{"id":6756,"date":"2014-03-03T19:28:54","date_gmt":"2014-03-03T19:28:54","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=6756"},"modified":"2014-12-02T08:25:32","modified_gmt":"2014-12-02T08:25:32","slug":"math-introduction-to-modular-clock-arithmetic","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2014\/03\/03\/math-introduction-to-modular-clock-arithmetic\/","title":{"rendered":"Math Introduction to Modular (Clock) Arithmetic"},"content":{"rendered":"

Overview<\/strong><\/p>\n

In modular arithmetic, numbers cycle rather than increase when operations are performed.\u00a0 Think of a clock with numbers equal to the cycle.\u00a0 Extensive applications include UPC numbers, the electronic numbers on the bottom of bank checks, and public key codes.<\/p>\n

How Does Modular Addition Work?<\/strong><\/p>\n

The most common real world example of modular arithmetic is that of a clock. A common clock goes from the numbers 1 to 12.\u00a0 When two numbers are added, the sum is no more than 12, because another cycle is made around the clock. The number line is a circle around the \u201cclock\u201d, rather than a straight line.\u00a0\u00a0If 10 + 6 is added in a mod 12 number system, the pointer would start at 10 and go 6 spaces, past the 12 again to end at 4.\u00a0 Imagine a clock that only went from 1 to 8.\u00a0 In that case, the cycle would be much shorter, so that adding 7 and 3 would cycle past 8 to 2.\u00a0 Another way to say this in equation form is 10 + 6 \u2261 4 mod 12.\u00a0 The symbol for equivalence looks like an equals sign with 3 lines instead of 2, and the word \u201cclock\u201d is dropped is dropped from the description of modular arithmetic.<\/p>\n

What about Multiplication?<\/strong><\/p>\n

Multiplication is repeated addition, with more cycles in modular arithmetic.\u00a0 Suppose the problem was (3 \u2219 7) + (2 \u2219 1) + 3 in mod 10.\u00a0 Now, 21 can be broken down to 10 + 10 + 1.\u00a0 In mod 10, 10 is just another cycle.\u00a0 In other words, 10 \u2261 0 mod 10.\u00a0 So 21 + 2 + 3 \u2261 1 + 2 + 3 mod 10 (or 6) mod 10. No matter how large the number, it can always be simplified by eliminating the modulus.\u00a0 A number such as 101000<\/sup>mod 10 is equivalent to 0.<\/p>\n

What Are UPC Numbers?<\/strong><\/p>\n

Modular arithmetic has several common applications is everyday life.\u00a0 One of the applications is the Universal Product Code (UPC) numbers, added as barcodes to scan everyday products.\u00a0 Modular arithmetic can be used to check that the UPC number is correct and in the correct order.\u00a0 Also, the nine-digit electronic number found at the bottom of bank checks can be verified by using modular arithmetic.<\/p>\n

How Is Modular Arithmetic Used to Decipher and Encode Data?<\/strong><\/p>\n

Public key codes use both modular arithmetic and prime numbers to decode messages from encrypted bank data to secret messages used in espionage.\u00a0 Only the intended receiver knows the numbers essential to the formula deciphering the numerical message.<\/p>\n

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Overview In modular arithmetic, numbers cycle rather than increase when operations are performed.\u00a0 Think of a clock with numbers equal to the cycle.\u00a0 Extensive applications include UPC numbers, the electronic numbers on the bottom of bank checks, and public key codes. How Does Modular Addition Work? The most common real world example of modular arithmetic […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[2615,2614,2616],"class_list":["post-6756","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-equivalence","tag-modular-clock-arithmetic","tag-upc-codes"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/6756","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=6756"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/6756\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=6756"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=6756"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=6756"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}