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Decimal, Binary, Octal and Hexadecimal Number Conversion
(Page 4 of 4)
Conversions From Decimal to Binary, Octal and Hexadecimal
Now let’s consider conversions
from decimal. These require that you perform the opposite
of the calculation above: you divide and subtract instead of multiplying
and adding.
Conversion From Decimal to Binary
The easiest of the three conversions
from decimal is to binary—since the maximum value of each digit
is one, there is no dividing, just subtraction. All you do is the following:
 Find the largest power of two that
is smaller than the number.
 Put a “1” in the digit place
for that power of two and subtract that power of two from the decimal
number.
 Repeat steps #1 and #2 until you are
reduced to zero.
This is easier to explain using an
example and a table, of course. Let's convert the decimal number 689,
as shown in Table 5.
Again, read the table starting from the upper left, and going down and
then across. We start by noticing that 1024 is not less than or equal
to 689, so the “1024s” place gets a 0. In the next place,
512 is less than 689, so we make the “512s” place a 1 and
subtract 512 from 689 to leave 177. The calculation continues, eventually
showing shows that 689 decimal is 1010110001 binary.
Table 5: Decimal to Binary Number Conversion
Decimal Value Before Considering
This Digit Place

689

689

177

177

49

49

17

1

1

1

1

Power of
Two

2^{10}

2^{9}

2^{8}

2^{7}

2^{6}

2^{5}

2^{4}

2^{3}

2^{2}

2^{1}

2^{0}

Value of Digit Place

1024

512

256

128

64

32

16

8

4

2

1

Value of
Digit Place Equal To or Less Than Current Decimal Number?

No

Yes

No

Yes

No

Yes

Yes

No

No

No

Yes

Subtraction Step

skip

689 512 = 177

skip

177 128 = 49

skip

4932 = 17

1716 = 1

skip

skip

skip

11 = 0

Binary Digits

0

1

0

1

0

1

1

0

0

0

1

Conversion From Decimal to Octal or Hexadecimal
The process for octal and hexadecimal
is almost the same, except you must divide by powers of two instead
of just subtracting:
 Start with the highest power of 16
(hexadecimal) or 8 (octal) that is smaller than the number.
 Divide the decimal number by that power,
keeping only the integer part of the result.
 Keep the remainder after the division
is done, for the next step.
 Repeat steps #1 to #3 until you get
to the “ones” place, and then put there whatever is left after
the higher digits were done.
Table 6
shows the same example as Table 5
but goes from decimal to hexadecimal instead of decimal to binary: 689
in decimal is 0x2B1 hexadecimal.
Table 6: Decimal to Hexadecimal Number Conversion
Decimal Value Before Considering
This Digit Place

689

689

177

1

Power of
16

16^{3}

16^{2}

16^{1}

16^{0}

Value of Digit Place

4096

256

16

1

Value of
Digit Place Smaller Than Current Decimal Number?

No

Yes

No

n/a

Division Step

skip

689/256 = 2.691;
use “2” for this digit.

177/16 = 11.0625;
use “B” for this digit.

n/a

Remainder
After Division

skip

177

1

n/a

Hexadecimal Digits

0

2

B

1

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