A lot of ones.

The symbol 1 for "one" seems to have been used in many languages for more than 5000 years. The ancient Egyptians cut it into stone and painted it on some of their pottery. Other ancient civilisations also used this numeral, which is thought to have been derived from the lifted finger, a natural way of indicating one object or the number one.

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Now one is not considered to be a prime, but it is definitely a number. However many early scholars did not consider one to be a number. The Flemish mathematician Simon Stevin (born 1548) argued that one is a number in the following way.

"If from a number there is subtracted no number the given number remains, but if from three we take one, three does not remain. Hence one is not no number."

Lots can be discovered mathematically using a string of ones. The numeral 1 is such a natural stroke of the pen, pencil or scribus, that it is often used to denote one object when making a tally. But a string of ones can also denote a numeral as in one thousand one hundred and eleven for example. Consider the following results from successive additions of a string of ones.

1 + 11 = 12

1 + 11 + 111 = 123

1 + 11 + 111+ 1111 = 1234

When does the pattern stop?

Next look at the following successive products of a string of ones.

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

and so on. When does the pattern break down? Investigate also 1 x 11, 11 x 111, etc.

A harder task is to ask your students to discover which sets of ones are prime? Start with

11 = 1 x 11 (prime)

111 = 3 x 37 (composite)

1111 = 11 x 101 (composite)

and hence it is easily seen that all numerals with an even number of ones are composite. But what about numerals with an odd number of ones? Your students can test numerals like 11111 using the ideas given in a previous Discovery article (de Mestre, 2008). Since [square root of 1111] [approximately equal to] 105.4, they will only need to check all the primes from 3 to 105 in this case to see if any divide evenly into 11111. Excel can handle this easily! They should find that

11111 = 41 x 271

and so 11111 is composite.

If your students persevere and check 1111111 they will find that it is also composite. Surprisingly there are very few numbers consisting of all ones which are prime. After 11 the next one has nineteen ones, and after that twenty three ones. It is known that 317 ones is also prime, but further investigations are still being carried out to find more. You could ask your students to check that

Seventeen ones = 2 071 723 x 5 363 222 357

and they will find that their calculator can only partially help.

Here is a pattern of operations which leads to all ones.

(1 x 9) + 2 = 11 (12 x 9) + 3 = 111 (123 x 9) + 4 = 1111

Again investigate when the pattern breaks down. Note also that

(12345679x9)= 111 111 111

and

(12345678 x 9) = 1 111 111 111 - 11 + 1

Fractions with just ones in the numerator and denominator produce patterns in their decimal form.

1/11 = 0.090909 ...

1/111 = 0.009009009 ...

1/1111 = 0.000900090009 ...

while

1/11 = 0.090909 ...

11/111= 0,099099099 ...

111/1111 = 0,099909990999 ...

Ask your students to investigate

11/1111, 11/111111, etc

Next investigate

111/11, 1111/11, 11111/11, etc

Two interesting results are obtained when ones are combined to infinity.

0.11111111 ... = 1/9

and 1 + 1/1+/1+/1+ ... = [square root of 2]

Try this using EXCEL with cell A1=1 and cell A2 =1 + 1 / (1 +A1) and fill down.

You might also like to consider lots of ones in different bases. In base 2 we have the following decimal equivalents.

11 = 3 (prime)

111 = 7 (prime)

1111 = 15 (composite)

11111 = 31 (prime)

111111 = 63 (composite)

and hence k lots of ones in base 2 equals [2.sup.k] -1 in base 10.

In base 3 we have

11 = 4 (composite and square)

111 = 13 (prime)

1111 = 40 (composite)

11111 = 121 (composite and square)

See if your students can find a general formula in base 10 for this pattern of ones in base 3. Geometric progressions will help.

Finally, you might like to set your students the task of finding as many numerals as possible using an increasing succession of ones and any of the operations +, -, x, =, (), and powers. Here are some examples.

(using 3 ones): (1 x 1) - 1 = 0,

11 + 1 = 12

(using 4 ones): (1+1+1)/1 = 3

[11.sup.(1+1)] = 121

See who can make the maximum number of numerals for each designated collection of ones from 2 lots onwards. Happy discoveries!

Reference

de Mestre, N. (2008). Prime numbers. The Australian Mathematics Teacher, 64 (2), 6-7.