BINARY NUMBER SYSTEM
The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base-10 system.
When you perform computations such as 3 + 2 = 5, or 21 – 7 = 14, you are using the decimal number system.
This system, which you likely learned in first or second grade, is ingrained into your subconscious; it’s the natural way that you think about numbers.
Evidence exists that Egyptians were using a decimal number system five thousand years ago.
The Roman numeral system, predominant for hundreds of years, was also a decimal number system (though organized differently from the Arabic base-10 number system that we are most familiar with).
Indeed, base-10 systems, in one form or another, have been the most widely used number systems ever since civilization started counting.
In dealing with the inner workings of a computer, though, you are going to have to learn to think in a different number system, the binary number system, also referred to as the base-2 system.
Consider a child counting a pile of pennies. He would begin: “One, two, three, …, eight, nine.”
Upon reaching nine, the next penny counted makes the total one single group of ten pennies.
He then keeps counting: “One group of ten pennies two groups of ten pennies three groups of ten pennies eight groups of ten pennies nine groups of ten pennies”.
Upon reaching nine groups of ten pennies plus nine additional pennies, the next penny counted makes the total thus far: one single group of one hundred pennies.
Upon completing the task, the child might find that he has three groups of one hundred pennies, five groups of ten pennies, and two pennies left over: 352 pennies.
More formally, the base-10 system is a positional system, where the rightmost digit is the ones position (the number of ones), the next digit to the left is the tens position (the number of groups of 10), the next digit to the left is the hundreds position (the number of groups of 100), and so forth.
The base-10 number system has 10 distinct symbols, or digits (0, 1, 2, 3,…8, 9).
In decimal notation, we write a number as a string of symbols, where each symbol is one of these ten digits, and to interpret a decimal number, we multiply each digit by the power of 10 associated with that digit’s position.
For example, consider the decimal number: 6349. This number is:
Consider: Computers are built from transistors, and an individual transistor can only be ON or OFF (two options).
Similarly, data storage devices can be optical or magnetic.
Optical storage devices store data in a specific location by controlling whether light is reflected off that location or is not reflected off that location (two options).
Likewise, magnetic storage devices store data in a specific location by magnetizing the particles in that location with a specific orientation.
We can have the north magnetic pole pointing in one direction, or the opposite direction (two options).
Computers can most readily use two symbols, and therefore a base-2 system, or binary number system, is most appropriate.
The base-10 number system has 10 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
The base-2 system has exactly two symbols: 0 and 1. The base-10 symbols are termed digits.
The base-2 symbols are termed binary digits, or bits for short. All base-10 numbers are built as strings of digits (such as 6349).
All binary numbers are built as strings of bits (such as 1101).
Just as we would say that the decimal number 12890 has five digits, we would say that the binary number 11001 is a five-bit number.
The Binary Number System
Consider again the example of a child counting a pile of pennies, but this time in binary.
He would begin with the first penny: “1.”
The next penny counted makes the total one single group of two pennies. What number is this?
When the base-10 child reached nine (the highest symbol in his scheme), the next penny gave him “one group of ten”, denoted as 10, where the “1” indicated one collection of ten.
Similarly, when the base-2 child reaches one (the highest symbol in his scheme), the next penny gives him “one group of two”, denoted as 10, where the “1” indicates one collection of two.
Back to the base-2 child: The next penny makes one group of two pennies and one additional penny: “11.”
The next penny added makes two groups of two, which is one group of 4: “100.”
The “1” here indicates a collection of two groups of two, just as the “1” in the base-10 number 100 indicates ten groups of ten.
Upon completing the counting task, base -2 child might find that he has one group of four pennies, no groups of two pennies, and one penny left over: 101 pennies.
The child counting the same pile of pennies in base-10 would conclude that there were 5 pennies.
So, 5 in base-10 is equivalent to101 in base-2.
To avoid confusion when the base in use if not clear from the context, or when using multiple bases in a single expression, We append a subscript to the number to indicate the base, and write:
510 =1012
Just as with decimal notation, we write a binary number as a string of symbols, but now each symbol is a 0 or a 1.
To interpret a binary number, we multiply each digit by the power of 2 associated with that digit’s position.
For example, consider the binary number 1101. This number is:
Since binary numbers can only contain the two symbols 0 and 1, numbers such as 25 and 1114000 cannot be binary numbers.
We say that all data in a computer is stored in binary—that is, as 1’s and 0’s.
It is important to keep in mind that values of 0 and 1 are logical values, not the values of a physical quantity, such as a voltage.
The actual physical binary values used to store data internally within a computer might be, for instance, 5 volts and 0 volts, or perhaps 3.3 volts and 0.3 volts or perhaps reflection and no reflection.
The two values that are used to physically store data can differ within different portions of the same computer.
All that really matters is that there are two different symbols, so we will always refer to them as 0 and 1.
A string of eight bits (such as 11000110) is termed a byte.
A collection of four bits (such as 1011) is smaller than a byte, and is hence termed a nibble.
This is the sort of nerd-humor for which engineers are famous.
The idea of describing numbers using a positional system, as we have illustrated for base-10 and base-2, can be extended to any base. For example, the base-4 number 231 is:
Converting Between Binary Numbers and Decimal Numbers
We humans about numbers using the decimal number system, whereas computers use the binary number system.
We need to be able to readily shift between the binary and decimal number representations.
Converting a Binary Number to a Decimal Number
To convert a binary number to a decimal number, we simply write the binary number as a sum of powers of 2.
For example, to convert the binary number 1011 to a decimal number, we note that the rightmost position is the ones position and the bit value in this position is a 1.
So, this rightmost bit has the decimal value of 1⋅20.
The next position to the left is the twos position, and the bit value in this position is also a 1.
So, this next bit has the decimal value of 1⋅ 21.
The next position to the left is the fours position, and the bit value in this position is a 0.
The leftmost position is the eights position, and the bit value in this position is a 1.
So, this leftmost bit has the decimal value of 1⋅23.
Thus:
The binary number 110110 as a decimal number. Solution:
For example, to convert the binary number 10101 to decimal, we annotate the position values below the bit values:
Then we add the position values for those positions that have a bit value of 1: 16 + 4 + 1 = 21.
Thus 101012 = 2110
You should “memorize” the binary representations of the decimal digits 0 through 15 shown below.
You may be wondering about the leading zeros in the table above.
For example, the decimal number 5 is represented in the table as the binary number 0101.
We could have represented the binary equivalent of 5 as 101, 00101, 0000000101, or with any other number of leading zeros.
All answers are correct.
Sometimes, though, you will be given the size of a storage location.
When you are given the size of the storage location, include the leading zeros to show all bits in the storage location.
For example, if told to represent decimal 5 as an 8-bit binary number, your answer should be 00000101.
Converting a Decimal Number to a Binary Number:
Method 2
The second method of converting a decimal number to a binary number entails repeatedly dividing the decimal number by 2, keeping track of the remainder at each step. To convert the decimal number x to binary:
Step 1. Divide x by 2 to obtain a quotient and remainder. The remainder will be 0 or 1.
Step 2. If the quotient is zero, you are finished: Proceed to Step 3. Otherwise, go back to Step 1, assigning x to be the value of the most-recent quotient from Step 1.
Step 3. The sequence of remainders forms the binary representation of the number.
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