It is a fertile field for the development of critical thinking, for the formation of the habit of scientific honesty, for objectivity, for rigor and for precision.

Perfect Numbers The Pythagoreans produced a theory of numbers comprised of numerology and scientific speculation.

In their numerology, even numbers were feminine and odd numbers masculine. The numbers also represented abstract concepts such as 1 stood for reason, 2 stood for opinion, 3 stood for harmony, 4 stood for justice, and so on.

Their arithmetica had a theory of special classes of numbers. A perfect number is a positive integer that is equal to the sum of it divisors. However, for the case of a perfect number, the number itself is not included in the sum. Although perfect numbers are regarded as arithmetical curiosities, their study has helped to develop the theory of numbers.

For example, if n assumes the value 2, 3, 5, or 7, the expression 2n-1 takes on the value 3, 7, 31, orall of which are prime. For these values of n we obtain the perfect numbers 6, 28,and 8, The Neoplatonists Nicomachus of Gerasa and Iamblichus of Chalcis listed these perfect numbers and concluded that they follow a pattern: They alternately end in a 6 or an 8, and there is one perfect number for each interval from 1 to 10, 10 toto 1, and 1, to 10, They conjectured that both parts of the pattern would continue, but in this they were wrong.

In addition, the sixth perfect number, like the fifth, ends with a six. Inthe twentieth perfect number was found. Today, thirty-seven perfect numbers are known.

The prime for the largest of these is 23,, which isdigits in length, and the largest perfect number is 1, digits in length It is not known whether there are an infinite number of perfect numbers.

It has also been proven that every even perfect number must end in six or eight and if it ends in six, the digit preceding it must be odd.

No one has as yet discovered an odd perfect number, but it is known that none exist below Square Numbers Square array of dots, probably formed with pebbles, led the Greeks to numbers that were perfect squares- that is to numbers which, when expressed in a various of ways as the products of two numbers, would have two equal factors.

The most complete discussion of square numbers was given by a Greek, Nicomachus of Gerasa c. Nicomachus was not the original mathematician, but he did organize previous generations of mathematics in a clear and precise manner. If you add any 2 consecutive triangular numbers, you will always make a square number.

As we see from the above figure, the first three triangular numbers are 1, 3, and 6. All square numbers end in either a 0, 1, 4, 5, 6 or a 9. Never a 2, 3, 7, or an 8. The National Council of Teachers of Mathematics.

Historical Topics for the Mathematics Classroom.Below you will find project ideas. I have divided them into 10 different topic areas that I think the project may use the most. Keep in mind that a few of these projects could have been classified in more than one topic.

Do you understand the difference between a formula, expression, identity and equation? A formula is a rule written using symbols that describe a . Later bash (starting from version ) re-implemented most of the advanced features of ksh93 plus a couple of its own.

Currently bash is restricted to integer arithmetic, while ksh93 can do floating-point arithmetic as well. I'm a learning programmer and I've run into a bit of a jumble. I am asked to write a program that will compute and display Fibonacci's Sequence by a user inputted start number and end number (ie.

startNumber = 20 endNumber = and it will display only the numbers between that range). Ask Math Questions you want answered Share your favorite Solution to a math problem Share a Story about your experiences with Math which could inspire or help others. 3 Answer ALL TWENTY SIX questions Write your answers in the spaces provided.

You must write down all the stages in your working. 1. Anil cycled from his home to the park.

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Presentation National High School – SCHEME OF WORK /