Expressing Values, and Bases - Practice Problems

Background

Let's work some more problems in bases!

Question

• Take the number 321 in base 10 and convert into the following bases: 5, 12, 16 
• Take the digits 321 in the following bases: 4, 6, and 14, and convert to base 10

Answer

• 321 base 10 = 2241 base 5; 229 base 12; 141 base 16
• 321 base 4 = 57 base 10; 321 base 6 = 131 base 10; 321 base 14 = 617 base 10

Analysis

Before we dive into the problems, it's important to talk about the digits available for any given base. The answer is that the digits available is equal to the base we're working with.

For example, in base 10, we use 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
For base 2, we use 0 and 1.
For base 5, we use 0, 1, 2, 3, 4.
For bases higher than 10, we use letters for the additional numbers.

For example, for base 11, we use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (where A = 10 in base 10).
For base 12, we use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B

and so on.

Question 1

Let's set up the places for the different bases first:

321 in base 10 to base 5




625 is bigger than we need so that gets a 0 (and since it's a 0 to the left of all the other digits, we can drop it).

How many times does 125 go into 321? Twice. (2 x 125 = 250 but 3 x 125 = 375 which is too much):



That's 250 done. We have 321 - 250 = 71 left to do, so now we move to the next place, the 25s. 25 x 2 = 50 but 25 x 3 = 75, which is too big. So that's 2 in the 25s:



We now have done 250 + 50 = 300. We have 321 - 300 = 21 left. We move to the next place, the 5s. 5 x 4 = 20 (and we can't do 5 x 5 = 25 because that's our next place over, in the 25s):



That's now 250 + 50 + 20 = 320. We have 1 left to do:



And so 321 base 10 is the same as 2241 base 5.

321 in base 10 to base 12

Again, let's set up the places first:



I've only done to 144 because the next place is 12 times higher, at 1728s (which is way too high).

Ok - how many 144s go into 321? 144 x 2 = 288 but 144 x 3 = 432. So we put in a 2:



We've done 288 and need to take care of the remaining 321 - 288 = 33

12 x 2 = 24 but 12 x 3 = 36:



And that leaves:

321 - 288 - 24 = 9



And so 321 base 10 = 229 base 12

321 in base 10 to base 16

Setting up the places:



256 x 1 = 256 but 256 x 2 = 512 so:



321 - 256 = 65

4 x 16 = 64 but 5 x 16 = 80, so



256 + 4 x 16 = 320.

321 - 320 = 1



Therefore, 321 base 10 = 141 in base 16

Question 2

321 base 4 to base 10

Let's first set up the places and the digits in them:



We can now look at the digits and do the math:

3 x 16 = 48
2 x   4 =  8
1 x   1 =  1
_________
           = 57

321 base 4 = 57 base 10

321 base 6 to base 10

We do the same the process for base 6:



3 x 36 = 108
2 x   6 =   12
1 x   1 =     1
___________
           = 131

321 base 6 = 131 base 10

321 base 14 to base 10



3 x 196 = 588
2 x   14 =   28
1 x     1 =     1
____________
             = 617

Vocabulary used:

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