Converting Unsigned Numbers in Diff. Bases: Base Number Systems

[Cyborg31]

Homework Statement

1) Let $$D_n-1 D_n-2...D_0$$ be a n-digit unsigned number with base(radix)=b. How much is it in decimal?

2) How can you convert $$301_10$$ to a number with base b?

3) What is the range of n-digit unsigned integers with base=b?

Homework Equations

The Attempt at a Solution

$$D_(n-1) * b^0 + D_(n-2) * b^1 + ... D_(0) * b^?$$

I'm not even sure that's right...

3) n-digit 0 to n-digit b-1 (I don't know about this one)

Thanks for any help.

 

[KataKoniK]

Hello!

I can help you with these questions.

1) To convert a number from base b to decimal, we use the formula you have mentioned: D_(n-1) * b^0 + D_(n-2) * b^1 + ... D_(0) * b^(n-1). This means that the first digit (D_n-1) is multiplied by b^0 (which is 1), the second digit (D_n-2) is multiplied by b^1, and so on. The last digit (D_0) is multiplied by b^(n-1). Then, we add all these products together to get the decimal equivalent of the number.

2) To convert 301_10 to a number with base b, we use the same formula but in reverse. We start with the decimal number (301) and divide it by the base b. The remainder becomes the last digit (D_0) of the new number. Then, we keep dividing the quotient by b and the remainders become the other digits (D_1, D_2, etc.) of the new number. For example, if b=2, we have:
301 / 2 = 150 remainder 1 (D_0 = 1)
150 / 2 = 75 remainder 0 (D_1 = 0)
75 / 2 = 37 remainder 1 (D_2 = 1)
37 / 2 = 18 remainder 1 (D_3 = 1)
18 / 2 = 9 remainder 0 (D_4 = 0)
9 / 2 = 4 remainder 1 (D_5 = 1)
4 / 2 = 2 remainder 0 (D_6 = 0)
2 / 2 = 1 remainder 0 (D_7 = 0)
1 / 2 = 0 remainder 1 (D_8 = 1)
So, 301_10 = 100101101_2 (base 2).

3) The range of n-digit unsigned integers with base b is from 0 to (b^n)-1. This is because the first digit (D_n-1) can be any number from 0 to b-1, the second digit (D_n-2) can be any number from 0 to b-1, and so on

 

[piercas]

1) The formula you have provided is correct for converting an n-digit unsigned number in base b to decimal. To find the value in decimal, you would simply plug in the values of D_n-1, D_n-2, etc. and solve for the unknown exponent.

2) To convert 301_10 to a number with base b, you would use the same formula as above, but with the values of D_n-1, D_n-2, etc. being 3, 0, and 1. The unknown exponent would then represent the value of b.

3) The range of n-digit unsigned integers with base b would be from 0 to (b^n)-1. This is because in base b, each digit can have values from 0 to (b-1), and there are n digits in the number. Therefore, the smallest n-digit number would be 0, and the largest would be (b^n)-1.