A Tutorial on Data Representation

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In the computer world " b inary dig it " is often shortened to the word " bit ". So, there are only two ways we can have a binary digit "0" and "1"or "On" and "Off" And without the leading 0s we have the first 16 binary numbers:.

Because we take all the previous possible values and match them with a "0" and a "1" like above. Or to put 5 bit binary numbers another way, it could show a number up to 1,,, note: There is an old Indian legend about a King who was challenged to a game of chess by a visiting 5 bit binary numbers. The King asked "what is the prize if you win? The Sage said he would simply like some grains of rice: The King was surprised by this humble request. On the first square: By the 30th square you can see it is already a lot of rice!

A billion grains of rice is about 25 tonnes 1, grains is about 25g Notice that the Total of any square is 1 5 bit binary numbers than the Grains on the next square Example: So the total of all squares is a formula: So, the power of binary doubling is nothing to be taken lightly billion 5 bit binary numbers is not light! By the way, in the legend the Sage reveals himself to be Lord Krishna and tells the King that he doesn't have to pay the debt at once, but can pay him over time, just serve rice to pilgrims every day until the debt is paid off.

Lastly, let us look at the special relationship between Binary and Hexadecimal. There are 16 Hexadecimal digits, and we already know that 4 binary digits have 16 possible values. Well, this is exactly how they relate to each other:. So, when people use computers which prefer binary numbersit is a lot easier to use the single hexadecimal digit rather than 4 binary digits.

For example, the binary number "" is "9B4" in hexadecimal. I know which I would prefer to write! Hide Ads About Ads. So, to fill all 64 squares in a chess board would need:

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The base 2 method of counting in which only the digits 0 and 1 are used. In this base , the number equals. This base is used in computers, since all numbers can be simply represented as a string of electrically pulsed ons and offs. In computer parlance, one binary digit is called a bit , two digits are called a crumb , four digits are called a nibble , and eight digits are called a byte.

An integer may be represented in binary in the Wolfram Language using the command BaseForm [ n , 2], and the first digits of a real number may be obtained in binary using RealDigits [ x , 2, d ]. Finally, a list of binary digits can be converted to a decimal rational number or integer using FromDigits [ l , 2]. The illustration above shows the binary numbers from 0 to 63 represented graphically Wolfram , p.

A negative number is most commonly represented in binary using the complement of the positive number , so would be written as the complement of , or This allows addition to be carried out with the usual carrying and the leftmost digit discarded, so gives. The number of times that a given binary number is divisible by 2 is given by the position of the first counting from the right. For example, is divisible by 2 twice, and is divisible by 2 zero times.

The number of times that 1, 2, OEIS A , which is the binary carry sequence. Real numbers can also be represented using binary notation by interpreting digits past the "decimal" point as negative powers of two, so the binary digits would represent the number. The sequence of binary digits for the integers , 1, Unfortunately, the storage of binary numbers in computers is not entirely standardized. Because computers store information in 8- bit bytes where a bit is a single binary digit , depending on the "word size" of the machine, numbers requiring more than 8 bits must be stored in multiple bytes.

However, a number represented as byte1 byte2 byte3 byte4 in a VAX would be read and interpreted as byte4 byte3 byte2 byte1 on a Sun. The situation is even worse for floating-point real numbers, which are represented in binary as a mantissa and characteristic , and worse still for long 8-byte reals! Binary multiplication of single bit numbers 0 or 1 is equivalent to the AND operation, as can be seen in the following multiplication table. Consider the cumulative digit sum of all binary numbers up to 1, 2, The first few terms are then 1, 2, 4, 5, 7, 9, 12, 13, 15, 17, 20, 22, This sequence in monotonic increasing left figure , but if the main asymptotic term is removed, a sequence of humped curves right figure; Trott , p.

A Foundation for Computer Science, 2nd ed. Endlessly Repeated Geometric Figures. Princeton University Press, pp. The Art of Scientific Computing, 2nd ed. Cambridge University Press, pp. The Mathematica GuideBook for Programming. A New Kind of Science. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: Thu Mar 29 Contact the MathWorld Team.

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