Hermine Smikle.
In the decimal system when a number is read from right to left each digit is multiplied by a progressively higher power of 10. These are commonly referred to as ones, tens, hundreds, thousands, and so on.
In the binary system the same concept applies except that the number being raised are powers of two, the digits therefore represents ones, twos, fours, eights and so on. In this counting system the only numbers that are used are zeros and ones.
Example of the comparison of Binary and Decimal Systems.
Decimal system

Binary System

10
^{
3
}
10
^{
2
}
10
^{
1
}
10
^{
0
}

2
^{
4
}
2
^{
3
}
2
^{
2
}
2
^{
1
}
2
^{
0
}

The arithmetic in the binary system employs the same operations as the decimal system but may be considered simpler. The addition involves grouping things in groups of twos with carrying to the next higher power.
Example
(a)

10 + 10


10


+ 10


100

carrying to the next higher power.
Number conversion
Binary to Decimal conversion: In a binary number each position corresponds to a power of two.
Example

(a) 110 means

2
^{
2
}
+ 2
^{
1
}
+ 0



=4 + 2 + 0



= 6


(b) 101 means 2
^{
2
}
+ 0 + 2
^{
0
}
(1)



=4 + 0 + 1



=5


(c) 11001

means 2
^{
4
}
+ 2
^{
3
}
+ 0
^{
2
}
+ 0
^{
1
}
+ 2
^{
0
}



=16 + 8 + 0 + 0 + 1



= 25

Multiplication is also a straight forward procedure, since each digit is either 0 Or 1; therefore each potential product is either zero or one.
Example

10 x 01


10

10


x01

x 10


10

00



10 partial products



100

The rule for multiplication is simply to write down the multiplicand shifted one place to the left for each of the multiplier that is a one the sum the numbers.
In summary then since binary operations uses the same concepts of value and positions of digits as the decimal system, the associated arithmetic is the same.
In addition we add column by column, carrying where necessary to higher positions. In subtraction we subtract column by column, borrowing where necessary from higher positions, and in division we do repeated subtractions just as in long division.
Examples

1. Addition

2. Subtraction


1110

1101


+1011

Ð1010 with borrowing


11001

0011


Multiplication


Division

(figure available in print form)
Division is done by counting how many times one number can be subtracted from the other; the counts is called the quotient, the quotient in is 0110.
Conversion from decimal to binary
.
To convert from binary to decimal we sum the values of the position of each of the digits ; but to convert a decimal number to its binary equivalent progressively divide the decimal number by two and record the remainders. The remainders written in the reverse order forms the binary equivalent.
Example

Change 18 to its binary equivalent

(figure available in printed form)
The importance of the binary system to technology is the convenience of the system to facilitate expressions in symbolic logic. In 1938 Claude Shannon pointed out that switching circuits could be used to evaluate logic statements. He used the dualities of on and off, high and low, and represent these by zero and ones. As a result logic systems like computers are designed to use the binary system.