1 1 1 0.111…6 = + + + ... = " = = 0.2. 6 62 63 6 1 5 1! 6

The following example demonstrates how to transform a decimal numeral to senary form by subsequent multiplication by 6.

1 1 3 3 3 9 1 Transforming to senary form. 6× = = 0 ; 6× = = 2 ; 16 16 8 8 8 4 4

1 3 1 1 6× = =1 ; 6× = 3. We collect the whole parts of all results to get a senary 4 2 2 2

1 numeral: 0.0213. So = 0.02136. 16

In the decimal system some fractions can be written as finite decimals and others as

1 1 infinite (recurring) decimals. For example, = 0.5 and = 0.04 are finite; 2 25

1 1 1 = 0.333… and = 0.1666… are infinite. In general, a fraction can be written 3 6 k as a finite decimal if and only if any prime factor of k is either 2 or 5. The reason is that 2 and 5 are the only prime factors of 10.

1 For base 6 the only prime factors are 2 and 3. So a fraction has a finite k expression in the senary system if and only if any prime factor of k is either 2 or 3.

Since multiples of 3 occur more often than multiples of 5, the fractions with finite expressions occur more often in the senary system than in the decimal system:

Senary Decimal

1 1 = 0.36 = 0.5 2 6 2

1 1 = 0.26 = 0.333… 3 6 3

1 1 = 0.136 = 0.25 4 6 4

1 1 = 0.111…6 = 0.2 5 6 5

1 1 = 0.16 = 0.1666… 10 6 6

1 1 = 0.0505…6 = 0.142857… 116 7

1 1 = 0.0436 = 0.125 12 6 8

1 1 = 0.046 = 0.111… 13 6 9

Base 60 seems even more practical for expressing fractions, since it has three prime factors: 2, 3 and 5.

5. Prime Numbers and Perfect Numbers in the Senary System

If a number in the senary system ends on 0, 2 or 4, then it is divisible by 2. If a number in the senary system ends on 0 or 3, then it is divisible by 3. So any prime number in the senary system (except 2 and 3) ends on 1 or 5. These are the first eleven prime numbers:

Decimal 5 7 11 13 17 19 23 29 31 37 41

Senary 56 116 156 216 256 316 356 456 516 1016 1056

We should emphasize that most properties of numbers (for example being a prime number) are independent of the numeral system because numerals are only symbols for expressing numbers. But some numeral systems express number properties in a simpler form than others.

Perfect numbers are another class of numbers that have a simpler form in the senary system. A perfect number is a natural number, which is a sum of its positive factors, excluding itself. Here are a few first perfect numbers:

6 = 1+2+3,

28 = 1+2+4+7+14,

496 = 1+2+4+8+16+31+62+124+248,

8128,

33550336.

These are their senary forms:

6 = 106,

28 = 446,

496 = 21446,

8128 = 1013446,

33550336 = 31550333446.

One can see that every numeral on the right of this list, except 6, ends on 44. It can be proven that in the senary system every perfect number (except 6) has 44 as its last two digits.

Thus, the senary system would make a better symbolic base for dealing with numbers in mathematics but due to historical reasons we use the decimal system and it is too late to change it now.

References

Ore, O. (1988). Number theory and its history. New York: Courier Dover Publications

Lam, L.Y.& Ang, T.S. (2005). Fleeting footsteps: tracing the conception of arithmetic and

algebra in ancient China. River Edge, N.J.: World Scientific.

Kachapova, F. and Kachapov, I. (2005) Senary Numeral System. Workshop at the 9-th

conference of NZ Association of Mathematics Teachers. Retrieved October 1, 2005 from

NZAMT website http://www.nzamt.org.nz/nzamt9/ka/Senary%20System.ppt

Senary. (2007). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2007,

from http://en.wikipedia.org/w/index.php?title=Senary&oldid =121979068

Perfect number. (2007). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2007, from

http://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=123382935