Converting Decimal into Binary |
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Converting Decimal into Binary
The Decimal or denary counting system uses the Base-10 numbering system where each digit
in a number takes on one of ten possible values from 0 to 9, eg 21310 (Two Hundred
and Thirteen). In a decimal system each digit has a value ten times greater than its previous number and this decimal
numbering system uses a set of symbols, b, together with a base, q,
to determine the weight of each digit. For example, the six in sixty has a lower weighting than the six in six hundred and
in a binary system we need some way of converting decimal into binary.
Therefore, a numbering system can be summarised by the following relationship:
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N = bi qi |
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where: |
N is a real positive number
b is the symbol
q is the base value
and integer (i) can be positive, negative or zero |
N = b2 q2 + b1 q1
+ b0 q0 + b-1 q-1 ... etc. |
For example: N = 616310
(Six Thousand One Hundred and Sixty Three) is equal to:
(6×103) + (1×102) +
(6×101) + (3×100) = 6163
Unlike the decimal numbering system which uses the Base-10 system, digital logic uses just two values or
states, logic level "1" or logic level "0", so each digit is considered as a single digit in a Base-2 or Binary
number. In the binary numbering system, each digit has a value twice that of the previous digit but can only have a value of
either "1" or "0" therefore, q = "2".
For example:
| Decimal Digit Value |
256 |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
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| Binary Digit Value |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
Adding together the value of all the "1" s gives us: (256) + (64) + (32) + (4) + (1) =
35710
Then, the number 1011001012 in binary is equivalent to
35710 in decimal or denary.
Another method of converting decimal into binary number equivalents is to write down the decimal
number and to continually divide by 2 (two) to give a result and a remainder of either a "1" or a "0" until the final
result equals zero.
Example. Convert the decimal number 29410
into its binary number equivalent.
| Number |
294 |
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Dividing each number by "2" gives a result plus a remainder. The binary result is
obtained by placing the remainders in order with the least significant bit (LSB) being at the top and the most
significant bit (MSB) being at the bottom.
|
| divide by 2 |
| result | 147 | remainder | 0 (LSB) |
| divide by 2 |
| result | 73 | remainder | 1 |
| divide by 2 |
| result | 36 | remainder | 1 |
| divide by 2 |
| result | 18 | remainder | 0 |
| divide by 2 |
| result | 9 | remainder | 0 |
| divide by 2 |
| result | 4 | remainder | 1 |
| divide by 2 |
| result | 2 | remainder | 0 |
| divide by 2 |
| result | 1 | remainder | 0 |
| divide by 2 |
| result | 0 | remainder | 1 (MSB) |
Then, the Decimal number 29410 is equivalent to
1001001102 in Binary format.
Then the main characteristics of a Binary Numbering System is that each "digit" or "bit"
has a value of either "1" or "0" with each digit having a weight or value
double that of its previous bit starting from the lowest or least significant bit (LSB).
Binary Number Names & Prefixes
Binary numbers can be combined into one of several size ranges depending upon the number of bits being
used and are generally referred to by the following more common names of:
| Number of Binary Digits (bits) | Common Name |
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| 1 | Bit |
| 4 | Nibble |
| 8 | Byte |
| 16 | Word |
| 32 | Double Word |
| 64 | Quad Word |
Today, as microcontroller or microprocessor systems become increasingly larger, the individual binary
digits (bits) are now grouped together into 8īs to form a single BYTE with most computer hardware
such as hard drives and memory modules commonly indicate their size in Megabytes or even
Gigabytes.
| Number of Bytes | Common Name |
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| 1,024 (210) | kilobyte (kb) |
| 1,048,576 (220) | megabyte (Mb) |
| 1,073,741,824 (230) | gigabyte (Gb) |
| a very long number! (240) | terabyte (Tb) |
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