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 …show more content…
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 …show more content…
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
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 …show more content…
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 …show more content…
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