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Number Systems

Published in: Mathematics
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  • Prantik S

    • Kolkata
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When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number. A value of each digit in a number can be determined using; The digit The position of the digit in the number The base of the number system (where base is defined as the total number of digits available in the number system).

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    Number Systems Topics Major Computer Components Bits, Bytes, and Words The Decimal Number System The Binary Number System Converting from Binary to Decimal Converting from Decimal to Binary The Hexadecimal Number System 1
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    Bits, Bytes, and Words A bit is a single binary digit (a 1 or O). A byte is 8 bits A word is 32 bits or 4 bytes Long word = 8 bytes = 64 bits Quad word = 16 bytes = 128 bits Programming languages use these standard number of bits when organizing data storage and access. What do you call 4 bits? (hint: it is a small byte) 2
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    Number Systems The on and off states of the capacitors in RAM can be thought of as the values 1 and 0, respectively. Therefore, thinking about how information is stored in RAM requires knowledge of the binary (base 2) number system. Let's review the decimal (base 10) number system first. 3
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    The Decimal Number System The decimal number system is a positional number system. Example: 5621 103 102 101 100 1 x 100 2 x 101 6 x 102 5 x 103 1 20 600 5000
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    The Decimal Number System The decimal number system is also known as base 10. The values of the positions are calculated by taking 10 to some power. Why is the base 10 for decimal numbers? 0 Because we use 10 digits, the digits 0 through 9.
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    The Binary Number System The binary number system is also known as base 2. The values of the positions are calculated by taking 2 to some power. Why is the base 2 for binary numbers? 0 Because we use 2 digits, the digits 0 and
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    The Binary Number System The binary number system is also a positional numbering system. Instead of using ten digits, 0 - 9, the binary system uses only two digits, 0 and 1. Example of a binary number and the values of the positions: 1001 101 26 25 24 23 22 21 20
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    Converting from Binary to Decimal 101 26 25 24 23 22 21 20 20 1 21 = 2 25 = 32 22 = 4 23 = 8 26 = 64 I x 20 ox 21 I X 22 I x 23 o x 24 ox 25 I x 26 4 = 64 8
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    Converting From Decimal to Binary Make a list of the binary place values up to the number being converted. Perform successive divisions by 2, placing the remainder of 0 or 1 in each of the positions from right to left. Continue until the quotient is zero. Example: 4210 25 24 32 16 1 23 22 21 20 21 9
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    Working with Large Numbers 0101000010100111 Humans can't work well with binary numbers; there are too many digits to deal with. Memory addresses and other data can be quite large. Therefore, we sometimes use the hexadecimal number system. 10
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    The Hexadecimal Number System The hexadecimal number system is also known as base 16. The values of the positions are calculated by taking 16 to some power. Why is the base 16 for hexadecimal numbers ? Because we use 16 symbols, the digits 0 and I and the letters A through F. 11
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    The Hexadecimal Number System Binary Decimal Hexadecimal Binary Decimal Hexadecimal 1010 1011 IIOO 1101 100 1110 101 110 1000 1001 12 15 12
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    The Hexadecimal Number System Example of a hexadecimal number and the values of the positions: 3 C 8B 051 166 165 164 163 162 161 160 13
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    Example of Equivalent Numbers Binary: 1010000101001112 Decimal: 2064710 o Hexadecimal: 50A716 o Notice how the number of digits gets smaller as the base increases.

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