# Number System Essay

2773 Words 12 Pages
Introduction I. Number Systems in Mathematics:
A Number system (or system of numeration) is a writing system for expressing numbers, that is a mathematical notation for representing number of a given set, using graphemes or symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
Ideally, a number system will: * Represent a useful set of numbers (e.g. all integers, or rational numbers) * Give every number represented a unique representation (or at least a standard representation) * Reflect the algebraic and arithmetic structure of the numbers.
For example, the
These are visually expressed by the digits 0 and 1. Every number expressed in the binary system is a combination of these two digits. b. History
The Indian scholar Pingala (circa 5th–2nd centuries BC) developed mathematical concepts for describing prosody, and in so doing presented the first known description of a binary numeral system. He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code. In the 11th century, scholar and philosopher Shao Yong developed a method for arranging the hexagrams which corresponds to the sequence 0 to 63, as represented in binary, with yin as 0, yang as 1 and the least significant bit on top.
Similar sets of binary combinations have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been widely applied in sub-Saharan Africa.
In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random
For each triplet, the octal conversion is the same as converting to a decimal number: 001 000 100 100 110 111 1 0 4 4 6 7 Therefore, 10001001001101112 = 1044678

II. Conversion of Decimal

Converting Decimal to binary, octal or hexadecimal, we have to follow the same procedure as follows: * Divide the decimal number by the desired target radix (2, 8 or 16). * Append the remainder as the next most significant digit. * Repeat until the decimal number has reached zero.

a. Decimal to Binary :
Let’s take a decimal number, say 179210. Now we can convert this number to binary using the above procedure. Decimal Number | Operation | Quotient | Remainder | 1792 | ÷ 2 = | 896 | 0 | 896 | ÷ 2 = | 448 | 0 | 448 | ÷ 2 = | 224 | 0 | 224 | ÷ 2 = | 112 | 0 | 112 | ÷ 2 = | 56 | 0 | 56 | ÷ 2 = | 28 | 0 | 28 | ÷ 2 = | 14 | 0 | 14 | ÷ 2 = | 7 | 0 | 7 | ÷ 2 = | 3 | 1 | 3 | ÷ 2 = | 1 | 1 | 1 | ÷ 2 = | 0 | 1 | 0 | done. | | | | Therefore, 179210 = 111000000002 ( interesting thing is that, arrangement of binary digit is started from last to fast

• ## Pascal's Triangle Essay

A literature of Pascal’s Triangle emerged in 1068 discovered by Hindu mathematician Bhattotpala (c.1068) who recorded the first 16 rows of the triangle (Wilson, 2013, p170). Meanwhile, in Persia, Al-Karaji (953-1029) also found the binomial theorem according to Pascal’s Triangle as well as several theorems related to it (Coolidge, 1949, p151). Although the original work from Al-Karaji had lost, the later Persian mathematician Khayyam (1048-1122) referred Al-Karaji’s work about Pascal’s Triangle and uses it to find the nth roots according to the binomial expansion. Also around 11th century, Chinese mathematician Jia Xian (1010-1070) used Pascal’s Triangle to extract square roots and cube roots – more details will be mentioned later in this paper. He wrote down his discovery in the book Shi Shuo Suan Shu [The key to Mathematics].…

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• ## Triangle Eequality Analysis

In particular, for ∥u − uh∥ L∞(Ω) = O(hp+1) a necessary condition is that all the kth partial derivatives at xi ∈ T satisfy (41) @ (u − uh)(xi) = O(hp+1−k); | | = k; 0 ≤ k ≤ p: In other words, we have a simultaneous approximation result. Here all smoothness refers to interior smoothness and {xi} is any collection of points, one from each element. Proof. Let Qh be a quasi-uniform subdivision on Ω in R2, and let u ∈ Wp+1 ∞ (Ω) and uI ∈ Php be such that ∥u − uI∥ L∞(Ω) ≤ Chp+1|u| Wp+1 ∞ (Ω): Let uh ∈ Qhp be given and to simplify the presentation, we will use shorthand notations: let | | = k and since we will treat one kth derivative at a time, there is no ambiguity in setting u(k) h = @ uh, u(k) I = @ uI , and u(k) = @ u. At a…

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• ## Alan Turing: The Father Of The Enigma Machine

To generate a public number, pick two prime numbers, such as 11 and 13. Calculate for m, in which m = (11 - 1)(13 - 1) = 10 x 12 = 120. Find a number not divisible by m, such as 7, the first public number. The second public number is derived from multiplying the two initial numbers, 11 x 13 = 143. The private number is formed using modular arithmetic with the first public number to find d, where 7d = 1 mod 120.…

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• ## Comparison Of Zermelo's Axiom Of Choice

However, we can measure the size using Borel set that we define its measure as a length. From this length, we can simply generalize the set to become Lebesgue measure Borel measures on R, B(R) is the smallest σ-algebra that contains the open intervals of R as stated in Wikipedia (2016). The Borel measure on real number, R is the choice of Borel measure which assigns μ ((a,b])= b-a for every half-open interval (a,b]. Besides, Weir (1974) mentioned that a Borel measure is any measure μ defined on the σ-algebra of Borel…

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• ## Buried Treasure: Case Study Quiz

We will continue to input numbers into b and d to create the equation to match x2 – 8x – 20 = 0. With this being said we must also keep in mind that a and c will be multiplied to equal positive 1 and that the middle will create the sum of ad and bc to create a -8. We will also have to remember that b and d when multiplied will have to be the same…

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• ## Lesson 6.7 Error Effecting Codes In Communication

8 information bits and 1 parity bit. Parity bit is an extra bit that is attached to the data bits. Parity bit is chosen so that the number of 1 bits in the code word is even or odd. Parity checking is a means of checking if the communication of a sequence of bits has been correctly received. The two types of most commonly used parity checking are simple parity and two-dimensional parity.…

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• ## Recall The Standard Statement Of The Chinese Remainder Theorem

Note that y −1 i mod ni exist since yi and ni are coprime by construction. Notice that when j 6= r, yjzj ≡ 0 mod nr and when j = r, yjzj ≡ 1 mod nr. Then x = X k i=1 aiyizi (7) satisfies all of the congruences in the statement of the theorem. 6 For example, suppose we want to find x such that x ≡ 1 mod 5 x ≡ 3 mod 7 x ≡ 2 mod 9. We then have that N = (5)(7)(9) = 315 and y1 = 315/5 = 63, y2 = 315/7 = 45, and y3 = 315/9 = 35.…

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• ## Percent Error For The Binary Problem 2

5. A single 10-bit, left justified conversion of 3.75V is complete in ATD0. Assume and i. Name the register(s) where the result of the conversion is found ii. What are the values in each of the 16-bits of this register after conversion is complete?…

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• ## Case Study: Descriptive Statistics

Use the same steps except select the Options button and change the Confidence level: to 99. MATH 221 FINAL…

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• ## Facial Recognition Essay

After this, the variance is found for all classes by using this mean value. Sigma2 = 1 / (n-k) * sum((x-mu)2). There are two additional steps required before making a final prediction using the LDA method, this paper will only look at the final function. Dk(x) = x * muk/〖sigma〗^2 -(〖muk〗^2/〖2sigma〗^2 +ln⁡(Plk)) where “Dk(x) is the discriminate function for class k given input x, the muk, sigma2 and Plk are all estimated from your data. (Browniee).…

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