Binary Numbers Overview
Binary Numbers Overview Binary is a number system used by digital devices like computers, cd players, etc.
Binary is Base 2, unlike our counting system decimal which is Base 10 (denary).
In other words, Binary has only 2 different numerals (0 and 1) to denote a value, unlike Decimal which has 10 numerals (0,1,2,3,4,5,6,7,8 and 9).
Here is an example of a binary number: 10011100
As you can see it is simply a bunch of zeroes and ones, there are 8 numerals in all which make this an 8 bit binary number. Bit is short for Binary Digit, and each numeral is classed as a bit.
The bit on the far right, in this case a 0, is known as the Least significant bit (LSB).
The bit on the far left, in this case a 1, is known as the Most significant bit (MSB)
notations used in digital systems:
4 bits = Nibble
8 bits = Byte
16 bits = Word
32 bits = Double word
64 bits = Quad Word (or paragraph)
When writing binary numbers you will need to signify that the number is binary (base 2), for example, let's take the value 101. As it is written, it would be hard to work out whether it is a binary or decimal (denary) value. To get around this problem it is common to denote the base to which the number belongs, by writing the base value with the number, for example:
1012 is a binary number and 10110 is a decimal (denary) value.
Once we know the base then it is easy to work out the value, for example:
1012 = 1*22 + 0*21 + 1*20 = 5 (five)
10110 = 1*102 + 0*101 + 1*100 = 101 (one hundred and one)
One other thing about binary numbers is that it is common to signify a negative binary value by placing a 1 (one) at the left hand side (most significant bit) of the value. This is called a sign bit, we will discuss this in more detail in the next part of the tutorial.
Electronically binary numbers are stored/processed using off or on electrical pulses, a digital system will interpret these off and on states as 0 and 1. In other words if the voltage is low then it would represent 0 (off state), and if the voltage is high then it would represent a 1 (on state).
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Converting binary to decimal To convert binary into decimal is very simple and can be done as shown below:
Say we want to convert the 8 bit value 10011101 into a decimal value, we can use a formula like that below:
128 64 32 16 8 4 2 1 1 0 0 1 1 1 0 1
As you can see, we have placed the numbers 1, 2, 4, 8, 16, 32, 64, 128 (powers of two) in reverse numerical order, and then written the binary value below.
To convert, you simply take a value from the top row wherever there is a 1 below, and then add the values together.
For instance, in our example we would have 128 + 16 + 8 + 4 + 1 = 157.
For a 16 bit value you would use the decimal values 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 (powers of two) for the conversion.
Because we know binary is base 2 then the above could be written as:
1*27 + 0*26 + 0*25 + 1*24 + 1*23 + 1*22 + 0*21 + 1*20 = 157. Converting decimal to binary To convert decimal to binary is also very simple, you simply divide the decimal value by 2 and then write down the remainder, repeat this process until you cannot divide by 2 anymore, for example let's take the decimal value 157:
157 ÷ 2 = 78
78 ÷ 2 = 39
39 ÷ 2 = 19
19 ÷ 2 = 9
9 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
1 ÷ 2 = 0 with a remainder of 1
with a remainder of 0
with a remainder of 1
with a remainder of 1
with a remainder of 1
with a remainder of 0
with a remainder of 0
with a remainder of 1 <--- to convert write this remainder first.
Next write down the value of the remainders from bottom to top (in other words write down the bottom remainder first and work your way up the list) which gives:
10011101 = 157
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Definition: The hexadecimal number system (also called base-16) is a number system that uses 16 unique symbols to represent a particular value. Those symbols are 0-9 and A-F. The number system that we use in daily life is called the decimal, or base-10, system and uses the symbols 0-9 to represent a value.
Many error messages and other various values used inside a computer are represented in the hexadecimal format.
For example, error codes called STOP codes, that display on a Blue Screen of Death, are always in hexadecimal format.
How To Count in Hexadecimal Counting in hexadecimal format is easy as long as you remember that there are 16 characters that make up each set of numbers.
In decimal format, we all know that we count like this:
0,1,2,3,4,5,6,7,8,9,10,11,12,13,... adding a 1 before beginning the set of 10 numbers over again (i.e. the number 10)
In hexadecimal format however, we count like this, including all 16 numbers:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13... again, adding a 1 before beginning the 16 number set over.
Here are a few examples of some tricky hexadecimal "transitions" that you might find helpful:
...17, 18, 19, 1A, 1B...
...1E, 1F, 20, 21, 22...
...FD, FE, FF, 100, 101, 102...
Also Known As: hex, base-16
http://pcsupport.about.com/od/termshm/g/hexadecimal.htm
Binary and hexadecimal numbers are two alternatives to the traditional decimal numbers we use in daily life. Critical elements of computer networks like addresses, masks, and keys all involve binary or hexadecimal numbers. Understanding how such binary and hexadecimal numbers work is essential in building, troubleshooting, and programming any network. Bits and Bytes This article series assumes a basic understanding of computer bits and bytes. Binary and hexadecimal numbers are the natural mathematical way to work with the data stored in bits and bytes. Binary Numbers and Base Two Binary numbers all consist of combinations of the two digits '0' and '1'. These are some examples of binary numbers:
Converting From Binary to Decimal Numbers All binary numbers have equivalent decimal representations and vice versa. Our handy Binary-Decimal Number Converter performs these calculations automatically for you. To convert binary and decimal numbers manually, you must apply the mathematical concept of positional values. The positional value concept is simple: With both binary and decimal numbers, the actual value of each digit depends on its position (how "far to the left") within the number.
For example, in the decimal number 124, the digit '4' represents the value "four," but the digit '2' represents the value "twenty," not "two." The '2' represents a larger value than the '4' in this case because it lies further to the left in the number.
Likewise in the binary number 1111011, the rightmost '1' represents the value "one," but the leftmost '1' represents a much higher value ("sixty-four" in this case).
In mathematics, the base of the numbering system determines how much to value digits by position. For base-ten decimal numbers, multiply each digit on the left by a progressive factor of 10 to calculate its value. For base-two binary numbers, multiply each digit on the left by a progressive factor of 2. Calculations always work from right to left.
In the above example, the decimal number 123 works out to:
For example, the decimal number 109 converts to binary as follows:
http://compnetworking.about.com/cs/basicnetworking/a/binaryhexnumber.htm
Binary is Base 2, unlike our counting system decimal which is Base 10 (denary).
In other words, Binary has only 2 different numerals (0 and 1) to denote a value, unlike Decimal which has 10 numerals (0,1,2,3,4,5,6,7,8 and 9).
Here is an example of a binary number: 10011100
As you can see it is simply a bunch of zeroes and ones, there are 8 numerals in all which make this an 8 bit binary number. Bit is short for Binary Digit, and each numeral is classed as a bit.
The bit on the far right, in this case a 0, is known as the Least significant bit (LSB).
The bit on the far left, in this case a 1, is known as the Most significant bit (MSB)
notations used in digital systems:
4 bits = Nibble
8 bits = Byte
16 bits = Word
32 bits = Double word
64 bits = Quad Word (or paragraph)
When writing binary numbers you will need to signify that the number is binary (base 2), for example, let's take the value 101. As it is written, it would be hard to work out whether it is a binary or decimal (denary) value. To get around this problem it is common to denote the base to which the number belongs, by writing the base value with the number, for example:
1012 is a binary number and 10110 is a decimal (denary) value.
Once we know the base then it is easy to work out the value, for example:
1012 = 1*22 + 0*21 + 1*20 = 5 (five)
10110 = 1*102 + 0*101 + 1*100 = 101 (one hundred and one)
One other thing about binary numbers is that it is common to signify a negative binary value by placing a 1 (one) at the left hand side (most significant bit) of the value. This is called a sign bit, we will discuss this in more detail in the next part of the tutorial.
Electronically binary numbers are stored/processed using off or on electrical pulses, a digital system will interpret these off and on states as 0 and 1. In other words if the voltage is low then it would represent 0 (off state), and if the voltage is high then it would represent a 1 (on state).
© Copyright 2001-2013 helpwithpcs.com
Converting binary to decimal To convert binary into decimal is very simple and can be done as shown below:
Say we want to convert the 8 bit value 10011101 into a decimal value, we can use a formula like that below:
128 64 32 16 8 4 2 1 1 0 0 1 1 1 0 1
As you can see, we have placed the numbers 1, 2, 4, 8, 16, 32, 64, 128 (powers of two) in reverse numerical order, and then written the binary value below.
To convert, you simply take a value from the top row wherever there is a 1 below, and then add the values together.
For instance, in our example we would have 128 + 16 + 8 + 4 + 1 = 157.
For a 16 bit value you would use the decimal values 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 (powers of two) for the conversion.
Because we know binary is base 2 then the above could be written as:
1*27 + 0*26 + 0*25 + 1*24 + 1*23 + 1*22 + 0*21 + 1*20 = 157. Converting decimal to binary To convert decimal to binary is also very simple, you simply divide the decimal value by 2 and then write down the remainder, repeat this process until you cannot divide by 2 anymore, for example let's take the decimal value 157:
157 ÷ 2 = 78
78 ÷ 2 = 39
39 ÷ 2 = 19
19 ÷ 2 = 9
9 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
1 ÷ 2 = 0 with a remainder of 1
with a remainder of 0
with a remainder of 1
with a remainder of 1
with a remainder of 1
with a remainder of 0
with a remainder of 0
with a remainder of 1 <--- to convert write this remainder first.
Next write down the value of the remainders from bottom to top (in other words write down the bottom remainder first and work your way up the list) which gives:
10011101 = 157
http://www.helpwithpcs.com/courses/binary-numbers.htm
Definition: The hexadecimal number system (also called base-16) is a number system that uses 16 unique symbols to represent a particular value. Those symbols are 0-9 and A-F. The number system that we use in daily life is called the decimal, or base-10, system and uses the symbols 0-9 to represent a value.
Many error messages and other various values used inside a computer are represented in the hexadecimal format.
For example, error codes called STOP codes, that display on a Blue Screen of Death, are always in hexadecimal format.
How To Count in Hexadecimal Counting in hexadecimal format is easy as long as you remember that there are 16 characters that make up each set of numbers.
In decimal format, we all know that we count like this:
0,1,2,3,4,5,6,7,8,9,10,11,12,13,... adding a 1 before beginning the set of 10 numbers over again (i.e. the number 10)
In hexadecimal format however, we count like this, including all 16 numbers:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13... again, adding a 1 before beginning the 16 number set over.
Here are a few examples of some tricky hexadecimal "transitions" that you might find helpful:
...17, 18, 19, 1A, 1B...
...1E, 1F, 20, 21, 22...
...FD, FE, FF, 100, 101, 102...
Also Known As: hex, base-16
http://pcsupport.about.com/od/termshm/g/hexadecimal.htm
Binary and hexadecimal numbers are two alternatives to the traditional decimal numbers we use in daily life. Critical elements of computer networks like addresses, masks, and keys all involve binary or hexadecimal numbers. Understanding how such binary and hexadecimal numbers work is essential in building, troubleshooting, and programming any network. Bits and Bytes This article series assumes a basic understanding of computer bits and bytes. Binary and hexadecimal numbers are the natural mathematical way to work with the data stored in bits and bytes. Binary Numbers and Base Two Binary numbers all consist of combinations of the two digits '0' and '1'. These are some examples of binary numbers:
- 1
10
1010
11111011
11000000 10101000 00001100 01011101
Converting From Binary to Decimal Numbers All binary numbers have equivalent decimal representations and vice versa. Our handy Binary-Decimal Number Converter performs these calculations automatically for you. To convert binary and decimal numbers manually, you must apply the mathematical concept of positional values. The positional value concept is simple: With both binary and decimal numbers, the actual value of each digit depends on its position (how "far to the left") within the number.
For example, in the decimal number 124, the digit '4' represents the value "four," but the digit '2' represents the value "twenty," not "two." The '2' represents a larger value than the '4' in this case because it lies further to the left in the number.
Likewise in the binary number 1111011, the rightmost '1' represents the value "one," but the leftmost '1' represents a much higher value ("sixty-four" in this case).
In mathematics, the base of the numbering system determines how much to value digits by position. For base-ten decimal numbers, multiply each digit on the left by a progressive factor of 10 to calculate its value. For base-two binary numbers, multiply each digit on the left by a progressive factor of 2. Calculations always work from right to left.
In the above example, the decimal number 123 works out to:
- 3 + (10 * 2) + (10*10 * 1) = 123
- 1 + (2 * 1) + (2*2 * 0) + (4*2 * 1) + (8*2 * 1)+ (16*2 * 1) + (32*2 * 1) = 123
For example, the decimal number 109 converts to binary as follows:
- 109 / 2 = 54 remainder 1
54 / 2 = 27 remainder 0
27 / 2 = 13 remainder 1
13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
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