[Binary to Decimal  Decimal to Binary (Method I)  Decimal to Binary (Method II)]
In everyday life, we normally use a numbering system that is constructed on multiples of ten. We call this numbering system the Base10 or decimal numbering system. Base10 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  "Base10"?).
Each placeholder can be represented by an exponent of ten. For instance, the expression 10^{0} represents the "ones" position, the expression 10^{1} represents the "tens" position, the expression 10^{2} 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 10^{0}*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: 10^{0}*8=1*8=8. Examine the following table to see exponential expressions for the other digits:
Number 
7  4  0  8 
Position 
Thousands  Hundreds  Tens  Ones 
Exponential

10^{3}*7  10^{2}*4  10^{1}*0  10^{0}*8 
Calculated

1000*7  100*4  10*0  1*8 
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 Base2 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 2^{0}! 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 
Eights  Fours  Twos  Ones 
Exponential

2^{3}*1  2^{2}*1  2^{1}*0  2^{0}*1 
Calculated

8*1  4*1  2*0  1*1 
So, how can I convert the binary number 1101 to a goodold 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!
Number 
1  1  0  1 
Number 
1  1  0  1 
ON/OFF 
ON  ON  OFF  ON 
Number 
1  1  0  1 
ON/OFF 
ON  ON  OFF  ON 
Exponential

2^{3}*1  2^{2}*1  0  2^{0}*1 
Number 
1  1  0  1 
ON/OFF 
ON  ON  OFF  ON 
Exponential

2^{3}*1  2^{2}*1  0  2^{0}*1 
Calculated

8*1  4*1  0  1*1 
Number 
1  1  0  1 
ON/OFF 
ON  ON  OFF  ON 
Exponential

2^{3}*1  2^{2}*1  0  2^{0}*1 
Calculated

8*1  4*1  0  1*1 
Solved

8  4  0  1 
Number 
1  1  0  1 
ON/OFF 
ON  ON  OFF  ON 
Exponential

2^{3}*1  2^{2}*1  0  2^{0}*1 
Calculated

8*1  4*1  0  1*1 
Solved

8  4  0  1 
Add to Calculate

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

2^{5}*1  0  2^{3}*1  2^{2}*1  0  2^{0}*1 
Calculated

32*1  0  8*1  4*1  0  1*1 
Solved

32  0  8  4  0  1 
Add to Calculate

32+0+8+4+0+1=45 
Why not try some on your own? Convert the following from binary to decimal. Click the answers link for each table for that table's correct answers:






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One method of converting from a decimal value to a binary value is to consider the values of the exponents that represent binary placeholders. Remember that each binary placeholder, like each decimal placeholder, can be represented by an exponential expression:
Placeholder

OneHundred

SixtyFours 
ThirtySeconds 
Sixteens 
Eights 
Fours 
Twos 
Ones 
Placeholder Exponential

2^{7}  2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
Calculated

128  64  32  16  8  4  2  1 
Okay, so how can we use the exponential expressions to convert from decimal to binary? For an example let's use the decimal number 97:
Decimal Number: 97  
Placeholder

2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
Calculated

64  32  16  8  4  2  1 
1/0 
1 
Decimal Number: 97  
Placeholder

2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
Calculated

64  32  16  8  4  2  1 
1/0 
1  1 
Expression 
1 or 0? 
Placeholder

Calculated

2^{6}+2^{5}+2^{4}?97
64+32+16?97 112?97 112>97 
0  2^{4}  16 
2^{6}+2^{5}+2^{3}?97
64+32+8?97 104?97 104>97 
0  2^{3}  8 
2^{6}+2^{5}+2^{2}?97
64+32+4?97 100?97 100>97 
0  2^{2}  4 
2^{6}+2^{5}+2^{1}?97
64+32+2?97 98?97 98>97 
0  2^{1}  2 
2^{6}+2^{5}+2^{0}?97
64+32+1?97 97?97 97=97 
1  2^{0}  1 
Decimal Number: 97  
Placeholder

2^{6}  2^{5}  2^{4}  2^{3}  2^{2}  2^{1}  2^{0} 
Calculated

64  32  16  8  4  2  1 
1/0 
1  1  0  0  0  0  1 
Binary Number: 1100001 
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:
Decimal Number=97  
Division Expression 
Quotient 
Remainder 
97/2  48  1 
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 
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 
Now, try some binary to decimal problems on your own. Try each of the following conversions. Below each are solutions to the conversions using Method I and Method II. Try each of the methods in doing the conversions: