How the binary number system works
To understand binary numbers, begin by recalling elementary school math. When we first learned how the binary number system works 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 carriedand 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. How the binary number system works 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 How the binary number system works. 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 how the binary number system works 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, 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 how the binary number system works gives We can divide by two again to get This is even, so we put a 0 in the 8's column. How the binary number system works we already knew how to how the binary number system works 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 indicatewe 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 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.
This is called the decimal number system and has base 10which means how the binary number system works this number system has 10 how the binary number system works digits to construct a number. But computers do not understand the decimal number system. Actually it's not their fault!
We humans have created them that way! Unlike humans, the insides of computers know only 2 digits - 0 and 1because in the simplest electrical systems, electricity can only be "on" or "off.
All the numbers are constructed with only 2 digits - 0 and 1. A digit in binary that's a 0 or a 1 is also called a bit — which means bi nary digi t. Computers use this number system to add, subtract, multiply, divide and do all their other maths. They even save data in the form of bits - well, they group them together into chunks of 8 bits.
And don't forget, this chunk of 8 bits is called a byte. In normal maths, we don't use binary. We were taught to use our how the binary number system works number system. Binary is much easier to do math in than normal numbers because you only are using how the binary number system works symbols - 1 and 0 instead of ten symbols - 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Computers use binary because they can only read and store an on or off charge. So, using 0 as "off" and 1 as "on," we can use numbers in electrical wiring. Think of it as this - if you had one color for every math symbol 0 to 9you'd have ten colors.
That's a lot of colors to memorize, but you have done it anyway. If you were limited to only black and white, you'd only have two colors.
It would be so much easier to memorize, but you would need to make a new way of writing down numbers. Binary is just that - a new way to record and use numbers which is true. In school, you were taught that we have a ones, tens, hundreds columns and so on they multiply by Binary also has columns, but they aren't ones and tens. The columns in binary are Binary is called base-2, because it uses two symbols. So what makes binary so easy? The answer lies in how we read the number.
If we had the number 52, we have a 2 in the ones column, adding 2 times 1 to the total 2. We have a 5 in the 10s column, multiply that together and get 50, adding that to the total.
Our total number is 52, like we expect. In binary, though, this is way simpler if you know how to read it fast. How do you write the number how the binary number system works from the base into a base-2 number? What about the rest?
Let's try writing the normal numbers from 1 to 10 in binary form, shall we? We said we'd write only the numbers from 1 to 10 into binary numbers, but look at that table!
It was so easy, we ended up converting the numbers until 16! But take note of how the binary numbers of 1, 2, how the binary number system works, 8 and 16 match the previous table above. Have you noticed a pattern in writing binary numbers? Study the table for 1 to 16 again until you understand why in binary. We have been trained to read these base numbers really quickly. Reading binary for humans is slower since we are used to base You are now only just starting to learn how to read base-2, so it will be slow.
You will get faster over time! We now have the number 52 as our total. The basics of reading a base-2 number is add each columns value to the total if there is a 1 in it. You don't have to multiply like you do in base to get the total like the 5 in the tens column from the above base examplewhich can speed up your reading of base-2 numbers.
Let's look at that in a table. The binary number isbut we don't know what it is. Let's go through the column-reading process to find out what the number is.
We are done, so the total is the answer. The answer is 11! Here are some more numbers for how the binary number system works to work out. Computers remember everything in binary. For example, if your name is "GEORGE" then the computer has some special binary word to store your name with only 0s and 1s. This is just like using sign language.
Every combination of gestures can mean a special word or number. Exactly in this manner, the computer has a different set of combinations for each letter or digit.
A bit defines a binary dual state On or Off, 0 or 1, True or False which can not be broken into more smaller units.
The name is short for binary digit, like the '1' in the binary number '10' representing decimal '2' in a similar fashion to how a decimal digit number that can have 10 distinct values how the binary number system works the default decimal base reference work, for instance the '9' and '8' that are part of the '98' decimal number. A byte is eight bits put together or a group of 8 bits.
It has to do with remembering letters, which you shall read later. From Wikibooks, open books for an open world. How Things Work This is the latest reviewed versionchecked on 25 How the binary number system works Retrieved from " https: How the binary number system works Read Edit View history.