Department of PG studies and Research in Mathematics St Aloysius College (Autonomous), Mangaluru
MESSAGE St Aloysius College(Autonomous) is a 141 year old premier educational institution imparting quality education in the coastal Karnataka region. The motto of our College is ‘Lucet et ardet’ which means shine to enkindle. At present the College has more than 6800 students studying both at undergraduate and post graduate levels. The College has 21 post-graduate programs. The department of post graduate studies in Mathematics was started in the year 2010.Though the department is young it conducts a number of activities in forming its students academically accomplished, morally upright and technologically sound individuals to face the challenges of the society and thus contribute to the growth of the Nation. Mathematics is the purest of sciences, often referred to as the Father of Sciences. All the applied sciences including Physics, Chemistry and Engineering rely on mathematical knowledge to understand their concepts. It is the methodical application of matter. Mathematics makes life orderly and minimises the chaos. Some of the intellectual qualities nurtured by Mathematics are power of reasoning, creativity, spatial thinking, critical thinking and problem solving ability. During the lockdown, the department of post graduate studies in mathematics has come up with the novel idea of E magazine. I congratulate the HOD Ms Anupriya Shetty and all other faculty members for this unique initiative. DR RICHARD GONSALVES DIRECTOR OF LCRI (Loyola Centre for Research and Innovation) BLOCK
MESSAGE Since 2010, the Department of Post Graduate Studies and Research in Mathematics has produced more than 300 post graduates in Mathematics, now pursuing their careers in academia and industry, who are competent enough to apply their knowledge, critical thinking and communication skills in the real-world of Magis - the challenge to strive for excellence. The department provides an environment for the students to gain knowledge of greater depth in the subject and helps them to develop logical thinking as well as analytical skills. Along with the regular classroom studies, we focus to enhance the all-round personality of our students through co-curricular activities such as Math Fiesta – an undergraduate level math event, seminars, workshops, social outreach, MATRIX- the departmental association etc., To ensure the continuity of these academic activities during the tough times of Covid 19 pandemic, we have adapted various e-learning platforms. The announcement of is one such attempt where-in students are given an opportunity to express their ideas, interests and creativity. I would like to congratulate Ms Laisha Laveena D’Souza, the staff in charge and the entire editorial team for their hard work and dedication that has resulted in the successful publication of this e-magazine. I am grateful to the faculty of our department for their guidance, all MSc Mathematics students for providing with the appropriate content and of course, our beloved alumni for expressing their opinion about the department. I would like to end with a quote by Prof Maryam Mirzakhani (1977-2017), an inspirational mathematician from Iran, the first and to-date only female to win the most prestigious award in Mathematics - the Field’s Medal.
“There are times when I feel like I'm in a big forest and don't know where I'm going. But then somehow, I come to the top of a hill and can see everything more clearly. When that happens it's really exciting.” MS ANUPRIYA SHETTY HEAD OF THE DEPARTMENT
MESSAGE Print media is one of the oldest means of disseminating information. Although quick technological tools exist for expression of ideas in today’s world, but significance of print media cannot be denied. Information and facts are very much available everwhere but analytical elucidation is developed when writing takes it form that leads to self-satisfaction. Magazine is a means to provide platform for students to come forward, identify their talent, discover their potential and move on the path of progress. It is supposed to garner diverse thoughts and expressions altogether. I am delighted to see that creative capacity of our students has been transformed into a tangible way in form of ‘п-zine’ as result of their tireless efforts. I sincerely thank the editorial team and the students of Mathematics Department whose diligent efforts and strong determination has made the dream of ‘п-zine ’ come true. I would like to thank the HOD of the department Ms. Anupriya Shetty for giving me the opportunity and encouraging me throughout. I congratulate the batch of 2019-2021 for the successive first edition of the magazine and wish them a very successful life in future. MS LAISHA LAVEENA D’SOUZA LECTURER
Message to our ex-HOD Ms Suma Devi P G Sometimes words become insu cient to thank someone. Our beloved Suma ma'am had to leave us due to the restriction in the pandemic rules, after enlightening us for two semester. She was not just the HoD by position but was also the motherly gure who guided and supported us in all our endeavors. We sincerely thank her for leading and encouraging us in the right path. Her absence will be felt always. Sreelekshmi S II MSc Mathematics
I used to always like Mathematics, but it was my Alma matter which turned this into my passion and now my profession.It’s indeed my privilege to be the alumnae of this pretistigious institution. ‘‘KEERTHI ALVA Head of the Department Dept. of Mathematics Canara college, Mangalore FOOT PRINTS ‘‘The Department of Post Graduate Studies and Research in Mathematics has given me a chance to gain a lot of knowledge on Mathematics. The department has a dedicated sta who render their sel ess service in imparting education to the students. The opportunities such as seminars, workshops, association and various programs organised by the department and college have helped in inculcating leadership quality and given me a path to overcome the challenges of life. AVINASH DANIEL DSOUZA
ST ALOYSIUS COLLEGE (Autonomous) Mangaluru - 575 003 Re-accredited by NAAC with ‘A’ Grade with a CGPA-3.62 Ranked 94 in College Category by NIRF, MHRD, Government of India Recognized by UGC as \" College with Potential for Excellence\" College with ‘STAR STATUS’ conferred by DBT, Government of India St Aloysius College belongs to a network of educational institutions administered by the members of the Society of Jesus, a religious order which runs schools, colleges and universities in more than 105 countries in the world. Established in 1880, St Aloysius College prides itself in its history of 140 years. Despite its status as a minority institution, the college has imparted high quality education to all sections of society regardless of caste, color or creed. The motto of the college, “Lucet et Ardet” which means “shine to enkindle” has inspired countless students to become men and women for and with others and thus bring light and joy into the lives of people. It is for this reason that Jesuit education has become a touchstone for evolving new paradigms in higher education.
About the Department Post Graduate Department of MATHEMATICS established in the year 2010 offers a two year PG Programme in MATHEMATICS. There is a great need of personnel in teaching profession and in pure research. The student who complete M.Sc. Mathematics will get respectable intellectual level seeking to expose scientific research orientation in Mathematics. The course is definitely at par with many of the famous institutes who offer a post graduate degree in mathematics. Objectives To generate manpower trained in Mathematics to meet the need of the industry and academia. To equip students with knowledge of principles, theories and concepts of Mathematics, which will help them to become responsible leaders in their respective fields. To train students to pursue research in the field of pure and applied Mathematics. To develop the personality of the induvidual by giving them the necessary skills. To lay the foundation for teaching for students who plan to enter the profession of teaching.
DEPARTMENT OF MATHEMATICSP G S T U D I E S & R E S E A R C H I N MS ANUPRIYA SHETTY HEAD OF THE DEPARTMENT DR D SHUBHALAKSHMI ASSISTANT PROFESSOR MS APOORVA SHETTY LECTURER MS LAISHA LAVEENA D’SOUZA LECTURER
STAFF EDITOR LAISHA LAVEENA D’SOUZA CHIEF EDITOR CO-EDITOR ASHWATH SHAHINSHA I I M S c M a t h e m a t i c s AITHAL B S I I M S c M a t h e m a t i c s THASVEER COORDINATOR PRINSTON MERINA II MSc Mathematics DSOUZA II MSc Mathematics JACOB ARISHA CHRISWIN I I M S c M a t h e m a t i c s ABDUL AZIZ I M S c M a t h e m a t i c s PREM VAS
RAKSHA BHAT
SHAHINSHA THASVEER
ZAINABATH AFREENA
ASHWATH AITHAL B S
Solve !t Find the missing number Using eight eights and addition only can ? 11 8 you make 1000 ? 56 9 2 The square of the 16 10 number 111,111,111 is amazing 7 8 What's that ? 41 9 Turn me on my side I 11 am everything ,but cut me half I am 5 nothing. Who am I ? Can you plug in either How many time you addition, subtraction, can subtract 10 from multiplication, division and 100? parenthesis among five 7s to make a target result Fathima Waseela number 50? I MSc Mathematics Find the missing number Find the missing number 19 16 If 9999 = 4, 8888 = 8, 9 17 5 9?6 1816 = 6, 1212 = 0 then 1919 =? 6 4 18 17 K A Deeksha 8 19 4 7 14 8 3 2 I MSc Mathematics
Madhuri Minz I MSc Mathematics Answers 14 888+88+8+8+8=1000 ((7×7×7)+7)/7 =50 15 12,345,678,987,654,321 2 8 ( if you turn , 0 if you cut) Once. Next time you would be subtracting 10 from 90.
Areeba Sara I MSc Mathematics Fun With Numbers 1)TRICK WITH NATURAL NUMBER 8: 1X8+1=9 12X8+2=98 123X8+3=987 1234X8+4=9876 12345X8+5=98765 123456X8+6=987654 1234567X8+7=9876543 12345678X8+8=98765432 123456789X8+9=987654321 2)Trick with natural number 9: 0x9+1=1 1x9+2=11 12x9+3=111 123x9+4=1111 1234x9+5=11111 12345x9+6=111111 123456x9+7=1111111 1234567x9+8=11111111 12345678x9+9=111111111 123456789x9+10=1111111111
3)Fun with multiplication: 9x0+8=8 9x9+7=88 9x98+6=888 9x987+5=8888 9x9876+4=88888 9x98765+3=888888 9x987654+2=8888888 9x9876543+1=88888888 9x98765432+0=888888888 4)Fun with 37: 37X3=111 37X6=222 37X9=333 37X12=444 37X15=555 37X18=666 37X21=777 37X24=888 37X27=999 5)Fun with 143: 143X7=1001 143X14=2002 143X21=3003 143X28=4004 143X35=5005 143X42=6006 143X49=7007 143X56=8008 143X63=9009
Areeba Sara I MSc Mathematics MATHEMATICS IN LIFE We must study mathematics- To Add noble qualities. To Subtract bad habits. To Multiply love and friendship. To Divide equal thoughts among friends. To Root out dreadful caste and creed. To Equate rich and poor in the society. To Eliminate social evils. To Differentiate learned from ignorant. To Integrate people of our country. To Maximize our IQ. To Minimize our ignorance. To Expand our unity with the world. To Simplify our difficulties and To be Rational and Dynamic.
Arisha Abdul Aziz II MSc Mathematics The Ramanujan Summation 1+2+3+4.....+ To begin with is summation, let’s see about a sum that is it may not be in the traditional sense, because all the series that is dealt naturally do not tend to a specific number. Thus talking about different sums namely the Cesaro summation, this summation assigns values to some infinite sums that do not converge in usual sense. “The Cesaro Sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first partial sums of the series”. We will also deal with the countable infinity concept, which is a different type of infinity that deals with the infinite set of numbers. For all those who are unfamiliar with this series which has come to be known as Ramanujan’s Summation, after a famous Indian Mathematician named Srinivasa Ramanujan. “It states that if you add up all the natural numbers, that is 1, 2, 3, and so on all the way to infinity, you will find that it is equal to .” Now let’s show the proof of this summation; To prove this we shall prove the two
equally crazy claims, i. 1-1+1-1+1-1+.......=½ ii.1-2+3-4+5-6+......=¼ Let’s begin with the claim (i.), without proving this other two are impossible. The first claim lets name it as A. i.e. A=1-1+1-1+..... Taking away A from 1, 1-A=1-(1-1+1-1+....) Simplifying the right hand side there something peculiar 1-A=1-1+1-1+1-1+.... Yes, the right hand side is A. So we have 1-A is A, Doing basic algebra we get, 1-A+A=A+A 1=2A A=½ This little beauty is Grandi’s series, named after an Italian mathematician, Philosopher and Priest Guido Grandi. Coming to the second claim, take it as B=1-2+3-4+5-6+..... Starting it similarly, instead of subtracting B from 1, we will subtract B from A. A-B=(1-1+1-1+....)-(1-2+3-4+5-..)
A-B=(1-1+1-1+...)-1+2-3+4-5+6..... Shuffling the terms we get, A-B = (1–1) + (–1+2) +(1–3) + (–1+4) + (1–5) + (–1+6)+.... A-B = 0+1–2+3–4+5+.... We see that the right hand side is same as and we know that A = 1/2 A-B = B A = 2B 1/2 = 2B 1/4 = B Booooom !!! Here’s your second claim. Now we come to the main part of the proof that’s the icing of the cake ! Considering the series C = 1+2+3+4+5+6..., Subtracting from we get, B-C = (1–2+3–4+5–6+...)-(1+2+3+4+5+6-....) Because Math is simply amazing we can re-arrange the order B-C = (1-2+3-4+5-6+...)-1-2-3-4-5-6+... B-C = (1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6)+... B-C = 0-4+0-8+0-12+... B-C = -4-8-12-... Take -4 as common out we get, B-C = -4(1+2+3+...) B-C = -4C
Where C=1+2+3... and taking B=¼ we have 1/4 = -3C 1/-12 = C or C = -1/12 Hence the Ramanujan’s Summation 1+2+3+4.....+
Aswathi K P II MSc Mathematics
Chriswin Prem Vas Suresh B I MSc Mathematics THE MATH JOKES THAT WILL MAKE “SUM” OF YOU GO LOL 1) What did the triangle say to the circle? “You’re pointless.” 2) Parallel lines have so much in common … It’s a shame they’ll never meet 3) I had an argument with a 90° angle. It turns out it was right. 4) Did you hear about the over-educated circle? It has 360°! 5) Why doesn’t anybody talk to circles? Because there’s no point. 6) Why was the obtuse triangle always upset? Because it’s never right. 7) Why did the student do multiplication problems on the floor? The teacher told him not to use tables. 8) Surgeon: Nurse, I have so many patients. Who do I work on first? Nurse: Simple, follow the order of operations.
9) Why did the girl wear glasses during math class? It improved di-vision. 10) What are ten things you can always count on? Your fingers. 11) What did the spelling book say to the math book? “I know I can count on you!” 12) What do you call two friends who love math? Algebros. 13) What did the student say about the equation she couldn’t solve? “This is derive-ing me crazy!” 2=1? Proof: Assume two variables a and b and that a = b Multiplying both sides by a, we get, a2 = ab Subtracting both sides by b2 we get, a2-b2 = ab-b2 i.e (a+b)(a-b) = b(a-b) dividing by (a-b) we get (a+b)=b since a=b the above equation becomes b+b =b i.e 2b=b which implies 2=1 The truth is we didn't actually prove that 2 = 1. Which, good news, means you can relax—we haven't shattered all that you know and love about math. Somewhere buried in that \"proof\" is a mistake. Actually, \"mistake\" isn't the right word because it wasn't an error in how we did the arithmetic manipulations, it was a
much more subtle kind of whoopsie-daisy known as a \"mathematical fallacy\". What was the fallacy in the famous faux proof we looked at? Like many other mathematical fallacies, our proof relies upon the subtle trick of dividing by zero. And we say subtle because this proof is structured in such a way that you might never even notice that division by zero is happening. Where does it occur? Take a minute and see if you can figure it out… It happened when we divided both sides by a - b in the fifth step. But, you say, that's not dividing by zero—it's dividing by a - b. That's true, but we started with the assumption that a is equal to b, which means that a - b is the same thing as zero! And while it's perfectly fine to divide both sides of an equation by the same expression, it's not fine to do that if the expression is zero. Because, as we've been taught forever, it's never OK to divide by zero! Inauspicious number 13? There is a popular maxim which states that the best way to predict future is to look into the future. History has time and again proven the number13 to be a diabolical entity, sneaking in corners, ready to cause bad omens and accidents, so much so that the fear of this number over the centuries has compounded manifold to the point of causing looks of n
horrors at the prospect of residing at the “13th floor” of a building. The fear of this number has its own personalized jargon known as “Triskaidekaphobia”. It is a combination of Greek words “tris”, meaning “three”, “kai”, meaning “and”, “deka” meaning “ten” and “phobia” meaning “fear”. Apollo 13 was launched on 11 April and it underwent explosion on 13th of April (2 thirteens). Zoroastrian tradition predicts chaos in the 13th millennium. Another example is the Columbia Space Shuttle. This one went to space on 1/16/2003. Add all the numbers and you will get the number 13. During the re-entry into Earth, it exploded. All the crew members died. Many hotels in China and America don’t have the 13th floor. After 12th, either they have 12 and half or 14th. Same goes to the number of houses too. Even Microsoft considers the number 13 as very unlucky and that is the reason why there is no version 13 of Microsoft office. Thus, the company skipped number 13 altogether. But just because some superstition is blindly acknowledged, does it make it
true? Does this number really deserve this notoriety? we beg to differ. In ancient Greece, Zeus is considered as the thirteenth and the most powerful God. This thirteenth God seems to be associated with totality, completion, and attainment. The ladder to eternity has 13 steps, on reaching the 13th step, it is assumed that your soul attains spiritual completion The number 13 is a prime number, which means it cannot be divided by any other number other than itself. Hence symbolizes qualities of incorruptible nature and purity. In one of the most powerful civilization of history, the Aztecs decided to have 13 days in a week as they considered 13 to be an extremely lucky number. Each day was ruled by one God. And the God who ruled the 13th day was associated with mystery, psyche, and magic. As we all know, 13 is the age of change or transition for every girl or boy. It is the age when children officially become teenagers. The US flag has 13 strips, that represent the union of 13 colonies to
fight the British rule, later these 13 colonies became the first thirteen states of United States of America. The Thai New Year (Songkran Day) is celebrated on the 13th April. It is considered to be a day of washing away all the bad omens, by splashing water on people, friends, and relatives. In Hindu mythology, Maha Shivratri is celebrated on 13th night of the Magha month, which is very sacred and holy night for all the Shiva devotees. In the sacred book of the Sikhs, the “Guru Grant Sahib”, the word “Waheguru” which means eternal guru appears 13 times. The list goes on. We being students of Mathematics, the first thing we learnt was that for a phenomenon to be true, there must be no exceptions. The phenomenon must be proved for every single condition. The fact that we are able to find numerous exceptions to this phenomenon is proof enough for it to be baseless superstition
Dhanya Adthale I MSc Mathematics Mathematics in Real Life CALCULUS 1)An architect uses calculus to calculate the material required to construct curved structures such as domes and arches. 2)Electrical Engineers use it to calculate the length of power cables between places that are miles apart. 3)Biologists use it to calculate the rate of growth in microbes. 4)In physics, calculus helps in finding of center of mass, center of gravity etc. And much more……………… ALGEBRA 1)Study of genes and DNA can be better explained in terms of equations. Their properties can be easily deduced with the help of X and Y chromosomes. 2)In economic, X and Y are used to determine the supply and demand of goods. 3)In computer graphics (CG), X and Y are used as coordinates to design object and their movements. And much more………………
T R I G N O M E T R Y: 1)Trignometric angles are used in the construction of marine lamps. 2)These are used to spot a location for navigation. 3)These angles are also used to determine the depth of the algae under water and sonar system. And much more……………… REFERENCE: Internet
Pallavi Acharya II MSc Mathematics UÀtÂvÀzÀ° ¥ÀÆtðvÉ UÀtÂvÀ«zÀÄ vÀªÀiÁµÉAiÀÄ ªÀiÁvÀ®è ¸ÀPÀ® ¸ËgÀªÀÄAqÀ®zÀ ¸ÀvÀåzÀ ªÀiÁAwæPÀ §UÉzÀµÀÄÖ §UÉ §UÉAiÀÄ ºÉƸÀ«µÀAiÀÄzÀ gÀ¸ÀzËvÀt ¸À« ¨sÉÆÃd£ÀPÉÌ ¸ÀASÁå¦æAiÀÄjUÉ ¨ÉÃgÉãÀÄ ¨ÉÃPÀtÚ ¸ÀAPÀ®£À, ªÀåªÀPÀ®£ÀzÀAvÀºÀ fêÀ£ÀzÀ M¼ÀUÀÄlÄÖ ZÀgÁPÀëgÀ, ¸ÀASÁå±Á¸ÀÛç, ¸À«ÄÃPÀgÀtUÀ¼À ªÉʲµÀÖöåªÀ PÀAqÀÄ ªÀÄÆgÀÄ PÀtÂÚ£À ¤Ã®PÀAoÀ£À gÀÆ¥À«zÀÄ Groups, Rings, Modules UÀ¼À£ÉƼÀUÉÆAqÀ gÀ¸ÀUÀÄ®è«zÀÄ 1D, 2D J£ÀÄßvÀ nD UÀ¼À £ÀAlÄ EzÀ ºÀÄqÀÄPÀÄvÀ PÁtĪÀÅzÀÄ £ÀªÀ¨sÁgÀvÀ«AzÀÄ ©AzÀÄ ©AzÀÄUÀ¼À eÉÆÃr¸ÀÄvÀ UÀtÂvÀzÀ°è£À §tÚ§tÚzÀ ¯ÉÆÃPÀ ªÀÄ.¸Á.C, ®.¸Á.C, wæPÉÆãÀ«Äw C£ÀÄ¥ÁvÀUÀ¼À ¥ÁPÀ Calculus, Algebra, Complex Analysis UÀ¼À ¸ÀzÀÄÝ ªÉÄzÀĽ£À £ÀgÀ§½îAiÀÄ° ºÉƸÀ C¯ÉAiÀÄ zÀAqÀÄ ¸ÀªÀÄ, ¨É¸À, ¸ÀA¨sÀªÀ¤ÃAiÀÄvÉAiÀÄ ªÉÄgÀÄUÀÄ EzÀ PÀ°AiÀÄÄvÀ £À¤ßà fêÀ£À ¸ÉƧUÀÄ - Pallavi Ach
Saubia Momin I MSc Mathematics DYSCALCULIA “Just because you can’t count, doesn’t mean you don’t count.” -Paul Moorcraft We often come across people who say that math is a difficult subject. Although it is not unusual for children to have a grueling time with math, if they have problems with numbers or score low in math but do well in other subjects, they might have a math learning disability called “Dyscalculia”. Dyscalculia is a math learning disorder that makes mathematical computation and reasoning difficult, in spite of average or greater intelligence, adequate education and proper motivation. It is a brain- related condition and is sometimes informally known as “math dyslexia”, though this can be misleading as dyslexia is a different condition from dyscalculia. Samantha Abeel, who has dyscalculia and is a 43 year old prize-winning author, once said, “I am twenty-five years old and I can’t tell the time. I struggle with dialing phone numbers, counting money, balancing my checkbook, tipping at restaurants, following directions, understanding distances, and applying
accomplished journalist and novelist, said “arithmetic was like being asked to speak in an unknown foreign language.” Symptoms of Dyscalculia: A young child with dyscalculia may: Need to use visual aids ( like fingers) to help count Lose track when counting Struggle to connect numerical symbols (10) with their corresponding words (ten) Have trouble in placing things in order and recognizing patterns Have a hard time in recognizing numbers Be delayed in learning to count And as math becomes a major part of the school day, kids with dyscalculia tend to: Have a hard time gauging how long it will take to fulfill a task Have serious difficulty learning basic math functions like addition and subtraction, multiplication tables and more Be unable to grasp the concepts behind word problems and other
non-numerical math calculations and have difficulty keeping at grade-level in math Have trouble with math homework assignments and tests Struggle to process visual-spatial ideas like graphs and charts The impact of math class doesn’t stop when math class ends. This disorder can also affect people outside of school. People with dyscalculia also: Struggle with remembering numbers such as zip codes and phone numbers. Get easily frustrated by games that require number strategies or counting and consistent score keeping Have difficulty reading clocks and telling time Struggle with money matters such as counting bills, making change, splitting a check, calculating a tip, or estimating the cost of something Have a hard time judging the length of distances and how long it will take to get from one location to another Struggle to remember directions and have a hard time telling left from right
Treatment of Dyscalculia: Although there are no medications that treat dyscalculia, there are a lot of ways to help people with dyscalculia to develop math skills. People with dyscalculia can benefit from specialized instructions from experts and the right tools. There are accomodations that can make it easier for a person with dyscalculia to work at the same level as his/her peers. Experts: Educational specialists or a math tutor, especially one who has experience working with students who learn differently, can help people with dyscalculia learn to approach math problems in a more effective manner. Tutoring will also allow this person to practice his/her math skills in a slower, less exacting environment. The right tools: A calculator he/she knows how to use Pencils (for erasing!) Graph paper to help him/her keep numbers and columns straight Math apps and games that allow him/her to practice essential skills in an enjoyable way Pre-set phone alarms and reminders to help him/her keep track of time
Accommodations: Extra time on tests and access to calculators, if possible, during class and tests The option to record lectures and access to the teacher’s notes In-school tutoring or homework assistance A quiet space to work Dyscalculia’s impact on day-to-day activities such as making correct change, playing board games or even reading clocks accurately can cause people to feel self-conscious and embarrassed. Helping people with dyscalculia understand their learning disorder can give them the tools they need to manage their dyscalculia, both emotionally and academically and make them understand that having a learning disability is nothing to be ashamed of. References: Butterworth, B. (2019). Low numeracy: from brain to education-Maths a Problem?. Retrieved from h t t p s : / / m a t h s n o p ro b l e m .c o m / w p - c o n t e n t / u p l o a d s / 2 0 1 9/ 0 5/ B r i a n - B u t t e r wo rth-PDF-compressed.pdf Hamilton-Newman, R. (2020). Dyscalculia and ADHD. Retrieved from https://www.dyscalculia.org/ WebMD. (2019). What Is Dyscalculia? What Should I Do if My Child Has It?. Retrieved from https://www.webmd.com/add-adhd/childhood-adhd/dyscalculia-facts Frye, D. (2020). How to Treat the Symptoms of Dyscalculia. Retrieved from https://www.additudemag.com/dyscalculia-treatment-accommodations-for -school-and-work/?src=embed_link Jacobson, R. (n.d.). How to Spot Dyscalculia. Retrieved from https://childmind.org/article/how-to-spot-dyscalculia/ Jacobson, R. (n.d.). How to Help Kids With Dyscalculia. Retrieved from https://childmind.org/article/how-to-help-kids-dyscalculia/
Shreya I MSc Mathematics Easy Maths Normally whenever we have to solve simple problems like addition, subtraction, multiplication and division we take the help of the calculator. Even though we know to solve them by ourself we need calculator, just to save our time. So here I am with some of the easy tricks to solve such problems like multiplication of so many numbers, and finding squares. 1.Finding Squares How will you find the square of a number which will end with 5 say 55,125,555 etc. Let me take 65 to find the square of this number. First write the square of 5 i.e 25 then multiply another number (6) with its succeeding number. (In this case we have 6 so succeeding number of 6 is 7, 6x7=42) Hence we got the answer 65² = 4,225 Let me take one more example, take 335 We know square of 5 =25 and succeeding number of 33 is 34 so 33x34=1122 So 335²=1,12,225.As I said this trick is only to find the squares of the number which ends with number 5.
Now we will see how to find the squares of numbers which is not ending with 5. Take an example say 78 First write squares of 7&8 togather i.e 4964 Next step is multiplying those two numbers with two i.e 7x8x2 = 112 Now add 4964 &112 we get 4964+112=6084 ( Remember that when you add these numbers the second number would be written how we write while do multiplication i.e leaving the space below the unit position) One more example, take 94 Squares of 9 and 4 = 8116 and 9x4x2 =72, now add them 8116 + 72 8836 2. Multiplication First let me take the numbers greater than 100 Say 105 and 107 now how to multiply them in a easy way? First subtract 100 from 105 and 107 105 -100 = 5 107-100 = 7 now add one of the the cross digits i.e either 105+7 or 107+5 you will get the same answer i.e 112 Now multiply 7&5 7x5=35 And we have the answer 11,235
105=5 107=7 11235 (add cross numbers and multiply 7&5) And if the number is less than 100 we have to subtract that number from 100 Take an example say 97 and 94 100-97=3 and 100-94=6 Now follow the same procedure as above but subtract the cross numbers (97-6=91 or 94-3=91) and 3x6=18 i.e 97= 3 94=6 9118 So these are the some of the easy tricks to solve easy multiplications using which we can avoid calculator and solve by ourself Reference: Internet
Shruthi Someshwar II MSc Mathematics Mersenne Prime Numbers Prime numbers are not just the numbers that can only be divided by themselves and one. They are the precious secrets that mathematicians have been trying to unravel since Euclid proved that there are infinitely many prime numbers. Largest prime number known to date, 24,862,048 digits long Mersenne prime number (2 -1) was discovered by the Great Internet Mersenne Prime Search (GIMPS) in December 2018. GIMPS is a largely distributed computing project in which volunteers run software to search for Mersenne Primes. They are named after Marin Mersenne , a French Minim friar who studied them in the early 17th century. Eg.3, 7, 31, 127, 8191,….. The Prime values of Mp appears to grow increasingly sparse as p increases.
For example, eight of the first 11 primes p, gives rise to a Mersenne Prime Mp while Mp is prime for only 43 of the first two million prime numbers. The method used for testing primality of a Mersenne Number is the Lucas-Lehmer primality Test. This Test makes use of the sequence (Lucas-Lehmer Sequence) whose first term is 4 and the following terms are got by squaring the previous number and subtracting two from it. Then the sequence will be 4, 14, 194, 37634, 1416317954, 2005956546822746114,…… To check whether Mp =2-1 is prime against the above sequence , find whether the number in the (p-1)th position of the above mentioned sequence is a multiple of Mp or not. If it is a multiple of Mp , then Mp=2 -1 is definitely a prime otherwise it is not a prime. Eg.2³-1=7, In this case p=3 and the number in (p-1)th position, i.e., 2nd position is 14, which is a multiple of 7. Thus 2³-1=7 is a prime. Lucas proved that 2 -1 is a prime by finding the 126th number in the sequence and also that 2 -1 is not a prime without even actually finding the factors of the numbers, which is a tedious job to do normally if we are finding whether a number is prime or not. This method helps us to find whether a number is prime or not without even bothering to find the factors of the number.
82589933 Imagine! To check whether is a prime or not, we need to find the 82589932th number in the Lucas-Lehmer Sequence and see if it a multiple of 82589933 o r e q u i v a l e n t l y c h e c k i f i t l e av e a r e m a i n d e r o n c e d i v i d e d by 82589933 . Now, as we proceed we keep track of the remainders of number in the sequence 82589933 (mod ) instead of the actual numbers in the Lucas-Lehmer sequence. In general, for 2-1 we get a new sequence with (p-1) terms, by squaring the remainders of the number in Lucas–lehmer Sequnce and then subtracting 2 from it. Eg:1 The sequence obtained in the case p=5, (25-1=31) is 4, 14, 8, 0 here, 4th term is zero which means that 4th term in the Lucas-Lehmar sequence is a multiple of 31. Hence 31=25-1 is a prime. Eg:2 The sequence obtained in the case p=7, (27-1=127) is 4, 14, 67, 42, 111, 0 here, 6th term is zero which means that 6th term in the Lucas-Lehmer sequence is a multiple of 127. Hence 127=27-1 is a prime. Lucas-Lehmer Test is used by GIMPS to locate large primes. This search has been successful in locating large primes known to date.
One of the most widely used application of prime number is the RSA encryption system which allows secures transmission of information such as credit card number. Disarranging data online requires the user to come up with two large prime numbers and multiply them. Larger the prime number safer the encryption. The difficulty of factorizing the resultant product is the only thing between hacker and one’s credit card number. Currently we need primes that are a few hundred digits long to keep our secrets safe. So the new Mersenne prime number is not going to be used for encryption any time soon. So why are people so much interested in finding these enormous primes? Chris K. Caldwell, a mathematician at University of Tennesse at Martin explains “Mersennes, in a way are kind of like a large diamond”. When he went to Washington, he says, he took his kids to see the Hope Diamond. That’s the 45-carat diamond that sits in a special case in the National Museum of Natural History, usually with crowds around it. “Nobody there looking at the Hope diamond ever asks, ‘why did they bother to dig it up?’ Or ‘what is it good for?’ even though it really isn’t good for much more than to just hang there and people look at it “, Caldwell says, “And in many ways the Mersennes play that same role- that they really are the jewels of Number Theory”.
REFERENCES: https://en.wikipedia.org/wiki/Mersenne_pr ime https://www.npr.org/templates/story/story .php?storyId=102876903 https://theconversation.com/why-do-we-n eed-to-know-about-prime-numbers-with- millions-of-digits-89878#:~:text=The%20la rger%20the%20numbers%2C%20the,and%20s o%20even%20more%20important.
Shwetha Hebbar II MSc Mathematics Happy Ending Problem The puzzle was given a memorable nickname, the “happy ending” problem for reason that had nothing to do with math. It was named by Paul Erdos as it led to the marriage of George Szekeres and Esther Klein. At the time, Klein was 23 years old and living in her hometown of Budapest, Hungary. One day she brought a puzzle to two of her friends, Paul Erdos and George Szekeres : “Given five points, and assuming no three fall exactly on a line, prove that it is always possible to form a convex quadrilateral”. Convex Quadrilateral : A four-sided shape that’s never indented (meaning that, as you travel around it, you make either all left turns or all right turns). Convex polygon Non - convex polygon
By 1935 Erdos and szekeres had solved this problem for shapes with three, four and five sides. If we put five points in the plane in general position and no three are in the straight line then we must get a subset of four points that forms 4-sided complex polygon. But if we take 4 points, it need not give 4-sided complex polygon. Example : Therefore it takes five points to guarantee a convex quadrilateral. They also knew it took three points to guarantee you could construct a convex triangle and nine points to guarantee a convex pentagon. In the same paper in which they presented these solutions, Erdos and Szekeres proposed an exact formula for the number of points it would take to guarantee a convex polygon of any number of sides: 2(n-2)+1, where n is the number of sides. Very recently people showed that it takes 17 points to guarantee a 6-sided convex polygon and 16 is not enough. Next case is going to be on computation. Reference: https://www.quantamagzine.org/mathematcian-dipro ves-group-algebra-unit-conjecture-20210412/ By KEVIN HARTNETT May 30, 2017
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