Binary Number System

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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. Counting in binary number system you know, the decimal system uses counting in binary number system digits to represent numbers.

The binary system works under the exact same principles as the decimal system, only it operates in base counting in binary number system 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 carriedand counting in binary number system 1 is recorded in the one's column of the sum.

Thus, the answer is Binary addition works on the same principle, counting in binary number system 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 counting in binary number system 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 counting in binary number system 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 counting in binary number system 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 Counting in binary number system 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 counting in binary number system, we would simply put a "1" rather than a "0" as the first bit: Counting in binary number system 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 beand 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 asand as To verify this, let's subtract 1 fromto get If we flip the bits, we getor 12 in decimal. In this notation, "m" indicates the total number of bits. Then convert back to decimal numbers.

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The binary number system is an alternative to the decimal base number system that we use every day. Binary numbers are important because using them instead of the decimal system simplifies the design of computers and related technologies.

The simplest definition of the binary number system is a system of numbering that uses only two digits—0 and 1—to represent numbers, instead of using the digits 1 through 9 plus 0 to represent numbers. To translate between decimal numbers and binary numbers, you can use a chart like the one to the left. Notice how 0 and 1 are the same in either system, but starting at 2, things change. For example, decimal 2 looks like 10 in the binary system. The 0 equals zero as you would expect, but the 1 actually represents 2.

In every binary number, the first digit starting from the right side can equal 0 or 1. But if the second digit is 1, then it represents the number 2. If it is 0, then it is just 0. The third digit can equal 4 or 0. The fourth digit can equal 8 or 0. If you write down the decimal values of each of the digits and then add them up, you have the decimal value of the binary number.

In the case of binary 11, there is a 1 in the first position, which equals 1 and then another 1 in the second position, so that equals 2. As numbers get larger, new digits are added to the left. To determine the value of a digit, count the number of digits to the left of it, and multiply that number times 2.

For example, for the digital number , to determine the value of the 1, count the number of digits to the left of the 1 and multiply that number times 2. The total value of binary is 4, since the numbers to the left of the 1 are both 0s. Now you know how to count digital numbers, but how do you add and subtract them? Binary math is similar to decimal math. Adding binary numbers looks like that in the box to the right above.

To add these binary numbers, do this: Start from the right side, just as in ordinary math. Write a 1 down in the solution area. According to our rule, that equals 0, so write 0 and carry the 1 to the next column.

Any time you have a column that adds up to decimal 3, you write down a 1 in the solution area and carry a 1. In the fifth column you have only the 1 that you carried over, so you write down 1 in the fifth column of the solution. Computers rely on binary numbers and binary math because it greatly simplifies their tasks.

Since there are only two possibilities 0 and 1 for each digit rather than 10, it is easier to store or manipulate the numbers. Computers need a large number of transistors to accomplish all this, but it is still easier and less expensive to do things with binary numbers rather than decimal numbers. The original computers were used primarily as calculators, but later they were used to manipulate other forms of information, such as words and pictures. In each case, engineers and programmers sat down and decided how they were going to represent a new type of information in binary form.

The chart shows the most popular way to translate the alphabet into binary numbers only the first six letters are shown. Although it is pretty complicated to do so, sounds and pictures can also be converted into binary numbers, too.

The result is a huge array of binary numbers, and the volume of all this data is one reason why image files on a computer are so large, and why it is relatively slow to view video or download audio over an internet connection.

Binary Numbers and Binary Math. Retrieved from " http: