Binary Numbers Explained – Beginners Guide

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In computing and electronic systems, binary-coded decimal BCD is a class of binary encodings of decimal numbers where each decimal digit is represented by a fixed number of bitsusually four or eight. Special bit patterns are sometimes used for a sign or for other indications e. In byte-oriented systems i.

The precise 4-bit encoding may vary however, for technical reasons, see Excess-3 for instance. BCD's main virtue is its more accurate representation and rounding of decimal quantities as well as an ease of conversion into human-readable representations, in comparison to binary positional systems. BCD's principal drawbacks are a small increase in the complexity binary code system pdf the circuits needed to implement basic arithmetics and a slightly less dense storage.

Although BCD per se is not as widely used as in the past and is no longer implemented in newer computers' instruction sets such as ARM ; x86 does not support BCD instructions in long mode any moredecimal fixed-point and floating-point formats are still important and continue to be used in financial, commercial, and industrial computing, where subtle conversion and fractional binary code system pdf errors binary code system pdf are inherent in floating point binary representations cannot be tolerated.

BCD takes advantage of the fact that any one decimal numeral can be represented by a four bit pattern. The most obvious way of encoding digits is "natural BCD" NBCDwhere each decimal digit is represented by its corresponding four-bit binary value, as shown in the following table.

This is also called "" encoding. Other encodings are also used, including so-called "" and ""—named after the weighting used for the bits—and " Excess-3 ". As binary code system pdf computers deal with data in 8-bit bytesit is possible to use one of the following methods to encode a BCD number:. As an example, encoding the decimal number 91 using unpacked BCD results in the following binary pattern of two bytes:.

Hence the numerical range for one unpacked BCD byte is zero through nine inclusive, whereas the range for one packed BCD is zero through ninety-nine inclusive. To represent binary code system pdf larger than the range of a single byte any number of contiguous bytes may be used. For example, to represent the decimal number in packed BCD, using big-endian format, a program would encode as follows:. Note that the most significant nibble of the most significant byte is zero, implying that the number is in actuality Binary code system pdf note how packed BCD is more efficient in storage usage as compared to unpacked BCD; encoding the same number with the leading zero in unpacked format would consume twice the storage.

Shifting and masking operations are used to pack or unpack a packed BCD digit. Other logical operations are used to convert a numeral to its equivalent bit pattern or reverse the process. BCD is very common in electronic systems where a numeric value is to be displayed, especially in systems consisting solely of digital logic, and not containing a microprocessor.

By employing BCD, the manipulation of binary code system pdf data for display can be greatly simplified by treating each digit as a separate single sub-circuit. This matches much more closely the physical reality of display hardware—a designer might choose to use a series of separate identical seven-segment displays to build a metering binary code system pdf, for example. If the numeric quantity were stored and manipulated as pure binary, interfacing to such a display would require complex binary code system pdf.

Therefore, in cases where the calculations are relatively simple, working throughout with BCD can lead to a simpler overall system than converting to and from binary. Most pocket calculators do all their calculations in BCD. The same argument applies when hardware of this type uses an embedded microcontroller or other small processor. Often, smaller code results when representing numbers internally in BCD format, since a conversion from or to binary representation can be expensive on such limited processors.

For these applications, some small processors feature BCD arithmetic modes, binary code system pdf assist when writing routines that manipulate BCD quantities. In packed BCD or simply packed decimaleach of the two nibbles of each byte represent a decimal digit. Most implementations are big endiani. The lower nibble of the rightmost byte is usually used as the sign flag, although some unsigned representations lack a sign flag.

As an example, a 4-byte value consists of 8 nibbles, wherein the upper 7 nibbles store the digits of a 7-digit decimal value and the lowest nibble indicates the sign of the decimal integer value. Other allowed signs are A and E for positive and B for negative. Most implementations also provide unsigned BCD values with a sign nibble of F. Burroughs systems used D for negative, and any other value is considered a positive sign value the processors will normalize a positive binary code system pdf to C.

No matter how many bytes wide a word is, there are always an even number of nibbles because each byte has two of them. Note that, like character strings, the first byte of the packed decimal — with the most significant binary code system pdf digits — is usually stored in the lowest address in memory, independent of the endianness of the machine. The extra storage requirements are usually offset by the need for the accuracy and compatibility with calculator or hand calculation that fixed-point decimal arithmetic provides.

Denser packings of BCD exist which avoid the storage penalty and also need no arithmetic operations for common conversions. Ten's complement representations for negative numbers offer an alternative approach to encoding the sign of packed and other BCD numbers. In this case, positive numbers always have a most significant digit between 0 and 4 inclusivewhile negative numbers are represented by the 10's complement of the corresponding positive number.

As a result, this system allows for, a bit packed BCD numbers to range from , to 49,, and -1 is represented as Binary code system pdf with two's complement binary numbers, the range is not symmetric about zero.

These languages allow the programmer to specify an implicit decimal point in front of one of the digits. The decimal point is not actually stored in memory, as the packed BCD storage format does not provide for it. Its location is simply known to the compiler and the generated code acts accordingly for the various arithmetic operations. If a decimal digit requires four bits, then three decimal digits require 12 bits.

However, since 2 10 1, is greater than 10 3 1,if three decimal digits are encoded together, only 10 bits are needed. The latter has binary code system pdf advantage that subsets of the encoding encode two digits in the optimal seven bits and one digit in four bits, as in regular BCD.

Some implementations, for example IBM mainframe systems, support zoned decimal numeric representations. Each decimal digit is stored in one byte, with the lower four bits encoding the digit in BCD form.

The upper four bits, called the "zone" bits, are usually set to a fixed value so that the byte holds a character value corresponding to the digit. For signed zoned decimal values, the rightmost least significant zone nibble holds the sign digit, which is the same set of values that are used for signed packed decimal numbers see above. These characters vary depending on the local character code page setting. The IBM series are character-addressable machines, each location being six bits labeled B, A, 8, 4, 2 and 1, plus an odd parity check bit C and a word mark bit M.

For encoding digits 1 through 9B and Binary code system pdf are zero and the digit value represented by standard 4-bit BCD in binary code system pdf 8 through 1. For most other characters bits B and Binary code system pdf are derived simply from the "12", "11", and "0" "zone punches" in the punched card character code, and bits 8 through 1 from the 1 through 9 punches.

A "12 zone" punch set both B and Aan "11 zone" set Band a "0 zone" a 0 punch combined with any others set A. Thus the letter Awhich is 12,1 binary code system pdf the punched card format, is encoded B,A,1. This allows the circuitry to convert between the punched card format and the internal storage format to be very simple with only a binary code system pdf special cases.

One important special case is digit 0represented by a lone 0 punch in the card, and 8,2 in core memory. The memory of the IBM is organized into 6-bit addressable digits, the usual 8, 4, 2, 1 plus Fused as a flag bit and Can odd parity check bit. BCD alphamerics are encoded using digit pairs, with the "zone" in the even-addressed digit and the "digit" in the odd-addressed digit, the "zone" being related to the 1211and 0 "zone punches" as in the series.

A variable length Packed BCD numeric data type is also implemented, providing machine instructions that perform arithmetic directly on packed decimal data. All of these are used within hardware binary code system pdf and processing units, and in software. The MicroVAX and later VAX implementations dropped this ability from the CPU but retained code compatibility with earlier machines by implementing the missing instructions in an operating system-supplied software library.

This is invoked automatically via exception handling when the no longer implemented instructions are encountered, so that programs using them can execute without modification on the newer machines.

The Intel x86 architecture supports a unique digit ten-byte BCD format that can be loaded into and stored from binary code system pdf floating point registers, and computations can be performed there.

The Motorola series had BCD instructions. In more recent computers binary code system pdf capabilities are almost always implemented in software rather than the CPU's instruction set, but BCD numeric data is still extremely common in commercial and financial applications.

There are tricks for implementing packed BCD and zoned decimal add or subtract operations using short but difficult to understand sequences of word-parallel logic and binary arithmetic operations.

Conversion of the simple sum of two digits can be done by adding 6 that is, 16 — 10 when the five-bit result of adding a pair of digits has a value greater than 9. Note that is the binary, not decimal, representation of the desired result. Also note that it cannot fit in a 4-bit number. In BCD as in decimal, there cannot exist a value greater than 9 per digit. To correct this, 6 is added to that sum and then the result is treated as two nibbles:.

The two nibbles of the result, andcorrespond to the digits "1" and "7". This yields "17" in Binary code system pdf, which is the correct result. This technique can be extended to adding multiple digits by adding in groups from right to left, propagating the second digit as a carry, always comparing the 5-bit result of each binary code system pdf sum to 9.

Some CPUs provide a half-carry flag to facilitate BCD arithmetic adjustments following binary addition and subtraction operations. Subtraction is done by adding the ten's complement of the subtrahend. To represent the sign of a number in BCD, the number is used to represent a positive numberand is used to represent a negative number.

The remaining 14 combinations are invalid signs. To illustrate signed BCD subtraction, consider the following problem: In signed BCD, is The ten's complement of can be obtained by taking the nine's complement ofand then adding one.

Since BCD is a form of decimal representation, several of the digit sums above are invalid. In the event that an invalid entry any BCD digit greater binary code system pdf exists, 6 is added to generate a carry bit and cause the sum to become a valid entry. So adding 6 to the invalid entries results in the following:. To check the answer, note that the first digit is 9, which means negative. To check the rest of the digits, represent them in decimal.

The binary code system pdf decimal scheme described in this article is the most common encoding, but there are many others. The following table represents decimal digits from 0 to 9 in various BCD systems:. In the case Gottschalk v. Bensonthe U. Supreme Court overturned a lower court decision which had allowed a patent for converting BCD encoded numbers to binary on a computer.

This was an important case in determining the patentability of software and algorithms. The Atari 8-bit binary code system pdf of computers used BCD to implement floating-point algorithms.

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To understand binary numbers, begin by recalling elementary school math. When we first learned about numbers, we were taught that, in the decimal system, things are organized into columns: H T O 1 9 3 such that "H" is the hundreds column, "T" is the tens column, and "O" is the ones column. So the number "" is 1-hundreds plus 9-tens plus 3-ones.

As you know, the decimal system uses the digits to represent numbers. The binary system works under the exact same principles as the decimal system, only it operates in base 2 rather than base In other words, instead of columns being. Therefore, it would shift you one column to the left. For example, "3" in binary cannot be put into one column. What would the binary number be in decimal notation? Click here to see the answer Try converting these numbers from binary to decimal: Since 11 is greater than 10, a one is put into the 10's column carried , and a 1 is recorded in the one's column of the sum.

Thus, the answer is Binary addition works on the same principle, but the numerals are different. Begin with one-bit binary addition:. In binary, any digit higher than 1 puts us a column to the left as would 10 in decimal notation. Record the 0 in the ones column, and carry the 1 to the twos column to get an answer of " The process is the same for multiple-bit binary numbers: Record the 0, carry the 1.

Add 1 from carry: Multiplication in the binary system works the same way as in the decimal system: Follow the same rules as in decimal division. For the sake of simplicity, throw away the remainder. Converting from decimal to binary notation is slightly more difficult conceptually, but can easily be done once you know how through the use of algorithms.

Begin by thinking of a few examples. Almost as intuitive is the number 5: Then we just put this into columns. This process continues until we have a remainder of 0. Let's take a look at how it works. To convert the decimal number 75 to binary, we would find the largest power of 2 less than 75, which is Subtract 8 from 11 to get 3. Thus, our number is Making this algorithm a bit more formal gives us: Find the largest power of two in D. Let this equal P. Put a 1 in binary column P.

Subtract P from D. Put zeros in all columns which don't have ones. This algorithm is a bit awkward. Particularly step 3, "filling in the zeros. Now that we have an algorithm, we can use it to convert numbers from decimal to binary relatively painlessly. Our first step is to find P. Subtracting leaves us with Subtracting 1 from P gives us 4. Next, subtract 16 from 23, to get 7.

Subtract 1 from P gives us 3. Subtract 1 from P to get 1. Subtract 1 from P to get 0. Subtract 1 from P to get P is now less than zero, so we stop. Another algorithm for converting decimal to binary However, this is not the only approach possible. We can start at the right, rather than the left. This gives us the rightmost digit as a starting point.

Now we need to do the remaining digits. One idea is to "shift" them. It is also easy to see that multiplying and dividing by 2 shifts everything by one column: Similarly, multiplying by 2 shifts in the other direction: Take the number Dividing by 2 gives Since we divided the number by two, we "took out" one power of two. Also note that a1 is essentially "remultiplied" by two just by putting it in front of a[0], so it is automatically fit into the correct column.

Now we can subtract 1 from 81 to see what remainder we still must place Dividing 80 by 2 gives We can divide by two again to get This is even, so we put a 0 in the 8's column. Since we already knew how to convert from binary to decimal, we can easily verify our result.

These techniques work well for non-negative integers, but how do we indicate negative numbers in the binary system? Before we investigate negative numbers, we note that the computer uses a fixed number of "bits" or binary digits. An 8-bit number is 8 digits long. For this section, we will work with 8 bits.

The simplest way to indicate negation is signed magnitude. To indicate , we would simply put a "1" rather than a "0" as the first bit: In one's complement, positive numbers are represented as usual in regular binary. However, negative numbers are represented differently. To negate a number, replace all zeros with ones, and ones with zeros - flip the bits. Thus, 12 would be , and would be As in signed magnitude, the leftmost bit indicates the sign 1 is negative, 0 is positive.

To compute the value of a negative number, flip the bits and translate as before. Begin with the number in one's complement. Add 1 if the number is negative.

Twelve would be represented as , and as To verify this, let's subtract 1 from , to get If we flip the bits, we get , or 12 in decimal. In this notation, "m" indicates the total number of bits. Then convert back to decimal numbers.