Binary Conversions

The Binary Numbering System

In everyday life, we normally use a numbering system that is constructed on multiples of ten. We call this numbering system the Base-10 or decimal numbering system. Base-10 numbering systems dictate that the numbering scheme begins to repeat after the tenth digit (in our case, the number 9). When we count, we usually count "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ..."

There's more to the numbering scheme than just counting, though. In grade school, we all were taught that each digit to the left and right of the decimal point is given a name which identifies that digit's placeholder. For right now, let's just consider digits to the left of the decimal, or positive numbers. Remember that the first digit to the left of the decimal point is called the "ones" digit. It is followed by the "tens" digit, followed by the "hundreds", followed by the "thousands", and on and on. What they probably didn't tell you in grade school is that each placeholder (ones, tens, hundreds, thousands, etc.) actually represents a multiple of ten (remember -- "Base-10"?).

Each placeholder can be represented by an exponent of ten. For instance, the expression 100 represents the "ones" position, the expression 101 represents the "tens" position, the expression 102 represents the "hundreds" position and so on.

We can begin to see this more clearly if we break down a number into exponents of ten. Let's take a look at the following number: 7408. Starting at the decimal point, we'll work our way left. The first digit to the left of the decimal point is 8. However, we can represent this using the arithmetic expression 100*8. Remember: Anything to the zero power is always equal to 1. If we were to calculate that last expression out it would look like this: 100*8=1*8=8. Examine the following table to see exponential expressions for the other digits:

Number

7

4

0

8

Position
Name

Thousands

Hundreds

Tens

Ones

Exponential
Expression

103*7

102*4

101*0

100*8

Calculated
Exponent

1000*7

100*4

10*0

1*8

Table 1: Decimal Placeholders

Like the decimal numbering system, binary numbering is also based on powers of a number. However, unlike the decimal system (which is based on multiples of ten), the binary numbering system is based on multiples of two. It is a Base-2 numbering system. Remember -- when counting in decimal, the numbering scheme repeats after the tenth digit (the number 9). In binary numbering the numbering scheme repeats after the second digit (the number 1). Let's count to five in binary: "0, 1, 10, 11, 100, 101"

Also like the decimal numbering system, binary numbering includes names for digit placeholders. Instead of "ones, tens, hundreds, thousands, etc.", binary has "ones, twos, fours, eights, sixteens, etc." If the binary system is based on powers of 2, why is there still a "ones" position? Remember: Anything to the zero power is always equal to 1. So, in binary, the "ones" position is represented by the exponential expression 20! Take a look at the following table to see how the binary number 1101 is broken into exponential expressions:

Number

1

1

0

1

Position
Name

Eights

Fours

Twos

Ones

Exponential
Expression

23*1

22*1

21*0

20*1

Calculated
Exponent

8*1

4*1

2*0

1*1

Table 2: Binary Placeholders

Binary to Decimal Conversions

So, how can I convert the binary number 1101 to a good-old decimal number? The best way to to this is construct a table in which you can do some simple arithmetic operations to solve the conversion! Let's try it!

1. First, I want to write the binary number in a row, separating the digits into columns:

Number

1

1

0

1

2. Next, I want to decide whether each digit placeholder is "ON" or "OFF." The reason for this will become a little clearer in a few minutes, but for right now just remember that a "1" is "ON" and a "0" is "OFF." When we calculate the exponential expressions, we don't have to calculate any digit placeholders that are turned off:

Number

1

1

0

1

ON/OFF

ON

ON

OFF

ON

3. In the third step, we write the exponential expressions ("powers of two") that represent each placeholder and multiply each expression by 1. We do this only for the placeholders that are turned ON. For the placeholders which are turned OFF, we simply bring down the zero from the number itself:

Number

1

1

0

1

ON/OFF

ON

ON

OFF

ON

Exponential
Expression

23*1

22*1

0

20*1

4. Now, we can calculate the exponents to get a simple multiplication expression for each placeholder. Again, we do this only for placeholders which are turned "ON." Again, we bring down the zero if the placeholder is turned "OFF":

Number

1

1

0

1

ON/OFF

ON

ON

OFF

ON

Exponential
Expression

23*1

22*1

0

20*1

Calculated
Exponent

8*1

4*1

0

1*1

5. In the fifth step, we solve the multiplication expressions from step #4. Again, we bring down any zeros for placeholders which are turned OFF:

Number

1

1

0

1

ON/OFF

ON

ON

OFF

ON

Exponential
Expression

23*1

22*1

0

20*1

Calculated
Exponent

8*1

4*1

0

1*1

Solved
Multiplication

8

4

0

1

6. In the final step, we add all the multiplication answers from step #5 together to get our decimal number!

Number

1

1

0

1

ON/OFF

ON

ON

OFF

ON

Exponential
Expression

23*1

22*1

0

20*1

Calculated
Exponent

8*1

4*1

0

1*1

Solved
Multiplication

8

4

0

1

Add to Calculate
Decimal Value

8+4+0+1=13

Let's take a look at another conversion. This time, we'll try 101101:

Number

1

0

1

1

0

1

ON/OFF

ON

OFF

ON

ON

OFF

ON

Exponential
Expression

25*1

0

23*1

22*1

0

20*1

Calculated
Exponent

32*1

0

8*1

4*1

0

1*1

Solved
Multiplication

32

0

8

4

0

1

Add to Calculate
Decimal Value

32+0+8+4+0+1=45

Why not try some on your own? Convert the following from binary to decimal.

1. 111

2. 1011

3. 10111

4.11100

Decimal to Binary Conversions - Using Division

The second method of converting from decimal to binary also involves constructing a table. This time, instead of using binary placeholder exponential expressions, we'll do some simple division. Again, let's use the decimal number 97 as our example:

1. The first step in the conversion is to take the decimal number and divide it by 2. Put the division expression in the upper left-most cell of our table. Take the quotient of the division result and put it in the second cell of the row. Put the remainder in the last cell of the row. Important: NEVER carry your divisions past the decimal point!

Decimal Number=97

Division Expression

Quotient

Remainder

97/2

48

1

2. For each subsequent row, we take the quotient from the previous row and divide it by two. We put the new quotient in the second cell of the row and put the remainder in the last cell of the row:

Decimal Number=97

Division Expression

Quotient

Remainder

97/2

48

1

48/2

24

0

24/2

12

0

12/2

6

0

6/2

3

0

3/2

1

1

1/2

0

1

3. The last step in the process is concerned only with the last column in our table -- the "Remainder" column. Notice that the remainder column only has ones or zeros. Also note that the cell in the remainder column of the last row must be a "1". If we read the 1s and 0s in the remainder column from the bottom to the top, we'll have our binary number!

Decimal Number=97

Division Expression

Quotient

Remainder

Direction

97/2

48

1

48/2

24

0

24/2

12

0

12/2

6

0

6/2

3

0

3/2

1

1

1/2

0

1

Binary Number=1100001

Binary addition

Adding binary numbers is a very simple task, and very similar to the longhand addition of decimal numbers. As with decimal numbers, you start by adding the bits (digits) one column, or place weight, at a time, from right to left. Unlike decimal addition, there is little to memorize in the way of rules for the addition of binary bits:

0 + 0 = 0

1 + 0 = 1

0 + 1 = 1

1 + 1 = 10

1 + 1 + 1 = 11

Just as with decimal addition, when the sum in one column is a two-bit (two-digit) number, the least significant figure is written as part of the total sum and the most significant figure is "carried" to the next left column. Consider the following examples:

. 11 1 <--- Carry bits -----> 11

. 1001101 1001001 1000111

. + 0010010 + 0011001 + 0010110

. --------- --------- ---------

. 1011111 1100010 1011101

The addition problem on the left did not require any bits to be carried, since the sum of bits in each column was either 1 or 0, not 10 or 11. In the other two problems, there definitely were bits to be carried, but the process of addition is still quite simple.

Sign and Magnitude

In Sign and Magnitude coding we use one bit to represent the sign of the number and the remaining (n-1) bits to represent the size (the magnitude).

The bit used to represent the sign of the number is called the sign bit. The bit used to represent the sign of the number is called the sign bit. The sign bit is the leftmost bit (ie. in the most significant position). If the sign bit is 0 then this means that it is a positive number, and if the sign bit is 1 then this means that it is a negative number.

Example

Using 8 bits and sign & magnitude:

29 would be represented as: 0001 1101

-29 would be represented as 1001 1101.

Converting back to Decimal

To convert from Sign & Magnitude back to decimal :

(1) Take off the leftmost bit. This will determine the Sign (0 for positive and 1 for negative).

(2) Convert the remaining n-1 bits from binary to decimal (the same way as unsigned binary).

Question

Using Sign & Magnitude, convert the following into decimal form: (use paper for working).

(1) 10101100

Two's Complement.

To convert positive integers using two's complement simply convert to binary and pad to the left with zeros to the required bit size.

To convert negative integers using two's complement:

· Convert to binary, then pad to the left with zeros to the require bit size.

· Invert all bits.

· Add 1.

Binary Coded Decimal

Short for binary-coded decimal, a format for representing decimal numbers (integers) in which each digit is represented by four bits (a nibble ). For example, the number 375 would be represented as:

0011 0111 0101

Below is a table of the decimal numbers 0 through 9 and the binary conversion.

Decimal

BCD

0

0000

1

0001

2

0010

3

0011

4

0100

5

0101

6

0110

7

0111

8

1000

9

1001

descriptive notes on all the conversion of binary