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I declare that the attached work is entirely my own (or when submitted to meet the requirements of an approved group assignment is the work of the group), except where materials cited, quoted or paraphrased are acknowledged in the text. I also declare that this work / assignment has not been submitted for assessment in any other course or university without due acknowledgement.

Question 1

The four (4) circles above are of different diameters. You are to measure its diameter and circumference respectively as accurately as possible and record the measurements in the Table below.

(a) (ii)  The pi () is a very interesting number in mathematics. All students will have to learn and use this symbol when learning mathematics. Search the web and write a short history of this symbol.

(b) Search the web and write a short summary on the history of the digit 'Zero'.

(a) The diagram below shows a simplified 1-4 abacus.

(i)Explain to a person the features of an abacus and how to use it.

(ii)Show to him/her the numbers in separate diagrams of the abacus (you can copy and paste the abacus diagram, I have 'grouped' all the beads, so you need to 'ungroup' the beads in order to move them)

(b)  What are the problems you would encounter when you are asked to use the abacus to find the sum of 53.7 + 9 + 306 + 1008

Explain how you would solve the problems you mentioned above.

Question 3

(a)The circle is a very unique shape and when we connect it with the multiplication process using the circle, we created beautiful shapes.

Question 3 (b)

(i)Copy and complete the Fibonacci sequence up to the 25th term.

(ii)Copy and complete the table of values as indicated in the Table below:

## Pi (π) History and Significance

Pi (π) is the ratio of the circumference of a circle to its diameter that means it is a constant number. For any circle, of any size, I will always be the same. Diameter is the distance measured through the center to the edges of a circle.

Babylonians in the ancient times found the area of a circle by taking three times the radius squared. This gave the value of pi to be pi = 3 and one their stone tablets found 1900-1680 BC had pi to an approximation of 3.125.

The Rhind Payrus in the year 1650 BC gave an insight of pi in the mathematics of the ancient Egypt. They calculated areas of circles that gave an approximate value of about 3.1605 for pi.

Archimedes of Syracuse who was one of the greatest mathematicians in the world did the first calculation of pi between 287-212 BC and he obtained the approximation 223/71 < π < 22/7. Before Archimede gave an indication of his proof, he used a lot of sophisticated inequalities as shown by his approximation. He never gave out the exact value of pi to be 22/7 because if we average the two bounds we get 3.1418. Here is his argument using Pythagoras theorem, he was able to approximate the area of the circle in order to find areas of two polygons as shown by the diagram below.

OA=1

AB= sin (π/K)

AT= tan (π/K)

From the diagram, area of the circles lies between the inscribed and circumscribed polygons and the polygons areas lie in the lower and upper bound of the circle.

The same approach was used by Zu Chongzhi, a Chinese mathematician and an astronomer in the year 429-501 BC. Little is known about his work but he was able to calculate the value of pi to be 355/113 and was done by inscribing a 24,576 regular polygon and did numerous calculations involving square roots in 9 decimal places.

In the 1700s, mathematicians started using the Greek letter π which was introduced by William Jones and this symbol was popularized by Leonard Euler who adopted the symbol in 1737.

In 1800s, George Buffon, a French mathematician devised a way to calculate pi in terms of probabilities.

• Search the web and write a short summary on the history of the digit 'Zero'.

Zero is one of the strange numbers that paradoxes human thoughts because it represents both everything and nothing at the same time. It is a number at the same time a numeral digit that’s used to represent numbers in numerals. Without zero, all branches of science would have no precise definitions. Zero as a digit is used as a placeholder in place values. Zero holds the highest value today. If zero was not invented, binary and the computer could not have existed. Therefore, zero is the greatest invention in which every calculation needs, no matter how small it is, it can never be ignored.

## Zero History and Significance

The history of zero dates back to the ancient Mesopotamia around 3rd B.C in Babylon. The Babylonians devised a number system that was based around values of 60 and developed a sign to make magnitudes the same way we today use zeros to differentiate tenths and hundreds differently. Besides, Mayans started using zero to mark their calendars, and that’s when it cropped to America around 350 A.D.

In India, zero was not seen to have great importance and thus was seen as a placeholder and not a number. Mathematicians lead by Brahmagupta used small dots to show zero and also viewed zero to be a null value and also, they were the first to see that if a number is subtracted by itself, it gives a zero. From India, zero went to China and back to the Middle East. It was taken by Mohammed Musa who a mathematician around 773. He showed that zero could be used a function in algebra equations and zero had made its way into the Arabic system in an oval shape as it is today.

The zero continued migrating, and it reached Europe in the 1100s where Italian mathematician Fibonacci embraced it and helped zero to be introduced into the mainstream and later used by Descartes and Sir Isaac Newton and Gottfried invention of the calculus. Since then zero has been playing an important role in development from physics, engineering, computing, and economics.

Question 2 (30 marks)

(a) The diagram below shows a simplified 1-4 abacus.

• Explain to a person the features of an abacus and how to use it.  [8 marks]

An abacus is an instrument that is used for performing addition, subtraction division and multiplication. It has the following key features;

• Ease of expense report creation which takes little time for beginners.
• Credit card integration whereby it can pull in receipts from all statements easily.
• Digital receipt management.
• Employee reimbursement directly to their accounts.
• Currency conversion feature.
• Ease of bookings such as flights and hotels easily.
• Creation of travel itineraries.
• Invoice creation.

This is how one can use abacus;

To read abacus, one has to look at which beads are moved where and each of the columns shows a different place values as shown by the diagram below.

The beads at the bottom represents numbers from 1 to 5 and those on top represent 5 and 10. You have to move the beads on top down and those on bottom up in order to represent a number. For example, if you push one top bead in hundreds column down will have the number 1000 and if you push three bottom beads in the ten’s column up, you have the number 30.

• Show to him/her the numbers in separate diagrams of the abacus (you can copy and paste the abacus diagram, I have 'grouped' all the beads, so you need to 'ungroup' the beads in order to move them)

(b)  What are the problems you would encounter when you are asked to use the abacus to find the sum of 53.7 + 9 + 306 + 1008
Explain how you would solve the problems you mentioned above?

## Abacus Features and How to Use It

The problems encountered when asked to use the abacus to compute the sum above is because am dealing with both decimal and whole numbers. It will be a bit tricky because I have to take the decimal as a whole number then I can compute the solution.

To solve the problem,

Step 1: we first add 9+306+1008

Step 2: In the one’s column, push the four beads up and the upper bead down to make 9 and make the number 306 by starting with the beads in hundred columns to add them to give you 315.

With the number 315 in place, go to thousand column and make 1008 and add with 315 to give a sum of 1323.

Finally, add the number with 53.7 to the sum obtained above.

Question 3 (40 marks)

• The circle is a very unique shape and when we connect it with the multiplication process using the circle, we created beautiful shapes.
The circumferences of the 2 circles below are divided into 32 equal segments. You are to name the points from 0 to 31 and write 32 beside the 0 and go on to indicate 33 next to 1, 34 next to 2 and so on …. continue to indicate the increasing numbers until you have done 4 rounds. Some numbers are already indicated in the diagram to assist you.
• Multiplication of 2 (2 x)
2 x 0 = 0
2 x 1 = 2

2 x 2 = 4

And so on ….

You are to connect a shape thin line from 1 to 2, 2 to 4, 3 to 6 and so on until a time that you discover no more numbers to connect because the points are already connected. (The first '0' has no line to connect, but when you are at second round, 2 x 32 = 64, there is a line to draw.)

What do you see in your diagram? [10 marks]

Solution

The resolution increases as it gets further from the focus and there is a curve taking shape.

It takes the same of a heart called cardiod.

• Multiplication of 3 (3 x)
3 x 0 = 0
3 x 1 = 3 and so on …. Do the same for the second circle.
What do you see in your diagram? [10 marks]

Solution

It forms a pattern that looks like the number 8.

Question 3 (b)

The Fibonacci sequence is given as 0 1 1 2 3 5 8 13 21 …… where the general formula is

• Copy and complete the Fibonacci sequence up to the 25th
[5 marks]

Solution

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368

• Copy and complete the table of values as indicated in the Table below:
 25th term =46368 24th term = 28657 1.61803 23rd term =17711 22nd term = 10946 1.61803 22nd term =10946 21st term = 6765 1.61803 19th term =2584 18th term = 1597 1.61803 13th term = 144 12th term = 89 1.61798

[5 marks]

• Describe the results obtained in the last column. Explain the significance of these results. [10 marks]

Solution

The squares of 12 and 9 obtained the results obtained in the last column.

Any Fibonacci number that’s a prime number has a subscript that is also a prime number, i.e., T(n)=P; if P is a prime number then n must be a prime number, but the opposite is not true.

Every third Fibonacci number is a multiple of 2.

Any multiple of three gives the fourth Fibonacci number.

Any multiple of five gives the fifth Fibonacci number.

Any multiple of eight gives the sixth Fibonacci number.

The sequence manifests itself in economics and also tracing the pedigree of male bees. It’s also used in computer science to generate random number through the algorithm called Pseudorandom Number Generators.

The sequence also plays a significant role in sorting algorithms by dividing the area into many proportions that are two consecutive Fibonacci numbers which are not equal parts.

Finally, this sequence is used in the derivation of other important mathematical identities.

References

The History of Pie from Egypt to Greece, Rome, Europe and Now – everythingPIES.com. (2018). Retrieved from https://www.everythingpies.com/history-of-pie/

Gibson, W. (2014). Zero history. New York: Berkley Books.

Jones, S. (2015). Abacus: The First 50 Years. Abacus51(4), 485-498.

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