Hi there. 

The previous post looked at even and odd numbers in a way that is easy and simple for young students. As students get older and learn more mathematics, the mathematics starts to get more structured and tougher with rules, math symbols, math notation, letters and abstract concepts. With that being said, here is a more technical look at even and odd numbers.

<center><img src="https://i.ytimg.com/vi/lcCCTzkQmWs/hqdefault.jpg" /></center>
<center><a href="https://i.ytimg.com/vi/lcCCTzkQmWs/hqdefault.jpg">Featured Image Source</a></center>

### A More Technical/Mathematical Version Of Even & Odd Numbers
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Given a natural number `k` where k = 1, 2, 3, 4, 5, 6, ..., an even number can take on the form as:

<center>http://quicklatex.com/cache3/a9/ql_f7b24ba980935b789ded08a6342389a9_l3.png</center>

If `k = 1`, I have the first even number as 2, if `k = 2` I then get `2 x 2 = 4` and so on. 

For the odd number case with k = 1, 2, 3, 4, 5, 6, ..., the form would be:

<center>http://quicklatex.com/cache3/e1/ql_8a848e69c90e09f62babb455848952e1_l3.png</center>

An alternate form for odd numbers would be:

<center>http://quicklatex.com/cache3/0d/ql_cd9f03695b2e61ad17051446105e560d_l3.png</center>

for k = 0, 1, 2, 3, ... (The variable k now starts at 0 versus 1.) An odd number can be thought of as an even number plus (or minus) 1.

### Some Properties
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<img src="http://quicklatex.com/cache3/cc/ql_37399511781dc793e3cbe1822c1b1bcc_l3.png" />
<a href="http://quicklatex.com/cache3/cc/ql_37399511781dc793e3cbe1822c1b1bcc_l3.png">Image Source</a>


**Proof Of 1)**

Let `k` and `m` be natural numbers from 1, 2, 3, 4 and so on. Define the first even number as http://quicklatex.com/cache3/fa/ql_004c18ddefa12e6fcb5b98b236e13bfa_l3.png and the second even number as http://quicklatex.com/cache3/9a/ql_b255c595ae0a8cb5896f153d7659589a_l3.png . Since two is a common factor the sum can be shown as follows.

<center><img src="http://quicklatex.com/cache3/ac/ql_e615cac1644204e46399eed6291015ac_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/ac/ql_e615cac1644204e46399eed6291015ac_l3.png">QuickLaTeX Image Source</a></center>

Two times any whole number is an even number. 

**Proof Of 2)**

Let `a` and `b` be natural numbers from 1, 2, 3, 4 and so on. Define the first number as a even number in the form of http://quicklatex.com/cache3/48/ql_7d540c427655c6706bb679f71ff22648_l3.png and the second number as an odd number as http://quicklatex.com/cache3/19/ql_9b37ee102bd783ad8f5542948b445919_l3.png . We show that the sum of an even and odd number is results in an odd sum. 

<center><img src="http://quicklatex.com/cache3/8d/ql_f22010908d5e6a9ef5abc301e646df8d_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/8d/ql_f22010908d5e6a9ef5abc301e646df8d_l3.png">QuickLaTeX Image Source</a></center>

Notice how the first two terms are factored and the minus one is unchanged. 


**Proof Of 4)**

Let `p` and `q` be natural numbers from 1, 2, 3, 4 and so on. Define the first even number as http://quicklatex.com/cache3/af/ql_41c7b12566532179900d6b34c37daaaf_l3.png and the second even number as http://quicklatex.com/cache3/58/ql_9be4ef21493f9c8a52fc5b6887083958_l3.png . The product of http://quicklatex.com/cache3/6e/ql_478fae55e8756d9dc97149d99f598b6e_l3.png and http://quicklatex.com/cache3/00/ql_ea03afd9b020071cb0fb450217321e00_l3.png would result in an even number.

<center><img src="http://quicklatex.com/cache3/00/ql_ba82223339ced33f7983dfb45aa10f00_l3.png" /></center>
<center><a href="http://quicklatex.com/cache3/00/ql_ba82223339ced33f7983dfb45aa10f00_l3.png">Image Source</a></center>

It does not matter what (natural) numbers `p` and `q` are as the product is multiplied by an even number of 4.

<center><img src="https://i.pinimg.com/736x/75/6e/96/756e9612182b99f4983a29626488700b.jpg" /></center>
<center><a href="https://i.pinimg.com/736x/75/6e/96/756e9612182b99f4983a29626488700b.jpg">Image Source</a></center>

### Notes
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* The rest of the proofs can be done as exercises.
* This type of topic was one of the first topics I was exposed to while learning math proofs.
* As you can see, the math you learn early on can be generalized. It becomes more theoretical and abstract.
* The math text in the form of images was done in LaTeX with QuickLaTeX.com
* I consider the number zero as neither even nor odd. A zero ending digit in a two digit number or higher is even though.