Finding Odd One

The odd number in arithmetic plays an important role as well in establishing math skills, for future purposes as well. Concerning that, the odd and even number up to 100, create a base for arithmetic foundation. In addition, discussion and significance of odd and even numbers, their properties and techniques give a clear aspect of having odd numbers in arithmetic.

Finding the odd number

The evidence of odd numbers has also left traces in the studies of Aristotle and Plato as well as Euclid’s Elements too. In the Pythagorean system, it was considered as the 1, 3, 5, 7, 9 are more important than even numbers, where 1 is considered as a deity, 3 is for trinity, 9 is trinity into trinity and so on.

Focusing on odd and even numbers

The odd and even numbers are both positive integers. The numbers which are considered as “even” can be divided by two, whereas odd numbers cannot be divided by two. For instance, if 1, 3, 5, 7, 9 are considered as odd, then 2, 4, 6, 8 are considered as even numbers, in respect to single-digit integers. In case of more than single-digit integers, the last or unit digit should be 1, 3, 5, 7, 9 of a number to consider it odd. The negative odd number are {…, -9, -7, -5, -3, -1}. To form an odd number n=2k+1 [k= integer] is considered. Like 3= (2*1+1) or (2*2+2) =5. Moreover, x² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1. In this case, 2k² + 2k is also an integer, so,x² = 2L + 1, (L = 2k² + 2k is an integer). As it proves x² is odd. The generating function for odd numbers is-

x (1 + x)/ (x-1)² = x + 3x² + 5x³ + 7x⁴ + …

Both positive and negative even numbers are divided by two, in this case, the congruence n=0 (mod 2), as it holds an even n. To form an even number n=2k [k= integer], for example 2= (2*1) or (2*2) = 4. The generating function for an even number is-

2x/ (x-1)²= 2x + 4x² + 6x³ + 8x⁴ + …

In this regard, the oddness in a number is called parity, which measures a number’s even or odd characteristics. The parity for odd numbers is 1 and for even numbers is 2. In this context, the product of an odd number and an even number is always obtained as an even number which can be written as-

2k (2l +1) = 2 [k (2l +1)]

[where it is divisible by 2, therefore it is considered as an even]

Discussion of odd numbers, especially from 1 to 100

There are a total of 50 odd numbers concerning even numbers, in 1-100. The smallest and largest single-digit odd numbers are 1 and 9. The smallest and largest two-digit odd numbers are 11 & 99. In this case, the formula of the sum of the odd numbers is Sn is for the sum of the first nth odd number.

Sn = Σni=1 (2i – 1)

Sn = n/2 [ 2a + (n – 1) d] OR n/2 [a + l]

[where, a=1, d=2 and l= 2n-1]

In that case,

Sn = n/2 [a + l]

= n/2 [1 + (2n – 1)]

= n/2 [2n] = n²

Therefore, Sn = n²

For Example, there are 50 odd numbers in 1-100. In that case, the sum of all consecutive odd numbers will be- S50 or square of 50, which is 2500. The average of all odd numbers is 50.

Definition of odd numbers

Previously, it was mentioned that the number which cannot be divided into 2, is considered as the odd number. For the division purpose, the divisor of Pn, where P= prime number 1, and the sum of divisors of Pn is counted as even numbers only if n is an odd number.

Properties and significance of odd numbers

This states as the addition of the first n odd numbers is always a square number, which can be formulated by-

Σnk=1 (2k – 1) =n²

In this regard, the difference of nth and (n-1)st square number is considered as the nth odd number, which can be formulated as-

n² – (n-1)² = 2n – 1

More property of odd numbers: