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What Are Perfect Numbers In Maths?

By Yann, published on 27/02/2019 We Love Prof > 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 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.

It may seem like a long time ago that you were learning about the maths basics that this knowledge is built upon. Since kindergarten 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 on 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.

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
  • Whoop! 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.

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

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 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.

Fun Ways 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.


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