<abstract xmlns="http://www.w3.org/1999/xhtml">Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye <xref ref-type="bibr" rid="j_forma-2018-0008_ref_007_w2aab3b7ab1b6b1ab1ab7Aa">[7]</xref>. Although the use of parity allowed to discover irrational numbers <xref ref-type="bibr" rid="j_forma-2018-0008_ref_006_w2aab3b7ab1b6b1ab1ab6Aa">[6]</xref>, there is a common opinion that this property is “not rich enough to become the main content focus of any particular research” <xref ref-type="bibr" rid="j_forma-2018-0008_ref_009_w2aab3b7ab1b6b1ab1ab9Aa">[9]</xref>.On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes.The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in <xref ref-type="bibr" rid="j_forma-2018-0008_ref_005_w2aab3b7ab1b6b1ab1ab5Aa">[5]</xref>, <xref ref-type="bibr" rid="j_forma-2018-0008_ref_004_w2aab3b7ab1b6b1ab1ab4Aa">[4]</xref>, and <xref ref-type="bibr" rid="j_forma-2018-0008_ref_003_w2aab3b7ab1b6b1ab1ab3Aa">[3]</xref> branches, formalized in Mizar <xref ref-type="bibr" rid="j_forma-2018-0008_ref_001_w2aab3b7ab1b6b1ab1ab1Aa">[1]</xref>, <xref ref-type="bibr" rid="j_forma-2018-0008_ref_002_w2aab3b7ab1b6b1ab1ab2Aa">[2]</xref>.</abstract>

Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although the use of parity allowed to discover irrational numbers [6], there is a common opinion that this property is “not rich enough to become the main content focus of any particular research” [9].On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes.The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in [5], [4], and [3] branches, formalized in Mizar [1], [2].

Parity as a Property of Integers

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Parity as a Property of Integers"}

Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although...

Parity as a Property of Integers

Published Online: Dec 24, 2018

Page range: 91 - 100

Accepted: Jun 29, 2018

DOI: https://doi.org/10.2478/forma-2018-0008

Keywords
divisibility, binary representation

© 2018 Rafał Ziobro, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Parity as a Property of Integers

Published Online: Dec 24, 2018

Page range: 91 - 100

Accepted: Jun 29, 2018

DOI: https://doi.org/10.2478/forma-2018-0008

Keywordsdivisibility, binary representation

© 2018 Rafał Ziobro, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
divisibility, binary representation