Varieties of Numbers In Mathematics

There are Varieties of Numbers in Mathematics, we will talk about them in this article. Such as Prefect number, Hardy-Ramanujan Number and so on.

As much as the ‘words’ have revolutionized mankind, the ‘numbers’ have played equally significant role. Numbers have an edge over words in a sense, both scientifically and psychologically speaking, that numbers are believed to determine everything that lies at the present and in the future. Numbers are fun too; you can add every numbers up to infinity and still get -1/12 as the final result (check for Ramanujan Summation over internet). So, here’s a list some unique numbers just for the sake of fun and information.

Varieties of Numbers In Mathematics:

1.  Singularity:

The singularity. The very first. You and no other. ‘1’ is a concept before it is a number. Give it a thousand names, but after all, when you have to start the journey to infinity, you always start with one single idea – the idea of ‘just one’.

2.     Zero:

 As much as the existence of something buzzes our mind, the concept of nothingness is equally perplexing. As a matter of fact, ancient Greek mathematicians and scholars were in dilemma if ‘nil’ could be counted as a number. They asked themselves “how can ‘nothing’ be something?”. The use of zero as a number can be found as back as in Mesopotamian and Egyptian civilization. But, it was in the Sanskrit language where ‘sunya’ was defined more clearly, referring to the concept of void. The documented mathematical use of zero dates back to 628 ADS by Brahmagupta. Later, the idea of using zero as a number spread to China and Middle East, and then to the whole world.

3.      Fibonacci Number:

Fibonacci number is a quite popular sequence of number when it comes to unique numbers. This sequence comprises the numbers starting from 1 and adding previous two numbers, except for the second ‘one’ which is only repeated.

1       1 2 3 5 8 13 21 34 55 89 144 …

 4.      Perfect number:

A perfect number is the sum of all of its proper positive divisors, excluding itself.

It is believed to be first documented by famous Swiss mathematician and physicist Leonhard Euler.

Examples:

6 = 1 + 2 + 3

28 = 1+ 2+ 3+ 4 +5 + 6+ 7

496 = 1 + 2 + 3 + 4 + 5+ … + 29 + 30 + 31

8128 = 1 + 2+ 3+ 4 + 5+ … + 125 + 126 + 127

33550336 = 1 +2 +3+ 4 +5 + … + 8189 + 8190+ 8191

5.     Hardy–Ramanujan number:

One time British mathematician G. H. Hardy went to visit his friend  Srinivasa Ramanujan. Hardy told Ramanujan that the taxi number, which was 1729, seemed dull to him. Ramanujan told Hardy that it was quite an interesting number. It was the smallest number expressible in as the sum of two cubes of positive numbers in two different way.

1729 = 1³ + 12³ = 9³ + 10³

But, if negative cubes are also to include, the smallest such number reduces to 91, which surprisingly is one of the factors of 1729.

91 = 6³ + (-5)³ = 3³ + 4³

Now, here’s what’s more interesting:

1 + 7 + 2 + 9 = 19

19*91 = 1729

6.     Kaprekar’s constant:

Kaprekar’s constant gets its name from Indian mathematician D. R. Kaprekar. It gets its popularity from its unique feature which is explained in few steps:

·      Take any four-digit number, excluding 1111, 2222, 3333, … 9999

·      Arrange the number once in ascending order and again in descending order.

·           Subtract the smaller number from the larger number.

·      Repeat these steps for few times. In at most 7 iterations, you’ll get the number 6174.

Do it for yourself for the proof! Comment and comment below!

7.     Transcendental Number:

In mathematics Transcendental numbers are those real or complex numbers which are not algebraic- that is, it is not a root of a non-zero polynomial equation with rational coefficient. The best examples of transcendental number are π and e.

Okay, a game at last. What’s the largest number you can make using only three digits?

999? No, it’s not.

The largest number you can make using 3 digits is 9^9^9.

Hope you liked this article “Varieties of Numbers in Mathematics” please do subscribe comment and share 😊.

Source: scientificmind.com.np

Zero Odd or Even?

                                                                                                     By: Suman Kandel

Have you ever wondered Zero is Odd or Even? In the following article we will go through basic definition of odd, even and finally will talk is Zero Odd or Even?

So, keep reading…

The number zero is beautiful and at the same time amazing and surprising number in modern mathematics. It has such properties and some wonders inside it that we all may want to know.

History

The word ‘zero’ came into existence as:
Śūnya →          ṣifr →         zefiro →      zero        → zero
(Sanskrit)     (Arabic)       (Italian)     (French)       (English)

A symbol for zero, a large dot, is used in the Bakhshali Manuscript, which contains problems of arithmetic, algebra and geometry. In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different dates, AD 224-383, AD 680-779, AD 885-993.

Even and Odd

Before defining even and odd numbers, let’s make us familiar to few mathematical terms:
Set of integers: The set of all positive and negative numbers along with zero are integers. It is denoted by ℤ and defined as, ℤ = {…,−3,−2,−1,0,1,2,3,…}
From above, we see that, the set of integers is an extended set of whole numbers where we duplicate each whole number by putting a minus (-) sign before it.

Odd numbers:

You already know what an odd number means. We all have heard that the number which is divisible by 1 and itself is an odd number and it is not divisible by 2. Examples are: 1, 3, 9, 217, etc.
That is an elementary definition. Let us try a mathematical one.
A number, p, is said to be odd if it can be expressed as;
p = 2q+1
where q ∈ ℤ.
Pick a number, let it be 1.
Now,
1 = 2*0 + 1
Where 0 ∈ ℤ
For 217,
217 = 2*108 +1
For 999997,
999997 = 2*499998 + 1
And so on.

Even numbers:

We are also familiar with primary level definition of even numbers. We were told that any number which is divisible by 2 is even. That’s true.
In mathematical form, it would be;
A number, p, is said to be even number if it can be expressed as;
p = 2q,
where q∈ ℤ.
Let’s try some numbers.
4 = 2*2, 2∈ ℤ.
Thus, 4 is even.
100 = 2*50, 50∈ ℤ.
Thus, 100 is even.
Now, take a big one, 25686.
25686 = 2* 12843, 12843∈ ℤ.
Thus, 25686 is even.

Zero Odd or Even?

To Zero again.

Now, try to make 0 fit for the above definitions. Which one does it satisfy?

The odd? NO.

It satisfies the definition of an even number.
0 = 2*0
Now, recall from the definition of set of integers that, 0∈ ℤ.
Thus, from the definition of odd and even, we clearly see that 0 is even.

Some more properties:

(1) An even number lies between two odd numbers. For example: 4 lies between 3 and 5, 100 lies between 99 and 101. Here 3,5,99 and 101 are odd.
What for zero?

From the number line, it can be known that 0 lies between -1 and 1. -1 and 1 are odd. Thus, we can say that 0 should be even.

(2) The purity of evens.
Pick out an even, 26.
Now, 26/2 = 13.
13 is an odd number.
For 82,
82/2 = 41
41 is an odd number.
Here, the even numbers, 26 and 82 are said to be singly even.
There are many singly even, like, 34, 142, 1002, etc.
Let’s take an even, say, 4.
Now, 4 /2 = 2
2 is an even.
2/2 = 1.
4 can be divided by 2, two times.
Also, for 12,
12/2 = 6
6 is an even.
6/2 = 3
12 can be divided by 2, two times.
Here, 4 and 12 can be referred being as doubly even.
Other double evens are, 24, 264, 1024, etc.
Again, for 0.
0/2 = 0, 0 is even.
0/2 = 0, 0 is even
0/2 = 0, 0 is even
0/2 = 0, 0 is even
0/2 = 0, 0 is even
0/2 = 0, 0 is even
And so on.
Thus we observe that 0 is the purest even.

(3) ‘0’ is neither positive nor a negative number. The whole concept of positive and negative number is defined on the basis of 0. Numbers greater than 0 are positive while numbers less than 0 are negative. But 0 equals 0, which makes us absurd to define zero as positive or negative.

Source: wikipedia.org

Prime Numbers

What are Primes? Are they really important part of mathematics? Okay before getting into this let us talk a little about chemistry, what is the basis of chemistry? Without what chemistry would be incomplete? Obviously, it is the elements, which makes whole chemistry. In similar manner Prime numbers are Building blocks of Mathematics. Let us talk in detail about this;

The concept of primes is that “A number greater than one is said to be prime if it is divisible by one and number itself.”  A few examples of prime numbers are 2,3,5,7,11 and so on. The history of prime shows earliest surviving records of explicit study of prime number comes from Ancient Greek Mathematics.

Among the small numbers primes are very common. Of the numbers 2 to 20, the numbers are 2,3,5,7,11,13,17,19 are prime a total of eight out of nineteen. The remaining numbers are all composite; that is, they are not prime, since each number can be evenly divided by some smaller numbers (apart from 1). As we look at larger and larger numbers the primes appear to be thin out. While there are 5 primes below 10, there are only 24 below 100 and just 168 below 1000. If we see the average rate at which prime appears, they have an average rate of 0.5 below 10, 0.24 below 100 and just 0.168 below 1000. The table of prime and average up to 1 million is:

N	10	100	1000	10000	100000	100000
Average	0.5	0.24	0.168	0.123	0.096	0.078

The farther we go the smaller the average becomes. From here we can arise three questions;

·      Does this thinning continue?

·      Or do we reach a point where it reverses and we find a lot of primes?

·      Or do we reach a point where we do not find primes at all?

One of these questions was answered by ancient Greek mathematician Euclid around 300 B.C, he proved that the primes continue forever and there are infinitely many of them. Later on, another mathematician showed that the average rate of primes is approximately equal to 1/ln(N). That is, we can say he noticed pattern of primes.

Characteristics of Primes:

A few characteristics of primes are as follows:

1.     Prime numbers have only two factors 1 and itself.

2.     The only even number that is prime is 2.

3.     There are infinitely many prime numbers.

4.     Every number can be uniquely expressed as product of prime numbers.

5.     The prime numbers can not be formed by product of two smaller natural number that are both smaller than it.

6.     No even number greater than 2 can be prime as it can be written as the product of 2*n/2.

7.     Except 2 all primes are odd.

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