Answer: i.) 667,411. ii.) 11D7 iii.) 158C. iv.) 16.
Explanation:
i.) Our decimal system, is a position-based one, where each digit, occupies a position represented by the power of 10 that corresponds to this position.
The number is built adding the products of each digit times the power of 10 corresponding to the relative position occupied by the digit in the number, going from right to left, as power indices increases.
For instance, 876 can be decomposed as follows:
8. 10² + 7. 10¹ + 6. 10⁰ (remembering that 10⁰ = 1).
In the same way, any number can be decomposed as a sum of digits (taking into account the particular base of the system, 10) times powers of the base (translated to decimal).
For instance, let's convert A2F13 to decimal:
As we have five digits, we know that we will have products including power of 16, from 0 to 4, as follows:
A. 16⁴ + 2. 16³ + F. 16² + 1 . 16¹ + 3. 16⁰
Replacing by the values we get;
(A2F13)₁₀ = 10. 65,536 + 2. 4,096 + 15. 256 + 16 + 3 = 667,411.
where A = 10₁₀ , F= 15₁₀, as for a system with base 16, we need to find new digits to replace the decimals 10,11,12,13,14 and 15, that have a different meaning in a hex system.
ii. ) In order to convert a decimal number in an hexadecimal , we should express the number in terms of powers of 16.
One way to do this, is finding the greatest power of 16, included in the decimal number (which means thas when divided by this power of 16, gives a remain not zero).
We find that 16³ = 4096, "fits" within 4567, as follows:
4567-4096 = 471.
Following with the same process, we have 471- 16² = 471-256 = 215
Now, we divide 215 between 16¹, as follows:
215/16 = 13. 4375
So, we know that the last digit (the resultant from the remain of the division) is less than the base, so we can find this digit taking the remain and multiplying by the base, as follows:
0.4375 . 16 = 7
So, we can build the number in powers of 16, as follows:
1. 16³ + 1. 16² + 13. 16 + 7. 16⁰
In Hex, the decimal 13 is replaced by the letter D, so finally we have the following:
(4567)₁₆ = 11D7
iii.) Let's take the number in binary first:
0001010110001100
If we want to convert it to decimal, we should repeat the former process, now, using power of 2.
But as we know the 2⁴ = 16, so we can use this fact in order to simplify the process.
If we divide the binary number in groups of 4, each of these groups represent a digit in hex, from 0 (0000) to F (1111).
So, dividing the number in groups of 4 , from left to right, we have:
0001 0101 1000 1100
Replacing each group by the digit in hex, we have:
158C (as C replaces to the 12₁₀).
We can test the result, expressing both numbers in decimal:
For the binary:
4,096 + 1024 + 256 + 128 + 8 + 4 = 5,516.
For the Hex:
1. 16³ + 5. 16² + 8. 16¹ + C. 16⁰ = 4,096 + 1,280 + 128 + 12 = 5,516, the same result as above.
iv. ) Using the same reasoning, if we have 64 binary digits, taking in groups of 4 (as 2⁴ = 16) , we will have 16 hexadecimal digits.
