Binary, Decimal and Hexadecimal Numbers Binary, Decimal and Hexadecimal Numbers Decimals To understand Binary and Hexadecimal numbers, it is best to know how Decimal Numbers work. Every digit in a decimal number has a "position", and the decimal point helps us to know which position is which. The position just to the left of the point is the "Units" position. Every position further to the left is 10 times bigger, and every position further to the right is 10 times smaller: Now, this is just a way of writing down a value. Other ways include Roman Numerals , Binary , Hexadecimal , and more. You could even just draw dots on a sheet of paper! The Decimal Number System is also called "Base 10". Because it is based on the number 10. And there are 10 symbols (0,1,2,3,4,5,6,7,8 and 9), but notice something interesting: there is no symbol for "ten".

"10" is actually two symbols put together, a "1" and a "0": In decimal you count "0,1,2,3,4,5,6,7,8,9,..." but then you run out of symbols! So you add 1 on the left and then start again at 0: 10,11,12, ... Counting with Different Number Systems But you don't have to use 10 as a "Base". You could use 2 ("Binary"), 16 ("Hexadecimal"), or any number you want to! Example: In binary you count "0,1,..." but then you run out of symbols! So you add 1 on the left and then start again at 0: 10,11 ... See how to count dots using Bases from 2 to 16 in this little demonstration: Try this: select a Base, watch it count for a while, then press "||" (Pause).

Now see if it has tallied the right number of dots, as in this example using base 2: Example: 1×16 + 1×8 + 1×1 = 16+8+1 = 25 So the general rule is: Count up until just before the "Base", then start at 0 again, but first you add 1 to the number on your left. Binary Numbers Binary Numbers are just "Base 2" instead of "Base 10". So you start counting at 0, then 1, then you run out of digits ... so you start back at 0 again, but increase the number on the left by 1. Like this: Hexadecimal | Math Wiki | Fandom powered by Wikia Template:Numeral systems In mathematics and computer science , hexadecimal (also base Template:Num , or hex) is a positional numeral system with a radix , or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a–f) to represent values ten to fifteen.

For example, the hexadecimal number 2AF3 is equal, in decimal, to (2 × 163) + (10 × 162) + (15 × 161) + (3 × 160) , or 10,995. Each hexadecimal digit represents four binary digits ( bits ) (also called a " nibble "), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital electronics. For example, byte values can range from 0 to 255 (decimal) but may be more conveniently represented as two hexadecimal digits in the range 00 through FF. Hexadecimal is also commonly used to represent computer memory addresses . Contents [ show ] Representing hexadecimal Template:Hexadecimal table In situations where there is no context, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 15910 is decimal 159; 15916 is hexadecimal 159, which is equal to 34510.

Other authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen: In URLs , character codes are written as hexadecimal pairs prefixed with %: where %20 is the space (blank) character (code value 20 in hex, 32 in decimal). In XML and XHTML , characters can be expressed as hexadecimal numeric character references using the notation &#xcode;, where code is the 1- to 6-digit hex number assigned to the character in the Unicode standard. Thus &#x2019; represents the curled right single quote (Unicode value 2019 in hex, 8217 in decimal). Color references in HTML and CSS and X Window can be expressed with six hexdigits (two each for the red, green, and blue components, in that order) prefixed with #: white, for example, is represented #FFFFFF . [1] CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange).

*nix (Unix and related) shells, and likewise the C programming language , which was designed for Unix (and the syntactic descendants of C [2] ) use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits: '\x1B' represents the Esc control character; "\x1B[0m\x1B[25;1H" is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters. [3] To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In the Unicode standard, a character value is represented with U+<code> followed by the hex value: <code>U+20AC is the Euro sign (€). In MIME (e-mail extensions) quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits (in ASCII) prefixed by an equal to sign =, as in Espa=F1a to send "España" (Spain).

(Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilde in the ISO/IEC 8859-1 character set.) In Intel-derived assembly languages , hexadecimal is denoted with a suffixed H or h: FFh or 05A3H. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit: 0FFh BBC BASIC and Locomotive BASIC use & for hex. [6] TI-89 and 92 series uses a 0h prefix: 0h5A3 The most common format for hexadecimal on IBM mainframes ( zSeries ) and midrange computers ( IBM System i ) running the traditional OS's ( zOS , zVSE , zVM , TPF , IBM i ) is X'5A3', and is used in Assembler, PL/I , Cobol , JCL , scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes. Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.

[7] Hexadecimal representations are written there in a typewriter typeface: 5A3 Any IPv6 address can be written as eight groups of four hexadecimal digits, where each group is separated by a colon (:). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334 ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary, quaternary (base-4) and octal numbers can be specified similarly. There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention. Early representations Bruce A. Martin's hexadecimal notation proposal The choice of the letters A through F to represent the digits above nine was not universal in the early history of computers. During the 1950s, some installations favored using the digits 0 through 5 with a macron character ("¯") to denote the values 10–15.

Users of Bendix G-15 computers used the letters U through Z. The Librascope LGP-30 used the letters F, G, J, K, Q and W. [8] Bruce A. Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous" and in a 1968 letter to the editor of the CACM proposed an entirely new set of symbols based on the bit locations, which did not gain much acceptance. [9] Soviet programmable calculators Б3-34 and similar used the symbols "−", "L", "C", "Г", "E", " " (space) on their displays. Verbal and digital representations There are no traditional numerals to represent the quantities from ten to fifteen—letters are used as a substitute—and most Western European languages lack non-decimal names for the numerals above ten.

Even though English has names for several non-decimal powers ( pair for the first binary power, score for the first vigesimal power, dozen , gross , and great gross for the first three duodecimal powers), no English name describes the hexadecimal powers (decimal 16, 256, 4096, 65536, ... ). Some people read hexadecimal numbers digit by digit like a phone number: 4DA is "four-dee-ay". However, the letter A sounds like "eight", C sounds like "three", and D can easily be mistaken for the "-ty" suffix: Is it 4D or forty? Other people avoid confusion by using the NATO phonetic alphabet : 4DA is "four-delta-alfa"; or the Joint Army/Navy Phonetic Alphabet ("four-dog-able"); or a similar ad hoc system. Hexadecimal finger-counting scheme. Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023 on ten fingers.

Another system for counting up to FF (255) is illustrated on the right; it seems to be an extension of an existing system for counting in twelves (dozens and grosses), that is common in South Asia and elsewhere. Signs The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210 and so on. However, some prefer instead to express the exact bit patterns used in the processor and consider hexadecimal values best handled as signed values. This way, the negative number −4210 can be written as FFFF FFD6 in a 32-bit CPU register , as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (assuming certain representation schemes, two's-complement in the 32-bit non-FPU instance and sign-magnitude in the FPU instances.) Hexadecimal exponential notation Just as decimal numbers can be represented in exponential notation so too can hexadecimal. By convention, the letter p represents times two raised to the power of, whereas e serves a similar pupose in decimal.

The number after the p is decimal and represents the binary exponent. Usually the number is normalised: that is, the leading hexadecimal digit is 1 (unless the value is exactly 0). Example: 1.3DEp42 represents 1.3DE16 × 242. Hexadecimal exponential notation is required by the IEEE 754 binary floating-point standard. This notation can be produced by some versions of the printf family of functions by using the %a conversion. Binary conversion Most computers manipulate binary data, but it is difficult for humans to work with the large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410). This example converts 11112 to base ten.

Since each position in a binary numeral can contain either a 1 or 0, its value may be easily determined by its position from the right: 00012 = 110 = 1510 With surprisingly little practice, mapping 11112 to F16 in one step becomes easy: see table in Representing hexadecimal . The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit. This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.

010111101011010100102 = 26214410 + 6553610 + 3276810 + 1638410 + 819210 + 204810 + 51210 + 25610 + 6410 + 1610 + 210 = 38792210 Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly: 010111101011010100102 5EB5216 The conversion from hexadecimal to binary is equally direct. The octal system can also be useful as a tool for people who need to deal directly with binary computer data. Octal represents data as three bits per character, rather than four. Converting from other bases Division-remainder in source base As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base.

Theoretically this is possible from any base but for most humans only decimal and for most computers only binary (which can be converted by far more efficient methods) can be easily handled with this method. Let d be the number to represent in hexadecimal, and the series hihi-1...h2h1 be the hexadecimal digits representing the number. i := 1 If d = 0 (return series hi) else increment i and go to step 2 "16" may be replaced with any other base that may be desired. The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously however, it is much more advisable to work with bitwise operators .

function toHex(d) { var r = d % 16; var result; if (d-r == 0) result = toChar(r); else result = toHex( (d-r)/16 ) + toChar(r); return result; } function toChar(n) { const alpha = "0123456789ABCDEF"; return alpha.charAt(n); } Addition and multiplication A hexadecimal multiplication table It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. I.e. to convert the number B3AD to decimal one can split the conversion into D (1310), A (1010), 3 (310) and B (1110) then get the final result by multiplying each decimal representation by 16p, where 'p' is the corresponding position from right to left, beginning with 0.

In this case we have 13*(160) + 10*(161) + 3*(162) + 11*(163), which is equal 45997 in the decimal system. Tools for conversion Most modern computer systems with graphical user interfaces provide a built-in calculator utility, capable of performing conversions between various radixes, generally including hexadecimal. In Microsoft Windows , the Calculator utility can be set to Scientific mode (called Programmer mode in some versions), which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 ( octal ) and 2 ( binary ); the bases most commonly used by programmers. In Scientific Mode, the on-screen numeric keypad includes the hexadecimal digits A through F which are active when "Hex" is selected. In hex mode, however, the Windows Calculator only supports integers. Real numbers As with other numeral systems, the hexadecimal system can be used to represent rational numbers , although recurring digits are common since sixteen (10h) has only a single prime factor (two): 0.0F where an overline denotes a recurring pattern.

For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system: Counting in base 3 is 0, 1, 2, 10 (three). Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1. Because the radix 16 is a perfect square (4²), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.

All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal : that is, any hexadecimal number with a finite number of digits has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.199999999999... in hexadecimal. However, hexadecimal is more efficient than bases 12 and 60 for representing fractions with powers of two in the denominator (e.g., decimal one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal, 0;3,45 in sexagesimal and 0.0625 in decimal). In decimal This also makes tetration easier when using two and four since: 32 = 24 = 10hex, 52 = 265536 = (1 followed by 16384 zeros)hex.

Cultural Etymology The word hexadecimal is composed of hexa-, derived from the Greek έξ (hex) for "six", and -decimal, derived from the Latin for "tenth". Webster's Third New International online derives "hexadecimal" as an alteration of the all-Latin "sexadecimal" (which appears in the earlier Bendix documentation). The earliest date attested for "hexadecimal" in Merriam-Webster Collegiate online is 1954, placing it safely in the category of international scientific vocabulary (ISV). It is common in ISV to mix Greek and Latin combining forms freely. The word " sexagesimal " (for base 60) retains the Latin prefix. Donald Knuth has pointed out that the etymologically correct term is "senidenary", from the Latin term for "grouped by 16".

(The terms "binary", "ternary" and "quaternary" are from the same Latin construction, and the etymologically correct term for "decimal" arithmetic is "denary".) [10] Schwartzman notes that the expected form from usual Latin phrasing would be "sexadecimal", but computer hackers would be tempted to shorten that word to "sex". [11] The etymologically proper Greek term would be hexadecadic (although in Modern Greek deca-hexadic (δεκαεξαδικός) is more commonly used). Use in Chinese culture The traditional Chinese units of weight were base-16. For example, one jīn (斤) (approximately 256 grams) in the old system equals sixteen liǎng (兩) (16g). The suanpan (Chinese abacus ) could be used to perform hexadecimal calculations. Common patterns and humor Template:Details Hexadecimal is sometimes used in programmer jokes because some words can be formed using hexadecimal digits. Some of these words are "dead", "beef", "babe", and with appropriate substitutions "c0ffee".

Since these are quickly recognizable by programmers, debugging setups sometimes initialize memory to them to help programmers see when something has not been initialized. An example is the magic number in Universal Mach-O files and java class file structure, which is "CAFEBABE". Single-architecture 32-bit big-endian Mach-O files have the magic number "FEEDFACE" at their beginning. " DEADBEEF " is sometimes put into uninitialized memory. Microsoft Windows XP clears its locked index.dat files with the hex codes: "0BADF00D". The Visual C++ remote debugger uses "BADCAB1E" to denote a broken link to the target system. Two common bit patterns often employed to test hardware are 01010101 and 10101010 (their corresponding hex values are 55h and AAh, respectively). The reason for their use is to alternate between off ('0') to on ('1') or vice versa when switching between these two patterns.

These two values are often used together as signatures in critical PC system sectors (e.g., the hex word, 0xAA55 which on little-endian systems is 55h followed by AAh, must be at the end of a valid Master Boot Record ). The following table shows a joke in hexadecimal: 3x12=36 2x12=24 1x12=12 0x12=18 The first three are interpreted as multiplication, but in the last, "0x" signals Hexadecimal interpretation of 12, which is 18. Another joke based on the use of a word containing only letters from the first six in the alphabet (and thus those used in hexadecimal) is... If only dead people understand hexadecimal, how many people understand hexadecimal? In this case, "dead" refers to a hexadecimal number DEAD (57005 base 10), as opposed to the state of not being alive. A Knuth reward check is one hexadecimal dollar, or $2.56. Primary numeral system Similar to dozenal advocacy , there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts usually propose pronunciation and/or symbology.

[12] Sometimes the proposal unifies standard measures so that they are multiples of 16. [13] [14] [15] An example of unifying standard measures is Hexadecimal time which subdivides a day by 16 so that there are 16 "hexhours" in a day. [15] Key to number base notation Simple key for notations used in article: Full Text Notation Numeral Systems - Binary, Octal, Decimal, Hex Numeral Systems n - can start from negative number if the number has a fraction part. N+1 - the number of digits Binary Numeral System - Base-2 Binary numbers uses only 0 and 1 digits. B denotes binary prefix.

Examples: 101012 = 10101B = 1×24+0×23+1×22+0×21+1×20 = 16+4+1= 21 101112 = 10111B = 1×24+0×23+1×22+1×21+1×20 = 16+4+2+1= 23 1000112 = 100011B = 1×25+0×24+0×23+0×22+1×21+1×20 =32+2+1= 35 Octal Numeral System - Base-8 Octal numbers uses digits from 0..7.

Examples: 278 = 2×81+7×80 = 16+7 = 23 308 = 3×81+0×80 = 24 43078 = 4×83+3×82+0×81+7×80= 2247 Decimal Numeral System - Base-10 Decimal numbers uses digits from 0..9. These are the regular numbers that we use. Example: 253810 = 2×103+5×102+3×101+8×100 Hexadecimal Numeral System - Base-16 Hex numbers uses digits from 0..9 and A..F. H denotes hex prefix.

2816 = 28H = 2×161+8×160 = 40 2F16 = 2FH = 2×161+15×160 = 47 BC1216 = BC12H = 11×163+12×162+1×161+2×160= 48146 Numeral systems conversion table Hexadecimals Hexadecimals A Hexadecimal Number is based on the number 16 16 Different Values There are 16 Hexadecimal digits. They are the same as the decimal digits up to 9, but then there are the letters A, B, C, D, E and F in place of the decimal numbers 10 to 15: Hexadecimal: So a single Hexadecimal digit can show 16 different values instead of the normal 10. Counting Choose "Hexadecimal" below and watch it count: After the "Ones" position is filled up (when we reach F) the ones start back at 0, and we add 1 to the next position on the left.

After that position reaches F, we do the same thing, and so on. As we move left, every number place is 16 times bigger. Example: What is the decimal value of the hexadecimal number "D1CE" = 53,248 + 256 + 192 + 14 This is 2×16×16 + 14×16 + 6 + 10/16 + 3/(16×16) =742.63671875 in Decimal Read below to find out why Numbers can be placed to the left or right of the point, to show values greater than one or less than one: The number just to the left of the point is a whole number. As we move left, every number place is 16 times bigger. The first digit on the right of the point means sixteenths (1/16). As we move further right, every number place is 16 times smaller (one sixteenth as big). More Examples Example 1: What is 4B5 (Hexadecimal)?

The "4" is in the "16×16" position, so that means 4 ×16×16 The "B" (11) is in the "16" position, so that means 11 ×16 The "5" is in the "1" position so that means 5. Answer: 4B5 = 4×16×16 + 11×16 + 5 (=1205) in Decimal Example 2: What is 2.3 (Hexadecimal)? On the left side is "2", that is the whole number part. The 3 is in the "sixteenths" position, meaning "3 sixteenths", or 3/16 So, 2.3 is "2 and 3 sixteenths" (=2.1875 in Decimal) The Word "Hexadecimal" The word "Hexadecimal" means "based on 16" (From Greek hexa: "six" and Latin decima: "a tenth part").

Number Bases: Octal and Hexadecimal | Purplemath Number Bases: Octal and Hexadecimal Purplemath Octal An older computer-based number system is "octal", or base eight. The digits in octal math are 0, 1, 2, 3, 4, 5, 6, and 7. The value "eight" is written as "1 eight and 0 ones", or 108. In technical terms, there are very many different computer-language protocols for octal, but we'll just be using the simple mathematical system. Content Continues Below MathHelp.com A few New-World tribes used base-8 numbering systems; they counted by using the eight spaces between their fingers, rather than the ten fingers themselves. The blue natives in the movie "Avatar" used octal because their hands had only four fingers. Let's dig right in: Convert 35710 to the corresponding base-eight number. I will do the usual sequential division, this time dividing by 8 at each step: Once I got to the "5" on top, I had to stop, because 8 doesn't divide into 5. Then the corresponding octal number is 5458.

Convert 5458 to the corresponding decimal number. I will follow the usual procedure, listing the digits in one row and then counting off the digits from the RIGHT in the next row, starting at zero: digits: Then I'll do the usual addition and multiplication: 5×82 + 4×81 + 5×80 = 5×64 + 4×8 + 5×1 = 320 + 32 + 5 Then the corresponding decimal number is 35710. Hexadecimal If you work with computer programming or computer engineering (or computer graphics, about which more later), you will encounter base-sixteen, or hexadecimal, math. Affiliate Advertisement As mentioned before, decimal math does not have one single solitary digit that represents the value of "ten". Instead, we use two digits, a 1 and a 0: "10". But in hexadecimal math, the columns stand for multiples of sixteen! That is, the first column stands for how many units you have, the second column stands for how many sixteens, the third column stands for how many two hundred fifty-sixes (sixteen-times-sixteens), and so forth.

In base ten, we had digits 0 through 9. In base eight, we had digits 0 through 7. In base 4, we had digits 0 through 3. In any base system, you will have digits 0 through one-less-than-your-base. This means that, in hexadecimal, we need to have "digits" 0 through 15. To do this, we would need single solitary digits that stand for the values of "ten", "eleven", "twelve", "thirteen", "fourteen", and "fifteen". But we don't. So, instead, we use letters. That is, counting in hexadecimal, the sixteen "numerals" are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F In other words, A is "ten" in "regular" numbers, B is "eleven", C is "twelve", D is "thirteen", E is "fourteen", and "F" is fifteen. It is this use of letters for digits that makes hexadecimal numbers look so odd at first.

But the conversions work in the usual manner. Convert 35710 to the corresponding hexadecimal number. Here, I will divide repeatedly by 16, keeping track of the remainders as I go. (You might want to use some scratch paper for this.) Reading off the digits, starting from the top and wrapping around the right-hand side, I see that: 35710 = 16516. Convert 16516 to the corresponding decimal number. List the digits, and count them off from the RIGHT, starting with zero: digits: As expected, F9BD16 = 6393310. Computer Graphics If you work on web pages and graphics programs, you may find it helpful to convert between the RGB values (for an image in your graphics program) and the hexadecimal values (for a matching background color on the web page). Affiliate WyzAnt Tutoring Graphics programs deal with the RGB (red-green-blue) values for colors. Each of these components of a given color have values somewhere between 0 and 255. These values may be converted to hexadecimal values between 00 and FF.

If you list the RGB components of a color as a string of three numbers, you might get, say, R:204, G:51, B:255, which translates into a light-purplish #CC33FF in HTML coding. Note that 20410 = CC16, 5110 = 3316, and 25510 = FF16. Affiliate On the other hand, if you have some coding for #990033, this would translate into a dark-reddish R:153, G:0, B:51 in your graphics program. That is, to convert between your graphics program and your web-page coding, deal with the hexadecimal number not as one six-digit number, but as three two-digit numbers, and convert these pairs of digits into the corresponding RGB values. For a discussion of the history of "web safe" colors, including why they involve only the hexadecimal equivalents of 0, 51, 102, 153, 204, and 255, look here .

For a demonstration of the different text and background colors in HTML, look here . URL: Convert Hexadecimal to Decimal - Steps, Chart & Examples | Math@TutorVista.com Top Convert Hexadecimal to Decimal In mathematics we use decimal numbers, that is, 0 to 9. But in digital system we use binary number system which has only 0 and 1. There are other number systems such as octal and hexadecimal which are used in digital systems. A decimal number has a base of 10, a binary number has a base of 2, an octal number has a base of 8 and a hexadecimal number has a base of 16. Hexadecimal number is way to represent the number, which base is 16. We are going to convert a number in base 16 to an equivalent number in base 10. It is generally use in digital electronic system. A hexadecimal number can be converted to a decimal number and vice versa. Back to Top Numbers to the base 16 is called as hexadecimal numbers. It uses digits from 0 to 9 and alphabets from A to F. That is, it will be a combination of numbers and alphabets.

As in the decimal system we have the the numbers 0 to 9, whereas in hexadecimal we have numbers 0 to 9, as well as numbers 10 to 15 are represented by alphabets A to F respectively. 10 - A, 11 - B, 12 - C, 13 - D, 14 - E, 15 - F A hexadecimal number A67 = $10 \times 16^2 + 6\times 16^1 + 7\times 16^0$. Examples: 1) 6FDA 2) 2A9B 3) EB2 Decimal System Back to Top Numbers with base 10 is called as decimal numbers. Decimal numbers include digits from 0 to 9. Decimal number system is the basic number system and is widely used in everyday mathematics. The decimal numbers work in power of 10. If a number is given as 567, it means $5\times10^2 + 6\times 10^1 + 7\times 10^0 = 500 + 60 + 7$.

The decimal system is used extensively in mathematics but when it comes to digital systems the binary, octal and hexadecimal systems are used. So, it becomes important to know conversion from decimal system to these systems and vice versa. Examples: 1) 25 2) 984 4) 21 How to Convert Hexadecimal to Decimal Back to Top See the below steps - Step 1: First we find the number of hexadecimal digits in the number. Let there be n numbers. Step 2: Then we multiply each hexadecimal digit with $16^{n-1}$, when n is equal to number of position from right side. Step 3: Then we add each number after multiplication. Step 4: The resultant is equivalent hexadecimal number of the given decimal number. If the given number contain decimal then Step 1: Take digits from the right of decimal point. Let the value of the position of digit to the right of decimal be m. Take a hexadecimal number 2.51. Step 2: Multiply each digit after decimal with $\frac{1}{16^m}$, where m is the number of position of digit from the decimal point.

2 = $2\times 16^0$ 5 = $5\times \frac{1}{16}$ 1 = $1\times \frac{1}{16^2}$ Step 3: Add all the obtained numbers.Obtained number is 2 + 0.3125 + 0.00390625 = 2.31640625. Step 4: The resultant is equivalent hexadecimal number of the given decimal number. Converting Hexadecimal to Decimal Examples Below are the examples on converting Hexadecimal to Decimal - Solved Examples Introduction to number systems and binary (video) | Khan Academy Introduction to number systems and binary How to count to 1000 on two hands Video transcript - [Voiceover] For as long as human beings have been around we've been counting things, and we've been looking for ways to keep track and represent those things that we counted.

So, for example if you were an early human and you were trying to keep track of the days since it last rained you might say okay let's see it didn't rain today so one day has gone by, and we now use the word one, but they might have not used it back then. Now another day goes by. Then another day goes by. Then another day goes by. Another day goes by. Another day goes by. Another day goes by, then it rained. And so when his friend comes he says, "Well, how long has it been since we last rained." Well you would say, "Well, this is how many days it's been." And your friend would say, "Okay, I think I have a general sense of that." And at some point they probably realized that it's useful to have names for these. So they would call this one, two, three, four, five, six, seven. Obviously every language in the world has different names for these. I'm sure there are lost languages that had other names for them. But very quickly you start to realize that this is a pretty bulky way of representing numbers. One it takes a long time to write down.

It takes up a lot of space, and then later if someone wants to read the number they have to sit here and count. It's hard enough with seven, but you could imagine if there were what we call 27 of it, or 1000 of it. Then it would take up, possibly, a whole page and even when you counted you might make a mistake. And to solve this human beings have invented number systems. And it's something that we take for granted. You might say, "Oh, isn't that just the way you've always counted? But hopefully over the course of this video you'll start to appreciate the beauty of a number system and to realize our number system isn't the only number system that is around. The number system that most of us are familiar with is the base 10 number system. Often called the decimal, the decimal number system. And why 10? Well probably because we have 10 fingers. Or most of us have 10 fingers. So, it was very natural to think in terms of bundles of 10 or to have 10 symbols. So however many bundles you have you can use your fingers and eventually your symbols to think about how many there are.

And since we needed 10 symbols we came up with zero, one, two, three, four, five, six, seven, eight, nine. These 10 digits, these are our 10 symbols that we use in the base 10 system. To just give us a little bit of a reminder how we use them imagine the number 231. So, 231. 231. What does this represent? Well, what's neat about number systems is we have place value. This place all the way to the right, this is the ones place. This is the ones place. This literally means one, one. One bundle of one. So, this is one, one right over here. This right over here, this is in the 10s place. This is in the 10s place. This three here, literally means three 10s. So this literally means three 10s. And this two here, this two here is in the 100s place. It's in the 100s place. So, this represent two 100s.

You add them together and once again I'm still thinking in base 10, you'd get 231. This is two 100s plus three 10s plus one. In our base 10 system notice every time we move to the left we're thinking in bundles of 10 of the space to the right. So, this is the ones place. You multiply by 10, you go to the 10s place. You want to go to the next place you multiply by 10 again. You get the 100s place. If you're familiar with exponents, one is the same thing as 10 to the zero power. 10 is the same thing as 10 to the first power. So this is the 10s place. Three tens. And 100 is the same thing as 10 to the second power. Obviously we could keep going on and on and on and on and on. That is the power of the base 10 system. So, you might be curious now. "Well, what if this wasn't 10 here? What if we did, let's just go as simple as we can.

You can almost view this as a base one system. You only have one symbol right over here. But what if we went to something slightly more complex, a base two system. You'd be happy to know that not only can we do this, but the base two system often called the binary system. This is called the decimal system. The base two system often called the binary system is the basis of all modern computing. It's the underlying mathematics and operations that computers perform are based on binary. And in binary you have two symbols. You have zero and you have one. The reason why this is useful for computation is because all the hardware that we use to make our modern computers, all of the transistors and the logic gates they either result in an on or an off state. On or an off state. And so what we do is when you use your calculator or whatever you might be operating in base 10, but underlying everything it is doing the operations in binary. But you might say well how do we actually think in terms of binary? Well, we can construct similar places here, but instead of them being powers of 10 they're going to be powers of two. So, let's set up some places here.

So, all the way on the right two to the zero power is still one. So we can still call that the ones place. Then we can move to the left of that. We can move to the left of that. That would be two to the first power. So we could call that the twos place, and I can even write it out if I want. Twos place instead of the 10s place. Then I could keep going. Instead of this space being the 10 to the second or the 100s place, it will be the two to the second, or the fours place. And I can keep going. I encourage you actually to pause the video and try to build this out for yourself. What would this be? Well this would be two to the third, or the eights place. Notice every time we're doing this we're multiplying by two. Everytime we go to the left, just like we multiplied by 10 here. So notice everywhere you see this 10s we're now dealing with twos. Let's keep going. Let's keep going and then we can actually represent this number using binary. So, let's do that.

So, this right over here I've already used that color. This right over here, this is two to the fourth. We could call that the 16s place. Then we could have -- I'll reuse some of these colors. This is two to the fifth. We could call this the 32s place. Then we can go two to the sixth. We can call that, multiply by two again, or two to the six is 64. So this is the 64s place. Tells us how many 64s we have. Zero or one 64s. We'll see that in a second. Then we can go over here. This would be two to the seventh. That would be the 128s place. And we can obviously keep going on and on and on, but this should be enough for me to represent this number. In future videos I will show you how to do that, but let's actually represent the number. It turns out that this number in decimal can be represented as 11100111 in binary. What does this mean?

This means you have one 128 plus one 64, plus one 32, plus no 16s, plus no eights, plus one four, plus one two, plus one one. So you can see that these are going to be the same thing. Notice, this is one 128. So it's 128, plus 64, plus 32. We have zero 16s, zero eights. So we're not going to add those. Plus four, one four. Plus one two. Plus one one. And add these together, and once again when we're doing this, when I'm writing it this way I'm kind of using the number system that we're most familiar with. We're most used to doing the operations in, but when you do it you will see that this is the exact same number as 231. This is just another representation. One isn't better than the other. The only reason why I converted this is this is what I'm used to thinking in. It's what I'm used to doing operations in. So, hopefully you find that pretty interesting.

To me, this kind of opened my mind to the power of even our decimal system. In future videos we'll explore other number systems. The most used ones, base 10 is used very heavily, binary and there's also hexadecimal where you don't have two digits or not 10 digits, but you have 16 digits. And we'll explore those in future videos and how to convert between or rewrite the the different representations and different bases. Number Bases: Introduction & Binary Numbers | Purplemath Number Bases: Introduction & Binary Numbers Intro & Binary Base-4 & Base-7 Octal & Hex Purplemath Converting between different number bases is actually fairly simple, but the thinking behind it can seem a bit confusing at first. And while the topic of different bases may seem somewhat pointless to you, the rise of computers and computer graphics has increased the need for knowledge of how to work with different (non-decimal) base systems, particularly binary systems (ones and zeroes) and hexadecimal systems (the numbers zero through nine, followed by the letters A through F). Content Continues Below MathHelp.com In our customary base-ten system, we have digits for the numbers zero through nine. We do not have a single-digit numeral for "ten".

(The Romans did , in their character "X".) Yes, we write "10", but this stands for "1 ten and 0 ones". This is two digits; we have no single solitary digit that stands for "ten". Instead, when we need to count to one more than nine, we zero out the ones column and add one to the tens column. When we get too big in the tens column -- when we need one more than nine tens and nine ones ("99"), we zero out the tens and ones columns, and add one to the ten-times-ten, or hundreds, column. The next column is the ten-times-ten-times-ten, or thousands, column. And so forth, with each bigger column being ten times larger than the one before. We place digits in each column, telling us how many copies of that power of ten we need. Affiliate The only reason base-ten math seems "natural" and the other bases don't is that you've been doing base-ten since you were a child. And (nearly) every civilization has used base-ten math probably for the simple reason that we have ten fingers.

If instead we lived in a cartoon world, where we would have only four fingers on each hand (count them next time you're watching TV or reading the comics), then the "natural" base system would likely have been base-eight, or "octal". Binary Advertisement Let's look at base-two, or binary, numbers. How would you write, for instance, 1210 ("twelve, base ten") as a binary number? You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or "places" for 100 = 1, 101 = 10, 102 = 100, 103 = 1000, and so forth. Similarly in base two, you have columns or "places" for 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, and so forth. The first column in base-two math is the units column. But only "0" or "1" can go in the units column.

When you get to "two", you find that there is no single solitary digit that stands for "two" in base-two math. Instead, you put a "1" in the twos column and a "0" in the units column, indicating "1 two and 0 ones". The base-ten "two" (210) is written in binary as 102. A "three" in base two is actually "1 two and 1 one", so it is written as 112. "Four" is actually two-times-two, so we zero out the twos column and the units column, and put a "1" in the fours column; 410 is written in binary form as 1002. Here is a listing of the first few numbers: ← swipe to view full table → decimal Then 1011001012 converts to 35710. Converting decimal numbers to binaries is nearly as simple: just divide by 2. Convert 35710 to the corresponding binary number. To do this conversion, I need to divide repeatedly by 2, keeping track of the remainders as I go.

Watch below: The above graphic is animated on the "live" web page. As you can see, after dividing repeatedly by 2, I ended up with these remainders: These remainders tell me what the binary number is. I read the numbers from around the outside of the division, starting on top with the final value and its remainder, and wrapping my way around and down the right-hand side of the sequential division. Then: 35710 converts to 1011001012. WyzAnt Tutoring Affiliate This method of conversion will work for converting to any non-decimal base. Just don't forget to include that first digit on the top, before the list of remainders. If you're interested, an explanation of why this method works is available here . You can convert from base-ten (decimal) to any other base. When you study this topic in class, you will probably be expected to convert numbers to various other bases, so let's look at a few more examples ... URL: Number Systems and Bases – BetterExplained Home › Programming ›Number Systems and Bases Base systems like binary and hexadecimal seem a bit strange at first.

The key is understanding how different systems “tick over” like an odometer when they are full. Base 10, our decimal system, “ticks over” when it gets 10 items, creating a new digit. We wait 60 seconds before “ticking over” to a new minute. Hex and binary are similar, but tick over every 16 and 2 items, respectively. Try converting numbers to hex and binary here: Way back when: Unary Numbers Way back in the day, we didn’t have base systems! It was uphill both ways, through the snow and blazing heat. When you wanted to count one, you’d write: l When you wanted 5, you’d write lllll And clearly, 1 + 5 = 6 l + lllll = llllll This is the simplest way of counting. Enter the Romans In Roman numerals, two was one, twice. Three was one, thrice: one = I two = II three = III However, they decided they could do better than the old tradition of lines in the sand. For five, we could use V to represent lllll and get something like l + V = Vl Not bad, eh?

And of course, there are many more symbols (L, C, M, etc.) you can use. The key point is that V and lllll are two ways of encoding the number 5. Give each number a name Another breakthrough was realizing that each number can be its own distinct concept. Rather than represent three as a series of ones, give it its own symbol: “3″. Do this from one to nine, and you get the symbols: 1 2 3 4 5 6 7 8 9 The Romans were close, so close, but only gave unique symbols to 5, 10, 50, 100, 1000, etc. Use your position Now clearly, you can’t give every number its own symbol. There’s simply too many But notice one insight about Roman numerals: they use position of symbols to indicate meaning. IV means “subtract 1 from 5″ and VI means “add 1 to 5″. In our number system, we use position in a similar way. We always add and never subtract. And each position is 10 more than the one before it.

So, 35 means “add 3*10 to 5*1″ and 456 means 4*100 + 5*10 + 6*1. Our choice of base 10 Why did we choose to multiply by 10 each time? Most likely because we have 10 fingers. One point to realize is you need enough digits to “fill up” until you hit the next number. Let me demonstrate. If we want to roll the odometer over every 10, so to speak, we need symbols for numbers one through nine; we haven’t reached ten yet. Imagine numbers as ticking slowly upward – at what point do you flip over the next unit and start from nothing? Enter zero And what happens when we reach ten? How do we show we want exactly one “ten” and nothing in the “ones” column? We use zero, the number that doesn’t exist. Zero is quite a concept, it’s a placeholder, a blank, a space, and a whole lot more. Suffice it to say, Zero is one of the great inventions of all time. Zero allows us to have an empty placeholder, something the Romans didn’t have.

Look how unwieldly their numbers are without it. George Orwell’s famous novel “1984″ would be “MCMLXXXIV”! Rolls right off the tongue, doesn’t it? Considering other bases Remember that we chose to roll over our odometer every ten. Our counting looks like this: 1 2 3 4 5 6 7 8 9 (uh oh, I’m getting full!) 10 (ticked over – start a new digit) What if we ticked over at 60 when we counted, like we do for seconds and minutes? 1 second 2 3 4 5 … 58 59 1:00 (60 seconds aka 1 minute. We’ve started a new digit.) Everything OK so far, right? Note that we use the colon (:) indicate that we are at a new “digit”. In base 10, each digit can stand on its own.

Try Base 16 If we want base 16, we could do something similar: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (we’re getting full) 1:00 (16 – we’ve started a new digit) However, we don’t want to write hexadecimal numbers with the colon notation (even though we could). We’d rather cook up separate symbols for 10-15 so we can just write numbers like we’re used to. We’ve run out of numbers (1-9 already used, with 0 as a placeholder) so we need some other symbols. We could use some squiggly lines or other shapes, but the convenions is to use letters, Roman style. Just like 5 became V, programmers use letters A-F to get enough digits up to 16.

That is, 1 2 3 4 5 6 7 8 9 A (10 – we’re using the symbol “A”) B (11) C (12) D (13) E (14) F (15 – uh oh, we’re getting full) 10 (16 – we start a new digit) Ahah! Now we can use one digit per “place”, and we know that 10 actually means we’ve “ticked over to 16″ once. 20 means we’ve ticked over to 16 twice (32). 25 means we’ve ticked over to 16 twice (giving us 32) and gone an extra 5. The total is 32 + 5 = 37. Quick review With me so far? This is pretty cool, right? We can count in any system we want. Also notice that base 16 is more “space efficient” in the sense we can write a number like 11 in a single digit: B. Base 16 really isn’t that different from base 10, we just take longer to fill up.

The wonderful world of binary We’ve seen plenty of base systems, from over-simple unary, to the unwiedly Roman numerals, the steady-going base 10 and the compact base 16. What’s great about binary? In the spirit of keeping things simple, it’s the simplest number system that has the concept of “ticking over”. Unary, where we just write 1, 11, 111… just goes on forever. Binary, with two options (1 and 0) looks like this: 1: 1 2: 10 (we’re full – tick over) 3: 11 4: 100 (we’re full again – tick over) 5: 101 6: 110 7: 111 8: 1000 (tick over again) … and so on. Because binary is so simple, it’s very easy to build in hardware. You just need things that can turn on or off (representing 1 and 0), rather than things that have 10 possible states (to represent decimal). Because it’s so simple, binary is also resistant to errors.

If your signal is “partially on” (let’s say 0.4), you can assume that’s a zero. And if it’s mostly on (say 0.8), then you can assume it’s a 1. If you’re using a system with 10 possible states, it’s difficult to tell when an error occurred. This is one reason digital signals are so resilient to noise. Other examples of bases We use other bases all the time, even dynamically changing bases. We usually don’t think of it that way: Hours, minutes, seconds: 1:32:04 We know this is 1 hour, 32 minutes, 4 seconds. In seconds, this is 16060 + 32*60 + 4. Feet and inches: 3′ 5″ This is 3 feet, 5 in or 3 * 12 + 5 inches. Pounds and ounces: 8 lbs, 5 oz Since a pound is 16 oz, This is 8 * 16 + 5 oz. We’ve been using a base 16 number system all along!

Parting thoughts “10″ in any number system indicates the base, and means we’ve ticked over once. 10 in binary means two, 10 in decimal means ten, and 10 in hexadecimal is sixteen. How do you keep these numbers apart? Programmers will often write “0b” in front of binary numbers. So 2 in binary is 0b10 Similarly, they’ll write 0x in front of hex numbers. So 16 in hex is: 0×10 If there aren’t any symbols (0b or 0x) in front, we assume it’s base 10, a regular number. Now go forth and enjoy your new knowledge!

Question: In maths the hexadecimal numeral system has a base of which number?
Answer: 16