Wax Treadmill Star Trac

1: The binary numbering

since man learned to use reason, found it necessary to have some way objects around him and, more particularly, those who had. The flourishing trade in ancient times aggravated the need using a numerical system accurate and easy to use.

Thus, the man began to count by tens (which is what we know today as the decimal system) influenced by the fact that he had ten fingers. As the numbers rose, each numerical unit received a different symbol (eg, 3, 4, 5 in Arabic numerals). After nine, he took the symbol representing the least amount of units (1) and add a zero, thus obtained the number ten. The operation began again until you get your count up to nineteen, after which it increased the figure to the left putting in a unit a zero to the right of it, repeating the process indefinitely. We can see that without the zero, would have required a different symbol for each number greater than nine (eg, the symbol A for ten, the symbol B for eleven, the symbol C for twelve, etc.).. Indeed, without zero, any number system is extremely complex and impractical (we can imagine the problems suffered by the Romans when in Rome numbering system tried to multiply a number by another, they tried something like XXIII multplicar instead LIV what for us is 23 by 54). Not for nothing has claimed the invention of zero as one of the most important advances in human history.

Our attention now turns to a philosophical problem. Suppose the man instead of having five fingers on each hand had had three. What would have been our way of telling?

A moment's reflection tells us that our number system in that case would not have been very different from the decimal system we know today. Having three fingers on each hand, our natural inclination would have been to have every six , in the same way that modern man with five fingers on each hand account of ten. When counting in sixes, numbering amount as follows:

We note that the number system based on six fingers, the base six number system , never use the symbol 6 , in the same manner as in the base ten number system (or decimal) there no special symbol to represent the number ten. We also note that the count up in the base six number system proceeds in a similar way to count up in the base ten number system. To get 5, take the symbol that represents the least amount of units (1) and add a zero, thus obtaining the following figure. The process is repeated indefinitely so similar to the process used in the decimal system. The number after 555, for example, would be 1000. Note that a collection of eight objects in the decimal system is represented by the number 8 while the base six number system is represented by the number 12 (this equivalence is represented symbolically as 8 _{10 6} = 12).

Oddly base six number system we like, we must remember that it we would not be so strange if we had three fingers on each hand.

Thus we see that the only reason why we have tens is because we have ten fingers on both hands. We also see that other systems are equally possible numerical not only the base six number system, but the base four number system, base seven number system, etc.

We can convert any number of our decimal base to a lower base (for example, a decimal number to its equivalent in three base system) by the method successive division. This method is carried out as follows:

(1) Divide the decimal number given from the base to which we turn to the issue, and highlights the residue obtained.

(2) The ratio obtained from the previous slice is split again between the base back to where we want to make the number, and highlights the residue thus obtained.

(3) The above procedure is repeated until no longer possible to continue dividing without obtaining a fraction with the decimal point. In reaching this stage, include the dividend and the residue obtained.

(4) The number of the smaller base is obtained by writing as the first digit of the dividend received in the last step above, and placing as the second digit (see right) obtained from the also the last step above.

(5) For the third digit, write to the right of the residue from the previous result of the penultimate division.

(6) The above step is repeated until who have exhausted all the digits.

To convert a number into a smaller base to decimal (for example, number seven in the base system to its equivalent in decimal), multiply the first digit of the lower base. The resulting product is added to the second digit and multiply again by the lower base. The procedure is continued until all the numbers, after which there will be the decimal number.

Of particular interest to us is the number system base two or binary .

If the man had had only one finger on each hand, then counting to go "up one at a time" in the system base two or binary system, and taking into account that just as in decimal or base ten system we are used there is no special symbol to represent the number ten either in the binary system there will be a special symbol to represent the number two, the binary count up "up" would proceed as follows:

The binary number 110 has been highlighted with a yellow background is that used to identify the symbol "6" in the habit of what we call a sixth object or a collection of six things. In a basket of apples, the object could be the sixth block, remains the same regardless of the symbols we use to identify it. The only thing that changes is our way of representing it, as we have seen is somewhat arbitrary. (This list of binary numbers has also been highlighted with cyan background, the binary number representing an eleventh object.) And so, in the binary system, perhaps by going to the market to buy some oranges we say to the put in charge of something like "please give me 101 oranges." And if this seems strange, you have to think that individuals of an alien civilization that they had seven fingers on each hand, giving a total of 14 fingers (With the numbering system you probably would base 14), our decimal counting system may seem very strange to them. It's all about perspective.

Why is so much interest for us to get into a numerical system as the binary system, as if we have enough problems with the decimal system?

trying to use electrical circuits to perform mathematical operations (or operations, control), we find that there are only two possible states that can be used to perform information processing. One is the state of on , which we can represent the number one (1 ). The other is the off , which we represent as zero ("0 ).

Imagine a row of five outbreaks, in which the first light (left) is off, the two foci following on, bulb off the fourth and fifth focus on. Representing the headlights on with a "1" and focuses each off with a "0" each, we obtain the following representation: 01101

This number represents the number 13 in the decimal system. Each digit binary number, on or off, is known as bit . A series of several bits in succession as shown above is commonly known as binary word or simply word. Thus, following the custom of the Arabs bequeathed Saracens in the binary numbering, as in the decimal numbering which goes under counting up the numbers of increasing magnitude for the units, tens, hundreds, etc. . they are written to the left, also in the base 2 numbering used to write binary numbers grow to the left, and in doing so the "bit" of lesser magnitude that is placed on the far right is known as the least significant bit (in English: Least Significant Bit or LSB ), while the "bit" of greater magnitude is placed on the left end and is known as the most significant bit (in English: Most Significant Bit or MSB ) .

Below is a table called table:

Using tables as it is possible to shorten the conversion of a number in binary to decimal and vice versa. For example, if you want to find the decimal equivalent of the word 10110, we note that:

10110 = 10000 + 100 + 10

= 16 + 4 + 2 =

22
Let's see this same from another point of view, from the standpoint of representing numbers using powers of two . The table above equivalence can be represented using powers of the number two (where by definition an exponentiation to the power of zero is taken as the unit):

2 ⁰ = 1

^{2 1 = 2}

2 2 = 2x2 = 4

<sup>2 3 = 2x2x2 = 8

2 4</sup> = 2x2x2x2 = 16

2 ⁵ = 2x2x2x2x2 = 32, etc. Taking

this in mind, we can build a table of powers of two as follows:

This table, based on the powers of the number two (where by definition the exponentiation to the power of zero is taken as equal to unit) is used as follows: Suppose we want to convert the decimal number 59 to its equivalent in binary. This number is greater than 32 but less than 64, so that the first amount that will be part of it will be 2.5 = 32. If we add the next lowest number of the table, 2.4 = 16, the cumulative amount will be 48, which shall not exceed the decimal number 59, so that we can add 2.4 to the cumulative sum. And if we add the next lowest number of the table, 2.3 = 8, the cumulative amount will be 56, which does not exceed the decimal number 59, so that we can add 2.3 to the cumulative sum. However, we can not add 2.2 = 4 because the cumulative sum is beyond the decimal number 59, so that ruled out 2.2 as possible component of the cumulative sum. Proceeding in this manner until all the table, we see that the decimal number 59 can be represented in powers of two as follows:

59 = 32 + 16 + 8 + 0 + 1 + 1 59 = 2

^{5 + 2 + 2 4} 3 + 0 + 2 1 + 2 0

With this, the representation of the number 59 in both bases (decimal base and base 2) must immediately consult the table:

_{59 10} = (100000) ₂ + (10000) ₂ + (1000 _{) 2} + (0) ₂ + (10) 2 + (1) 2

_{59 10} = 111011 ₂

This result can be verified with the method of successive division.

For the reverse, ie convert a certain number based on their equivalent decimal system, we do so easily carried out by expanding the number of representation in the powers of base number in its orignal. For example, if we convert the binary number 101001 to the equivalent decimal system, the expansion on powers of two will take place as follows: 101001

₂ = ( 1 ) 2 ⁵ + ( 0 ) 2 ⁴ + ( 1) 2 3 + ( 0 ) 2 ² + ( 0 ) 2 1 + (1 ) 2 ⁰

101001 2 +0 = 32 + 8 + 0 + 0 + 1 2

101001 = 41 ₁₀

Now, we can add, subtract, multiply and divide in the binary system in the same way in which we conduct these operations in the decimal system.

there a special way to represent decimal numbers using the binary system, so that they will look a little more to the numbers we use (although not pure binary notation). Each decimal digit is represented by its equivalent separately, without performing any conversion. For example, the number 3497 is represented as follows:

This form of representation code is known as binary coded decimal BCD (English Binary Coded Decimal).

Now we will consider another philosophical dilemma slightly different problem with which we began this chapter: Suppose the man instead of having five fingers on each hand had been eight . What would have been our way of telling? (The case is not as hypothetical as you might think, some people are born carrying a genetic flaw that causes them something called polydactylism , which is a medical term to describe the presence of extra digits either in the hands or feet, and although might seem that there is some advantage in having a greater number of fingers on both hands or feet that the five we currently have, for some reason evolution has not favored a greater number of fingers).

Again, a moment's reflection tells us that our number system in that case would not have been very different from the decimal system we know today, except that we would be counting on sixteen sixteen . By having an abundance of fingers on both hands, most likely would have invented a unique symbol as the symbol A to represent in the number system base-16 what today we denote as ten to two symbols (10). To represent the equivalent of the decimal number 11 our twelfth finger could have represented another new symbol, the symbol B . Thus, we would have had a different symbol to represent each number up before reaching the number 16 (decimal). And when they reached what would be the equivalent of decimal number 16, would take the symbol that represents the least amount of units (1) and would add a zero, thus obtaining the following figure. The process is repeated indefinitely so similar to the process used in the decimal system. A count-up in the hexadecimal number system proceeds as follows:

_____ Base 10 Base 16

0 0 __________ __________
1 1 2
2 __________ __________
3 3
4 __________ __________ 4
5
5 6 6
__________ __________ 7
7 8 8
__________ __________ 9 9 10
A
__________ __________ B 11 12
C __________ __________
13 14
D E
__________ __________ 15 F
16 10
__________ __________ 17
11 18 12
__________ __________ 19 13 20
__________ __________ 14
21
15 22 16
__________ __________ 23
17 __________ 24 18 25
19 __________ __________
26
27 1A 1B
__________ __________ 28
1C 1D 29
__________ __________ 30
1E 1F 31 __________ 32
__________ 20

To highlight a number as a number that is based in the hexadecimal system, we use a letter h either end of the number or the beginning of the number or the number subscribed. So hexadecimal number 19 is coming highlighting one of the following representations: 19h

_{19 h}

Strange as it may appear, this hexadecimal number system is widely used the area of \u200b\u200bcomputer science. The reason for its enormous usefulness lies in the fact that there is a simple relationship between the representations of a pure binary number system and its equivalent in hexadecimal when the binary number is a multiple of four bits :

aaaaa ___ Binary Hexadecimal

________ 0000 0 0001
________ 1
0010 ________ 2
0011 ________ 3
0100 ________ 4
0101 ________ 5
0110 ________ 6
7 0111 ________
1000 ________ 8
1001 ________ 9 1010
________ A
1011 ________ B
1100 ________ C
1101 ________
D 1110 E ________ ________
1111 F

which greatly simplifies the conversion of a numeric system to another. For example, if we find the hexadecimal equivalent of the following binary number: 11000101000001101000000101011000

all we have to do is "split" the number groups of four binary bits:

0000 0101 1100 0110 0001 1000 0101 1000

after which we can directly convert each individual group in its hexadecimal equivalent:

C 5 0 6 8 1 5 8

To convert a hexadecimal number to binary, simply apply the reverse procedure. If we convert the hexadecimal number AF37 to its binary equivalent, we do take into account that A = 1010, F = 1111 , and 3 = 0011 7 = 0111 . Thus, the hexadecimal number example is:

1010 1111 0011 0111

or abbreviated form (although slightly less clearly): 1010111100110111

Since it requires many bits to represent a number of moderate size, reading a 32-bit number stored in a register as follows:

1010 1111 0101 0111 0110 0001 0001 1011

is much faster and easier for humans to write or read:

AF57611B

shown that the binary number.

As in the decimal numbering exist and are managed with often negative numbers , preceded by a minus sign (-) position to the left thereof, in binary numbers are also handled frequently and negative numbers. However, in the binary numbering to distinguish a positive one negative number is not customary to do it with a minus sign (-). One way of carrying out some kind of distinction is predating the binary number with a " 0 " if the figure is positive (+) or with a " 1" if the figure is negative (-). If you reserve the first bit to the left to represent the sign of the binary number, then the seven remaining bits in a word bit of a "byte" are not sufficient to encode decimal numbers with sufficient precision, and in this case requires at least two bytes to represent decimal numbers to 32 000. Under the universal convention of the sign just given:

0 0000001 represents the decimal number 1

1 0000010 represents the decimal number -2

A disadvantage of this representation is that the binary numbers of different signs can not be added in the usual manner as is customary to do so since If we add the two previous binary will result 10000011, or -3, which is incorrect (the correct answer should be -1). Anyway, keep this performance until we find in later chapters one that will enable us to properly perform arithmetic operations on numbers of different signs in the binary system always getting the correct size with the correct sign. Anyway, what will not change is the use of the first bit denoting the sign reserving the quantity.

We talked about the use of binary numbers to be integers counting one by one in the base-2 system. It is possible that here there is some reader might wonder: Is it possible to also use the binary system for counting and measuring fractions, numbers less than unity, as we do in the decimal system? The answer is yes, and to achieve that we must introduce into the binary numbering the same trick we use to distinguish integers numbers less than unity: the point that in this case instead of the decimal point is the point binary.

A fraction represents a division. Like as in the decimal system, fractions in the binary system can be written with a numerator and a denominator separated by a horizontal bar:

In the decimal fractions can be written with a decimal point. Examples include:

Similarly, the fractions in the binary system can also be written using a point to it, but instead of talking about a decimal point we are talking about a binary point . Thus:

Put another way, to represent fractions in the binary system, the principle remains the same. Decimal symbols to fractional quantities are constructed based on tenths, hundredths (tenths of tenths), mills (tenths of tenths of tenths), thousandths, and so on. The binary symbols are constructed out of halves, halves of halves, halves of halves of halves, and so on. This allows us to build the following table:

and so on. Other fractions can be represented as combinations of these key issues on the table of equivalences. Thus:

.11 = 1 / 2 + 1 / 2 = 3 / 4

.101 = 1 / 2 + 1 / 3 = 5 / 3

can corroborate these findings are as follows, representing the fraction as the ratio of two binary integers:

.11 = 11/100 = 3 / 4

.101 = 101/100 = 5 / 3

addition binary numbering system, the system BCD, and hexadecimal system, there are other number systems, including some prominence is the octal system or base-8 system. For comparative purposes, the following is a list of the first ten numbers to its decimal equivalent, an octal, and binary equivalent:

The role of the octal system in the development of digital systems circuits based on binary has to do with the simple relationship between symbols and binary octal symbols. To better understand this relationship, examine the following symbols equal to amounts slightly larger:

For best viewing, each equivalent binary has been separated into groups of three digits (following an order from right to left ), which allows us to discover that each group of three digits correspond to the equivalent octal digit in the same position . Thus, a binary number as 10001001 can be separated into groups of three digits 10,001,001 , Which allows us to immediately determine an octal as 211. The binary number 10001001 equivalent to the decimal number 137, and we can verify that the octal number 211 also equals the decimal number by the usual tactic of assigning to each octal digit positional value in the decimal system: 211

₈ = 2 (8 2) + 1 (8 1) + 1 (8 ⁰ ) 211

₈ = 2 ( 64) + 1 (8) + 1 (1) 211

8 = 128 + 8 + 1 = 137 10

The purpose of the octal number (as hexadecimal numbering) is a bridge between the decimal number system we is so familiar and least understood binary system. Decimal symbols are our daily means of arithmetic work, but the language of "ones" and "zeros" is the natural language with which the machines work. The disadvantage of binary numbers is that it requires a long series of "ones" and "zeros" to represent an amount in the decimal system can be represented more compactly as number 10001001 is equivalent to the decimal number 137. The advantage of using octal symbols is that they are convenient abbreviations of binary symbols, and using octal numbers rather than the more familiar decimal numbers represents a natural step to bridge the gap between a computer "human" used to work in the system decimal and machine.