Answer:
Explanation:
Decimal Binary
0 0
1 1
2 10
3 11
4 100
7 111
8 1000
10 1010
16 10000
20 10100
8 × 100 = 8 × 1 = 8
Using the number 18 for comparison:
(1 × 101) + (8 × 100) = 10 + 8 = 18
In binary, 8 is represented as 1000. Reading from right to left, the first 0 represents 20, the second 21, the third 22, and the fourth 23; just like the decimal system, except with a base of 2 rather than 10. Since 23 = 8, a 1 is entered in its position yielding 1000. Using 18, or 10010 as an example:
18 = 16 + 2 = 24 + 21
10010 = (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (0 × 20) = 18
The step by step process to convert from the decimal to the binary system is:
Find the largest power of 2 that lies within the given number
Subtract that value from the given number
Find the largest power of 2 within the remainder found in step 2
Repeat until there is no remainder
Enter a 1 for each binary place value that was found, and a 0 for the rest
Using the target of 18 again as an example, below is another way to visualize this:
2n 24 23 22 21 20
Instances within 18 1 0 0 1 0
Target: 18 18 - 16 = 2 → 2 - 2 = 0
EX: 10111 = (1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (1 × 20) = 23
24 23 22 21 20
1 0 1 1 1
16 0 4 2 1
Hence: 16 + 4 + 2 + 1 = 23.
Binary Addition
Binary addition follows the same rules as addition in the decimal system except that rather than carrying a 1 over when the values added equal 10, carry over occurs when the result of addition equals 2. Refer to the example below for clarification.
Note that in the binary system:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0, carry over the 1, i.e. 10
EX:
10 11 11 10 1
+ 1 0 1 1 1
= 1 0 0 1 0 0
The only real difference between binary and decimal addition is that the value 2 in the binary system is the equivalent of 10 in the decimal system. Note that the superscripted 1's represent digits that are carried over. A common mistake to watch out for when conducting binary addition is in the case where 1 + 1 = 0 also has a 1 carried over from the previous column to its right. The value at the bottom should then be 1 from the carried over 1 rather than 0. This can be observed in the third column from the right in the above example.
Binary Subtraction
Similar to binary addition, there is little difference between binary and decimal subtraction except those that arise from using only the digits 0 and 1. Borrowing occurs in any instance where the number that is subtracted is larger than the number it is being subtracted from. In binary subtraction, the only case where borrowing is necessary is when 1 is subtracted from 0. When this occurs, the 0 in the borrowing column essentially becomes "2" (changing the 0-1 into 2-1 = 1) while reducing the 1 in the column being borrowed from by 1. If the following column is also 0, borrowing will have to occur from each subsequent column until a column with a value of 1 can be reduced to 0. Refer to the example below for clarification.
Note that in the binary system:
0 - 0 = 0
0 - 1 = 1, borrow 1, resulting in -1 carried over
1 - 0 = 1
1 - 1 = 0
EX1:
-11 20 1 1 1
– 0 1 1 0 1
= 0 1 0 1 0
EX2:
-11 2-10 0
– 0 1 1
= 0 0 1
Note that the superscripts displayed are the changes that occur to each bit when borrowing. The borrowing column essentially obtains 2 from borrowing, and the column that is borrowed from is reduced by 1.
Binary Multiplication
Binary multiplication is arguably simpler than its decimal counterpart. Since the only values used are 0 and 1, the results that must be added are either the same as the first term, or 0. Note that in each subsequent row, placeholder 0's need to be added, and the value shifted to the left, just like in decimal multiplication. The complexity in binary multiplication arises from tedious binary addition dependent on how many bits are in each term. Refer to the example below for clarification.
Note that in the binary system:
0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
EX:
1 0 1 1 1
× 1 1
1 0 1 1 1
+ 1 0 1 1 1 0
= 1 0 0 0 1 0 1
As can be seen in the example above, the process of binary multiplication is the same as it is in decimal multiplication. Note that the 0 placeholder is written in the second line. Typically the 0 placeholder is not visually present in decimal multiplication. While the same can be done in this example (with the 0 placeholder being assumed rather than explicit), it is included in this example because the 0 is relevant for any binary addition / subtraction calculator, like the one provided on this page. Without the 0 being shown, it would be possible to make the mistake of excluding the 0 when adding the binary values displayed above. Note again that in the binary system, any 0 to the right of a 1 is relevant, while any 0 to the left of the last 1 in the value is not.
EX:
1 0 1 0 1 1 0 0
= 0 0 1 0 1 0 1 1 0 0
≠ 1 0 1 0 1 1 0 0 0 0