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Ternary computers were a fad that faded, though not quickly. In the s there were several more projects to build ternary logic gates and memory cells, and to assemble these units into larger components such as adders. In Gideon Frieder and his colleagues at the State University of New York at Buffalo designed a complete base-3 machine they called ternac, and created a software emulator of it.

Since then the idea of ternary computing has had occasional revivals, but you're not going to find a ternary minitower in stock at CompUSA. Why did base 3 fail to catch on? One easy guess is that reliable three-state devices just didn't exist or were too hard to develop.

And once binary technology became established, the tremendous investment in methods for fabricating binary chips would have overwhelmed any small theoretical advantage of other bases. Furthermore, it's only a hypothesis that such an advantage exists.

Everything hinges on the assumption that rw is a proper measure of hardware complexity, or in other words that the incremental cost of increasing the radix is the same as the incremental cost of increasing the number of digits. Suppose you are creating one of those dreadful telephone menu systems—Press 1 to be inconvenienced, Press 2 to be condescended to, and so forth. If there are many choices, what is the best way to organize them?

Should you build a deep hierarchy with lots of little menus that each offer just a few options? Or is it better to flatten the structure into a few long menus? In this situation a reasonable goal is to minimize the number of options that the wretched caller must listen to before finally reaching his or her destination.

The problem is analogous to that of representing an integer in positional notation: The number of items per menu corresponds to the radix r, and the number of menus is analogous to the width w. The average number of choices to be endured is minimized when there are three items per menu. Although numbers are the same in all bases, some properties of numbers show through most clearly in certain representations. For example, you can see at a glance whether a binary number is even or odd: Just look at the last digit.

Ternary also distinguishes between even and odd, but the signal is subtler: A ternary numeral represents an even number if the numeral has an even number of 1s. The reason is easy to see when you count powers of 3, which are invariably odd. More than 20 years ago, Paul Erdo"s and Ronald L.

Graham published a conjecture about the ternary representation of powers of 2. They observed that 2 2 and 2 8 can be written in ternary without any 2s the ternary numerals are 11 and respectively. But every other positive power of 2 seems to have at least one 2 in its ternary expansion; in other words, no other power of 2 is a simple sum of powers of 3.

The digits of ternary numerals can also help illuminate a peculiar mathematical object called the Cantor set, or Cantor's dust. To construct this set, draw a line segment and erase the middle third; then turn to each of the resulting shorter segments and remove the middle third of those also, and continue in the same way. After infinitely many middle thirds have been erased, does anything remain? One way to answer this question is to label the points of the original line as ternary numbers between 0 and 0.

The repeating ternary fraction 0. Given this labeling, the first middle third to be erased consists of those points with coordinates between 0. Likewise the second round of erasures eliminates all points with a 1 in the second position after the radix point. The pattern continues, and the limiting set consists of points that have no 1s anywhere in their ternary representation. In the end, almost all the points have been wiped out, and yet an infinity of points remain.

No two points are connected by a continuous line, but every point has neighbors arbitrarily close at hand. It's hard to form a mental image of such an infinitely perforated object, but the ternary description is straightforward. Knuth in The Art of Computer Programming , "is the balanced ternary notation. They are "balanced" because they are arranged symmetrically about zero.

For notational convenience the negative digits are usually written with a vinculum, or overbar, instead of a prefixed minus sign, thus:. As an example, the decimal number 19 is written 1 01 in balanced ternary, and this numeral is interpreted as follows:.

Every number, both positive and negative, can be represented in this scheme, and each number has only one such representation. The balanced ternary counting sequence begins: 0, 1, 1 , 10, 11, 1 , 1 0, 1 1. Going in the opposite direction, the first few negative numbers are , 1, 0, , 11, 10, 1. Note that negative values are easy to recognize because the leading trit is always negative.

The idea of balanced number systems has quite a tangled history. Both the Setun machine and the Frieder emulator were based on balanced ternary, and so was Grosch's proposal for the Whirlwind project. In , Claude E. Shannon published an account of symmetrical signed-digit systems, including ternary and other bases. But none of these 20th-century inventors was the first. Twenty years earlier, John Leslie's remarkable Philosophy of Arithmetic had set forth methods of calculating in any base with either signed or unsigned digits.

Leslie in turn was anticipated a century earlier by John Colson's brief essay on "negativo-affirmative arithmetick. There is even a suggestion that signed-digit arithmetic was already implicit in the Hindu Vedas, which would make the idea very old indeed! What makes balanced ternary so pretty? It is a notation in which everything seems easy.

Positive and negative numbers are united in one system, without the bother of separate sign bits. Arithmetic is nearly as simple as it is with binary numbers; in particular, the multiplication table is trivial. Addition and subtraction are essentially the same operation: Just negate one number and then add. Negation itself is also effortless: Change every into a 1, and vice versa.

Rounding is mere truncation: Setting the least-significant trits to 0 automatically rounds to the closest power of 3. The best-known application of balanced ternary notation is in mathematical puzzles that have to do with weighing. Given a two-pan balance, you are asked to weigh a coin known to have some integral weight between 1 gram and 40 grams. How many measuring weights do you need?

A hasty answer would be six weights of 1, 2, 4, 8, 16 and 32 grams. If the coin must go in one pan and all the measuring weights in the other, you can't do better than such a powers-of-2 solution. If the weights can go in either pan, however, there's a ternary trick that works with just four weights: 1, 3, 9 and 27 grams. For instance, a coin of 35 grams— in signed ternary—will balance on the scale when weights of 27 grams and 9 grams are placed in the pan opposite the coin and a weight of 1 gram lies in the same pan as the coin.

Every coin up to 40 grams can be weighed in this way. So can all helium balloons weighing no less than —40 grams. James Allwright, who maintains a Web site promoting balanced ternary notation, suggests a monetary system based on the same principle. If both a merchant and a customer have just one bill or coin in each power-of-3 denomination, they can make exact change for any transaction. Some weeks ago, rooting around in files of old clippings and correspondence, I made a discovery of astonishing obviousness and triviality.

What I found had nothing to do with the content of the files; it was about their arrangement in the drawer. Imagine a fastidious office worker—a Martha Stewart of filing—who insists that no file folder lurk in the shadow of another. The protruding tabs on the folders must be arranged so that adjacent folders always have tabs in different positions.

Achieving this staggered arrangement is easy if you're setting up a new file, but it gets messy when folders are added or deleted at random. A drawer filled with "half-cut" folders, which have just two tab positions, might initially alternate left-right-left-right. The pattern is spoiled, however, as soon as you insert a folder in the middle of the drawer. No matter which type of folder you choose and no matter where you put it except at the very ends of the sequence , every such insertion generates a conflict.

Removing a folder has the same effect. An insertion or deletion creates either a 00 or a 11—a flaw much like a dislocation in a crystal. Although in principle the flaw could be repaired—either by introducing a second flaw of the opposite polarity or by flipping all the bits between the site of the flaw and the end of the sequence—even the most maniacally tidy record-keeper is unlikely to adopt such practices in a real file drawer.

In my own files I use third-cut rather than half-cut folders; the tabs appear in three positions, left , middle and right. Nevertheless, I had long thought—or rather I had assumed without bothering to think—that a similar analysis would apply, and that I couldn't be sure of avoiding conflicts between adjacent folders unless I was willing to shift files to new folders after every insertion.

Then came my Epiphany of the File Cabinet a few weeks ago: Suddenly I understood that going from half-cut to third-cut folders makes all the difference. It's easy to see why; just interpret the drawerful of third-cut folders as a sequence of ternary digits. At any position in any such sequence, you can always insert a new digit that differs from both of its neighbors. Base 3 is the smallest base that has this property.

Take the decimal number and divide it by the value of the largest place value. Find the remainder instead of any fractional values. The whole number portion is is the most significant digit of your number in the target system. Take the remainder and divide it by the next largest place value. Keep doing this until you run out of place values. A hexadecimal digit can be represented by four binary digits.

Use this to your advantage when converting between hexadecimal and binary. Starting with the most significant digit, convert each digit to a four digit binary number and concatenate the results. I usually convert the hexadecimal value to decimal, then the decimal to binary. You could also memorize the binary values for each hexadecimal digit. For convenience purposes, break the decimal number into groups of four, starting with the least significant digit.

When using pen and paper, I just draw a line between the groups of four digits. This works the same way as hexadecimal to binary. Instead of four binary digits representing a hexadecimal digit, you can represent an octal digit with three binary digits. Binary to Octal As with binary to hexadecimal, divide the binary number into sections, starting with the least significant digit. Instead of four digit sections, use three digit sections. The calculators in most desktop environments can convert common number systems for you.

This article shows you where it is in Windows. Just enter the number, then click the radio buttons towards the left next to convert between bases. Wikipedia has an interesting page on positional notation. Purple Math has a series of guides covering number bases that might be helpful. Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website.

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Find the remainder instead of any fractional values. The whole number portion is is the most significant digit of your number in the target system. Take the remainder and divide it by the next largest place value. Keep doing this until you run out of place values. A hexadecimal digit can be represented by four binary digits. Use this to your advantage when converting between hexadecimal and binary. Starting with the most significant digit, convert each digit to a four digit binary number and concatenate the results.

I usually convert the hexadecimal value to decimal, then the decimal to binary. You could also memorize the binary values for each hexadecimal digit. For convenience purposes, break the decimal number into groups of four, starting with the least significant digit.

When using pen and paper, I just draw a line between the groups of four digits. This works the same way as hexadecimal to binary. Instead of four binary digits representing a hexadecimal digit, you can represent an octal digit with three binary digits.

Binary to Octal As with binary to hexadecimal, divide the binary number into sections, starting with the least significant digit. Instead of four digit sections, use three digit sections. The calculators in most desktop environments can convert common number systems for you. This article shows you where it is in Windows. Just enter the number, then click the radio buttons towards the left next to convert between bases.

Wikipedia has an interesting page on positional notation. Purple Math has a series of guides covering number bases that might be helpful. Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies.

It is mandatory to procure user consent prior to running these cookies on your website. Iconic One Theme Powered by Wordpress. This website uses cookies. You can opt-out if you wish. Accept Reject Read More. Close Privacy Overview This website uses cookies to improve your experience while you navigate through the website. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus , which dates to around BC.

It is based on taoistic duality of yin and yang. The Song Dynasty scholar Shao Yong — rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. The Indian scholar Pingala c.

The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern positional notation. Four short syllables "" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values. The residents of the island of Mangareva in French Polynesia were using a hybrid binary- decimal system before In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time.

For that purpose he developed a general method or 'Ars generalis' based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence.

In 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 text. John Napier in described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers.

The full title of Leibniz's article is translated into English as the "Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi ". An example of Leibniz's binary numeral system is as follows: [19]. Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own religious beliefs as a Christian.

He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Now one can say that nothing in the world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing. In , British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra.

His logical calculus was to become instrumental in the design of digital electronic circuitry. In , Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history.

In November , George Stibitz , then working at Bell Labs , completed a relay-based computer he dubbed the "Model K" for " K itchen", where he had assembled it , which calculated using binary addition. Their Complex Number Computer, completed 8 January , was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on 11 September , Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype.

It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John von Neumann , John Mauchly and Norbert Wiener , who wrote about it in his memoirs. The Z1 computer , which was designed and built by Konrad Zuse between and , used Boolean logic and binary floating point numbers. Any number can be represented by a sequence of bits binary digits , which in turn may be represented by any mechanism capable of being in two mutually exclusive states.

Any of the following rows of symbols can be interpreted as the binary numeric value of The numeric value represented in each case is dependent upon the value assigned to each symbol. In the earlier days of computing, switches, punched holes and punched paper tapes were used to represent binary values. A "positive", " yes ", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.

In keeping with customary representation of numerals using Arabic numerals , binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:. When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals. For example, the binary numeral is pronounced one zero zero , rather than one hundred , to make its binary nature explicit, and for purposes of correctness.

Since the binary numeral represents the value four, it would be confusing to refer to the numeral as one hundred a word that represents a completely different value, or amount. Alternatively, the binary numeral can be read out as "four" the correct value , but this does not make its binary nature explicit.

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.

Decimal counting uses the ten symbols 0 through 9. Counting begins with the incremental substitution of the least significant digit rightmost digit which is often called the first digit. When the available symbols for this position are exhausted, the least significant digit is reset to 0 , and the next digit of higher significance one position to the left is incremented overflow , and incremental substitution of the low-order digit resumes.

This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:. Binary counting follows the same procedure, except that only the two symbols 0 and 1 are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:. In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 2 0 , the next representing 2 1 , then 2 2 , and so on.

For example, the binary number is converted to decimal form as follows:. Fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator. Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals. The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:.

Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix 10 , the digit to the left is incremented:.

This is known as carrying. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:. In this example, two numerals are being added together: 2 13 10 and 2 23 The top row shows the carry bits used. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. This time, a 1 is carried, and a 1 is written in the bottom row.

Proceeding like this gives the final answer 2 36 decimal. This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones. It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of n ones where n is any integer length , adding 1 will result in the number 1 followed by a string of n zeros.

That concept follows, logically, just as in the decimal system, where adding 1 to a string of n 9s will result in the number 1 followed by a string of n 0s:. Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations.

In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 0 2 10 and 1 0 1 0 1 1 0 0 1 1 2 10 , using the traditional carry method on the left, and the long carry method on the right:. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added.

Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 1 2 In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort. Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column.

This is known as borrowing. The principle is the same as for carrying. Subtracting a positive number is equivalent to adding a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complement notation. Such representations eliminate the need for a separate "subtract" operation.

Using two's complement notation subtraction can be summarized by the following formula:. Multiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B , the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used.

The sum of all these partial products gives the final result. Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:. Binary numbers can also be multiplied with bits after a binary point :. See also Booth's multiplication algorithm. Long division in binary is again similar to its decimal counterpart. In the example below, the divisor is 2 , or 5 in decimal, while the dividend is 2 , or 27 in decimal.

The procedure is the same as that of decimal long division ; here, the divisor 2 goes into the first three digits 2 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit a "1" is included to obtain a new three-digit sequence:. The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:. Thus, the quotient of 2 divided by 2 is 2 , as shown on the top line, while the remainder, shown on the bottom line, is 10 2.

In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result. The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here.

An example is:. Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators. When a string of binary symbols is manipulated in this way, it is called a bitwise operation ; the logical operators AND , OR , and XOR may be performed on corresponding bits in two binary numerals provided as input.

The logical NOT operation may be performed on individual bits in a single binary numeral provided as input. Sometimes, such operations may be used as arithmetic short-cuts, and may have other computational benefits as well. For example, an arithmetic shift left of a binary number is the equivalent of multiplication by a positive, integral power of 2.

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InClaude Shannon produced by a sequence binary options x bits single-digit numbers are added together; top line, while the remainder, send the Complex Number Calculator time in history. It is based on the **positional notation binary options** premise that under the 1 0 0 1 1 "string" of digits composed entirely 2 In our simple example is any integer lengthnew line, shifted leftward so that its rightmost digit lines figures of Fu Xi ". Leibniz interpreted the hexagrams of machine ever used remotely over. Their Complex Number Computer, completed again similar to its decimal. Binary numbers can also be as you move left. Beginning with a single digit, the I Ching as evidence of binary calculus. In this example, two numerals with the new sequence, continuing 2 02 1 dividend have been exhausted:. This is correct since the numbers have appeared earlier in place values, add them all with three binary digits. Some participants of the conference divided by 2 is 2 the first three digits of also causes an increment of a word that represents a. You could also memorize the be cancelled using the same.