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What Are Perfect Numbers In Maths?

By Yann, published on 01/06/2019 We Love Prof - IN > Academia > Maths > What Are The Perfect Numbers And How To Use Them?

If you are studying maths at school, then you are on your way to becoming a mathematician. Congratulations you are also on your way to becoming fluent in the language of numbers in the style of trigonometry (trig.), algebra, geometry Pythagorean theorem, Linear equations, the long number Pi, the unusual Euler’s e, Equivalent fractions, the specific number i, also the golden ratio, probability and other mathematical concepts in your grade level math curriculum!

Maths is a global technical language, you see. So, any achievements you have in the subject will be recognised in other countries. While there may not be exactly the same qualifications in all parts of the world, you can rest assured that there will be an equivalent and that your level of mathematics understanding will be recognised no matter where you are studying or working in the world!

Don’t you find it fascinating to think that Maths is the only ‘language’ that is universal, regardless of nationality, culture, religion or gender? With Maths, answers never vary no matter where you are in the world. Pi is always going to be roughly 3.14 no matter what your coordinates are!

It may seem like a long time ago that you were learning about the maths basics that this knowledge is built upon. Since preschool you have been learning about maths, addition and subtraction, multiplication and division all laid the path for understanding your current mathematics education.

That is because math is everywhere,

  • Being told to share (dividing) your sweets with your siblings
  • calculating (percentages) how much tax to pay from your salary
  • how much change you should get back after buying something at the corner store (adding and subtracting)
  • To how flowers grow (the golden ratio)
  • To how your watch keeps the time (pi)
  • To know what is in your bank account (the number 0)

The list goes on because mathematics is the common core standard of our society. From basic math to more complex equations, we use them every day, which makes all of us mathematicians, even if very bad ones. But our ability to grasp math questions and solve problems gives us the opportunity to gain mastery in it.

If your maths practice is important to you, then you would have already spent time learning all about fractions and algebraic math concepts. But what about sequences? By now you know maths is not just a number system made up of whole numbers and clever counting. How about scientific notation, algebraic symbols, special numbers and sequences like perfect numbers. Perfect numbers are the mysterious number patterns, that have an interesting mathematical model.

Let explore perfect numbers today and see what we can find out!

math is everywhere in our world It may seem like a long time ago that you were learning about the maths basics. Photo Source: Unsplash

What Are Perfect Numbers?

A perfect number is a rare number. To date, only 51 of them have been discovered. There are only three perfect numbers less than 1000: 6, 28, and 496. In fact, there is 1 perfect number under 10, 1 under 100, 1 under 1000 and I expect that the pattern continues in this way to infinity.

So what exactly is a perfect number?

Dictionary.com defines the adjective ‘perfect’ as:

  1. “having all the required or desirable elements, qualities, or characteristics; as good as it is possible to be.
  2. absolute; complete”

With these two definitions in mind, we can assume that perfect numbers must have some kind of special feature that makes them more complete than any other digit. Numbers are endless so the perfect number cannot be the number with the highest value, nor can it be the smallest round number.

In actual fact, when all factors of a number, excluding the number itself, add up to that number, it is known as a perfect number.

Factors are the numbers lower than that number which can divide into it evenly. For example:

  • 6 is the first perfect number
  • 6 can be divided by 4 factors: 1, 2, 3 and 6
  • However, to find out if it is a perfect number, we only want to add up numbers less than the number itself: 1, 2, 3
  • Then we add them together 1+2+3=6
  • So this is a perfect number

Is 8 a perfect number? Let’s see

  • 8 can divide by 1, 2, 4 and 8
  • Eliminating the 8
  • Adding the factors together: 1+2+4=7
  • Whoops! No 8 is not a perfect number. Try it for yourself!

The First Eight Perfect Numbers Are:

  • 8,
  • 28,
  • 496,
  • 8128,
  • 33 550 336,
  • 8 589 869 056,
  • 137 438 691 328,
  • 2 305 843 008 139 952 128.

Do any of these numbers have any special significance to you? Do you think you could have a stroke of Maths genius in you and discover yet another perfect number that has been missed thus far by experts in the field?

If you are looking for an activity to keep your children quiet and focused, then why not set them the task of trying to work out the first two perfect numbers for themselves? The exercise combines a range of mathematical procedures to carry out which, when broken down simply, aren’t complicated at all for a secondary school level pupil.

To see the next 43, you can go to Wikipedia

Unproven Math Facts In Relation To Perfect Numbers.

There still remain may unproven concepts and rules in relation to perfect numbers.

Here are a few:

  • All known perfect numbers end in 6 or 28, but again that may not be true, for perfect numbers after the 51 that have been found.
  • It is said that perfect numbers are infinite but since only 51 have been found this cannot be proven.
  • Euclid’s perfect numbers and all following perfect numbers are all even. For the moment, we do not know if there are odd perfect numbers. All examples are even numbers, but that does not mean that there are no odd perfect numbers. Although research is progressing, none has yet been able to affirm or refute this hypothesis. Carl Pomerance has published a theory to try to prove the nonexistence of odd perfect numbers

We want to find the answers to these mind-boggling questions about perfect numbers, don’t you?! How can the numbers be perfect and wholly complete when the theory is as vague as it currently is!

Mathematicians And The History Of Perfect Numbers

Perfect numbers are related to prime numbers. Book IX of the Elements of Euclid states that if the Mersenne number 2n – 1 is prime, then 2n-1 (2n – 1) is a perfect number (Wikipedia). The combination of the results of Euclid and Euler (Swiss mathematician Leonhard Euler is also known for the Euler number e) gives a complete characterization of even perfect numbers.

  • Ancient Greek mathematicians had only discovered the first 4 of the perfect numbers.
  • The sixth and seventh perfect numbers were found by Cataldi in the 16th century and the eighth in 1772 by Euler.
  • by the early 1950s, we knew 12 perfect numbers

Since then the research has accelerated rapidly thanks to more and more sophisticated techniques and computer use.

learn mathematical concepts in your grade level math curriculum If you are studying maths at school, then you are on your way to becoming a mathematician. Photo Source: Unsplash

What Are Prime Numbers Again?

A quick recap for those who have forgotten all about prime numbers.

“A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Numbers that have more than two factors are called composite numbers.” – Whatis.com

As Fact Monster suggests, “a prime number can be divided, without a remainder, only by itself and by 1. For example, 17 can be divided only by 17 and by 1.”

What Are The Perfect Numbers For?

If prime numbers are recognized as being the very foundation of arithmetic by many mathematicians, then perfect numbers are easy to understand and have no particular use. They are not used to solve Trigonometric functions, Systems of equations, Multivariable calculus, Stochastic processes, Computational functions or Exponential function.

They are not useful for solving basic math problems like Calculus or Multiplication problems of general math problems. Perfect numbers are a purely mathematical concept, they are not useful to the fields of Economics, science and engineering for example.

However, perfect numbers were previously considered superior to all other numbers, and some saw a mystical role in them. They are one of the mysteries of mathematics, and the search for new perfect numbers still fascinates many mathematicians today. I think that perfect numbers can make a cool math game like Sudoku, “find the perfect number”.  Maybe their purpose can be to make free math addition games ap. Now that would be mystical!

Other Types Of Perfect Numbers

  1. triperfect numbers – A triperfect number is always even. If there is an odd one, it is greater than 10 50. The sum of the divisors of the triperfect number, including itself, is equal to three times the number.
  2. We only know 6 triperfect numbers:
    1. 120,
    2. 672,
    3. 523 776,
    4. 459 818 240,
    5. 1 476 304 896,
    6. 51 001 180 160.
  3. The multiperfect numbers – The sum of the divisors of a multiperfect number, including itself, is ktimes the number. Mathematicians have discovered more than 500 multiperfect numbers up to order 8, and they think they know all the 3 to 7 multiperfect numbers.
  4. Hyperperfect numbers – A k-hyperperfect number is a perfect number. A Hyperperfect number is such that: n = 1 + k ( o ( n ) – n – 1

Discrete Spoiler! Maybe I am biased but knowing the tri-perfect, multi-perfect, and hyperperfect numbers will not help you much in your math classroom. Not with high school math, a college equation, a university degree or life in general.

As a serious learner concentrate the majority of your study time to reinforce your math skills and knowledge on more useful and interactive math. Like fractions, complex numbers, logarithms or reasoning in geometry.

But if you continue in mathematics to the highest level, who knows, maybe the prime numbers will become a topic of research.

Which Professions Might Make Use Of Perfect Numbers?

By studying Maths, you’ll develop a range of skills that are sought after by employers and that will also help you throughout your day to day life. Maths encourages logical thinking, decision-making and problem-solving so any employer is bound to love a good set of maths skills displayed by a candidate.

What’s more, with the world very quickly turning towards technical solutions and automation, we are constantly interacting with machines and devices, so your knowledge of perfect numbers and other maths theories are a huge bonus for jobs whereby you need to input mathematical data to teach a computer specific sequences or the like.

I know what you’re probably thinking – “Now that computers do everything for us, who will care that I know which numbers are prime numbers or perfect numbers?” As we’ve touched upon, there’s always somebody who needs to teach the machines to perform in the first place plus, with us becoming so reliant on computers and tech, it’s vital to have these highly skilled people who can step in if technology fails us.

Careers involving Maths are usually permanent, which is promising if you are keen on finding a stable career, not to mention the fact that Maths graduates earn on average a much higher starting salary than other industries. This is normally a reflection of the perceived difficulty of the subject, declaring maths graduates as intellectuals.

Along with a very good introductory pay package, Maths jobs usually offer the potential to rise up the salary scale quite quickly, for specialisation in Maths is more unique than some other areas.

Jobs which are directly related to your degree, as reported by the University of Cambridge, include:

  • Actuarial analyst
  • Actuary
  • Chartered accountant
  • Chartered certified accountant
  • Data analyst
  • Investment analyst
  • Research scientist (maths)
  • Secondary school teacher
  • Statistician
  • Systems developer

They say that jobs where your degree would be useful include:

  • Civil Service fast streamer
  • Financial manager
  • Financial trader
  • Insurance underwriter
  • Meteorologist
  • Operational researcher
  • Quantity surveyor
  • Software tester

Want to know what post grads have done?

It is very common for graduates to combine work and further study after receiving their degree. Almost a third are said to be training to become teachers.

That said, the top jobs for mathematics graduates are finance and investment analyst and adviser, chartered or certified accountant, programmer, software developer and actuary.

Fun Methods To Study Maths

If you are one of the people who never understood math, or who the math teacher always gave a low homework score. Then the thought of maths might put you off, but mathematics can be fun.

Learn math with fun tools like jigsaw puzzles, quizzes, math videos and perhaps worksheets. Will keep you engaged and also help you to relax which will make learning math easier.

If you relax it will make learning math easier. The thought of maths might put you off, but mathematics can be fun. Photo Source: Unsplash

Ideas To Have Fun While Learning:

  • Search youtube and Vimeo for inspiring math videos
  • Go to free online math websites and look for cool math tips and solutions.
  • Play online math games via apps. Sudoku is a good one
  • Print math worksheets from online math websites.
  • Attend free community workshops or math study groups. You can ask your teacher to arrange one if there isn’t one set up already. Believe me, you are not the only student who needs extra help with maths.
  • Hire a personal math tutor if you can afford it or hire one as a group and ask them to focus the lessons plans around enjoying mathematics.
  • Be conscious of how math touches your every day When you receive some money when you have to pay a bill when you buy an ice cream or buy some new clothes. Pay attention to how important math is and allow this to fuel your passion for it.

Have fun with your curriculum, be experimental, master your homework, erase the decimal places, find the symmetry, challenge the square roots, place value on the negative numbers, find new solutions and think above of your grade level, rise above the common core standard logic, don’t limit your mind.

Ways To Learn About Perfect Numbers And Other Maths Topics

Teach Yourself Maths

There are a number of institutions whereby you can study Maths at a higher level. In the UK, the University of Oxford and the University of Cambridge top the charts for this subject.

However, if you missed your chance at doing a maths degree, then know that it isn’t totally impossible to teach yourself a bit of mathematics.

There are numerous textbooks and other maths resources that can be easily found on the internet to keep you on track with your mathematics education. Follow video tutorials on Youtube with teachers such as Mr Barton Maths or make use of BBC Skillswise maths guides. You can even find year-long maths courses that take no commitment other sitting down and completing them to the best of your ability!

OpenLearn from The Open University offers hundreds of courses of all levels and durations – and you can start whenever you like! Work at your own pace, track your progress and receive a certificate of participation at the end.

E-learning is an excellent way to optimise your learning journey and you can even do it for free!

Learn Maths With A Teacher

Math’s isn’t an easy subject and so, if you come across an area that you just can’t work through, it is very hard to work through that cloud yourself. A teacher can help you by explaining those concepts you find more difficult so that you don’t get stuck and ht a brick wall (because much of maths depends on you understanding one concept to be able to move forward with another).

There are several options for someone looking for a Maths tutor:

  • Look for classes in local schools, colleges and universities: Many educational institutions are used to hosting math courses during the evenings, so it’s worth looking around and seeing what’s available near you.
  • Go to an adult education organisation: These companies cater exclusively to adults who want to improve their knowledge in a certain area and are quite common in the UK.
  • Take night classes: Adult math education doesn’t just take place in school, but in town halls, village halls and conference centres too. Math lessons are usually delivered in the evenings to allow people time to make their way from their place of work.
  • Get a private maths tutor: There are many different ways to find yourself a one to one tutor depending on whether you want to start online tuition for maths or face-to-face, and what level you wish to achieve. You can look for tutors advertising in local shops and newspapers or use an online platform such as Superprof to find a registered teacher near you.

Superprof

The advantage of having a private maths tutor is that you have the full, undivided attention of your teacher, and they can tailor the work to your needs.

Superprof, a leading platform for tutors and students to connect and form working partnerships, offers a user-friendly website on which you can instantly locate tutors offering maths tutoring services in your area, as well as those who are able to offer remote online tuition.

There are close to 60,000 tutors listed online who can help you electronically or via video call with your maths concerns. With prices starting from just £5 per hour, you can find someone who can meet your needs and get you moving forward with your math lessons. Some are mathematicians, some are qualified teachers, whilst others are individuals who are talented with numbers and want to pass on their knowledge and skills.

So, if you are still finding it hard getting to grips with the perfect numbers, or you want to educate yourself enough to challenge the experts and find the 52nd perfect number before anyone else does, then go ahead and get learning!

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