Converting a number from any number system to a decimal number is quite easy. Using that rule, we can convert a number from any number system to a decimal number. The generalized format of converting this number to decimal number is written as:.
For example, if we want to covert a base 3 number to a base 7 number, first step is to convert a base 3 number to decimal and then to convert this decimal number to a base 7 number.
We can use the same procedure to convert a binary number to octal or hexadecimal number, but there are some useful shortcuts that will make the process easier. To convert a binary number to a hexadecimal number, we could simply break a binary number into groups of 4 digits starting at the right and adding leading zeros if one runs out of digits and then reinterpreting these groups of 4 as hexadecimal values listed in the table above.
That being said, we have:. Similar to this, to convert a binary number to an octal number, we could simply break a binary number into groups of 3 digits and the rest of the procedure is the same as converting binary number to a hexadecimal number.
Reversing the process is even easier. From the table we can read binary values for each digit of the hexadecimal number:. Twitter response:. Non-Positional Number System To explain non-positional number system, we will take Roman numerals as an example. You may ask is there some pattern to form all of the other symbols? The answer is yes. When a symbol with a smaller value is placed after a symbol having an equal or larger value, the values are added.
Examples are shown in the table below. When a symbol with a smaller value is placed before a symbol having a larger value, the smaller value is subtracted from the larger one. Positional Number System A positional number system allows the expansion of the original set of symbols so that they can be used to represent any arbitrarily large or small value. Binary Number System The binary number system contains two unique numerals 0 and 1. Octal Number System The octal number system contains 8 unique numerals 0, 1, 2, 3, 4, 5, 6, 7.
Hexadecimal Number System The hexadecimal number system contains 16 unique numerals. Converting to Decimal Converting a number from any number system to a decimal number is quite easy. Example 1. Converting from Decimal to any other base We can convert a decimal number to any other base using just a few simple steps: Divide the decimal number to be converted by the value of the new base.
Write a remainder aside Divide the quotient of the previous divide by the new base. Write a remainder aside Repeat Steps 3 and 4 until the quotient becomes zero in Step 3. The required number is then consisting of remainders written from bottom to top, left to right. Example 2. Convert 25 to a binary number.
Example 3. Convert to a hexadecimal number. Remember, equivalent to number 11 in hexadecimal number system is letter B. Equivalence between different number systems. The Magic Square Learn about the History of pi The number Pi has Most downloaded worksheets Vectors measurement of angles What is Mathemania? Contact Us. This website uses cookies to ensure you get the best experience on our website.
Got it! More info. Click here to sign up. Download Free PDF. Binary, octal, decimal and hexadecimal numbers. Usman Itopa. A short summary of this paper. A digital computer contains elements that can be in either of two states: on or off, magnetized or not magnetized, and so on. For such devices, calculations are most conveniently done using binary numbers. In this chapter we learn what binary numbers are and how to convert between binary and decimal numbers. However, binary numbers are hard to read, partly because of their great length.
To represent a nine-digit Social Security number, for example, requires a binary number 29 bits long. So, in addition to binary numbers, we also study other ways in which num- bers can be represented. Hexadecimal, octal, and binary-coded decimal systems allow us to express binary numbers more compactly, and they make the transfer of data between computers and people much easier. Here we learn how to convert numbers between each of these systems, and between decimal and binary as well.
A binary number having a fractional part contains a binary point also called a radix point , as in the number A byte is a group of 8 bits, and a word is the largest string of bits that a computer can handle in one operation. The number of bits in a word is called the word length. Different computers have dif- Note that this is different from the usual ferent word lengths, with 8, 16, or 32 bits being common for desktop or personal meaning of these prefixes, where kilo computers. The longer words are often broken down into bytes for easier handling.
Writing Binary Numbers A binary number is sometimes written with a subscript 2 when there is a chance that the binary number would otherwise be mistaken for a decimal number.
The value of the entire number is then the sum of these products. Thus it took 15 binary digits to represent 5 decimal dig- its. As a rule of thumb, it takes about 3 bits for each decimal digit. Stated another way, if we have a computer that stores numbers with 15 bits, we should assume that the decimal numbers that it prints do not contain more than 5 significant digits.
We then repeat this process until the quotient is zero. Solution: We divide 59 by 2, getting a quotient of 29 and a remainder of 1. Then dividing 29 by 2 gives a quotient of 14 and a remainder of 1. To convert a decimal fraction to binary, we first multiply it by 2, remove the integer part of the product, and multiply by 2 again. We then repeat the procedure. Solution: We multiply the given number by 2, getting 1. We remove the integer part, 1, leaving 0. We repeat the computation until we get a product that has a fractional part of zero, as in the following table: Integer Part 0.
The column Note that this is the reverse of what we containing the integer parts is now our binary number, with the digits at the top did when converting a decimal integer appearing at the left of the binary number. So to binary. This is not always possible, as shown in the following example. Solution: We follow the same procedure as before and get the following values.
Integral Part 0. This inability to make an exact conversion is an unavoidable source of inaccuracy in some compu- tations. To decide how far to carry out computation, we use the rule of thumb that each decimal digit requires about 3 binary bits to give the same accuracy.
Thus our original 3-digit number requires about 9 bits, which we already have. The result of our conversion is then 0. Solution: From Table 1, 0.
Retain 8 bits to the right of the binary point when the conversion is not exact. Keep 8 bits to the right of the binary point when the conversion is not exact. Some calculators can convert directly between binary and decimal numbers.
Some will do this only for integers. Check your calculator manual to see if you can perform these operations, and if you have this capability, use it to solve any of the problems in this exercise set. A hex number one-fourth the length of the binary number is thus obtained.
Base 16 Since a 4-bit group of binary digits can have a value between 0 and 15, we need 16 symbols to represent all of these values. The base of hexadecimal numbers is thus We use the digits from 0 to 9 and the capital letters A to F, as shown in Table 2. Decimal Binary Hexadecimal Octal 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 Table 2 also shows octal numbers, 8 8 10 which we will use later.
Then assign to each group the appropriate letter or number from Table 2. Solution: Grouping from the binary point gives us 10 E to binary. Solution: We write the group of 4 bits corresponding to each hexadecimal symbol. E To convert from hex to decimal, first replace each letter in the hex number by its decimal equivalent. Write the number in expanded notation, multiplying each hex digit by its place value.
Add the resulting numbers. F to decimal. Solution: We replace the hex B with 11, and the hex F with 15, and write the num- ber in expanded form.
The remainders form our hex number, the last remainder obtained being the most significant digit. B2
0コメント