If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
019. Place-value System with Other Bases
Question: `q001. There are 7 questions in this set.
The calculations of the preceding qa were done in our standard base-10 place value system. We can do similar calculations with bases other than 10.
For example, a base-4 calculation might involve the number 3 * 4^2 + 2 * 4^1 + 1 * 4^0. This number will be expressed as 321{base 4}.
What would this number be in base 10?
Your solution:
Confidence Assessment:
Given Solution:
In base 10, 3 * 4^2 + 2 * 4^1 + 1 * 4^0 = 3 * 16 + 2 * 4 + 1 * 1 = 48 + 8 + 1 = 57.
STUDENT COMMENT:
I am not understanding this.
INSTRUCTOR RESPONSE
statement 1: 321{base 4} means 3 *
4^2 + 2 * 4^1 + 1 * 4^0.
statement 2: 3 * 4^2 + 2 * 4^1 + 1 * 4^0 = 57.
What is it you do and do not understand about the above two statements?
Self-critique (if necessary):
Self-critique Rating:
Question: `q002. What would the number 213{base 4} be in base 10 notation?
Your solution:
Confidence Assessment:
Given Solution:
213{base 4} means 2 * 4^2 + 1 * 4^1 + 3 * 4^0 = 2 * 16 + 1 * 4 + 3 * 1 = 32 + 4 + 3 = 39.
Self-critique (if necessary):
Self-critique Rating:
Question: `q003. Suppose we had a number expressed in the form 6 * 4^2 + 7 * 4^1 + 3 * 4^0. In base 4 every term needs to be expressed in the highest possible power of 4. This is not the case for the given number, since for example the coefficient 7 can be expressed as 1 * 4^1 + 3 * 4^0.
How would the number 6 * 4^2 + 7 * 4^1 + 3 * 4^0 be expressed without using any coefficients greater than 3?
Your solution:
Confidence Assessment:
Given Solution:
7 = 4 + 3 so 7 * 4^1 can be written as 4 * 4^1 + 3 * 4^1 = 4^2 + 3 * 4^1 Since 6 = 4 + 2, we have 6 * 4^2 = 4 * 4^2 + 2 * 4^2. Since 4 * 4^2 = 4^3, this is 4^3 + 2 * 4^2. Thus
6 * 4^2 + 7 * 4^1 + 3 * 4^0 =
(4 * 4^2 + 2 * 4^2) + (4 * 4^1 + 3 * 4^1) + 3 * 4^0
=4^3 + 2 * 4^2 + 4^2 + 3 * 4^1 + 3 * 4^0 =
1 * 4^3 + 3 * 4^2 + 3 * 4^1 + 3 * 4^0. This number would then be 1333 {base 4}.
STUDENT COMMENT
I understand the answer, but not the first paragraph of the explanation.
INSTRUCTOR RESPONSE
Here is an expanded version of the first line:
7 * 4^1 = (4 + 3) * 4^1 = 4 * 4^1 + 3 * 4^1.
Since 4 * 4^1 = 4^2, it follows that 7 * 4^1 = 4^2 + 3 * 4^1.
STUDENT QUESTION
I can’t figure this out, because I’m not
sure what a coefficient is. Even after looking up the definition, I’m not sure.
When you start off writing 7 as 4 * 4˄2, isn’t that first four a
coefficient? If so, it is more than 3.
INSTRUCTOR RESPONSE
<h3>@&
To clarify the terminology of 'coefficient':
In the expression
6 * 4^2 + 7 * 4^1 + 3 * 4^0
6 is the coefficient of 4^2,
7 is the coefficient of 4^1 and
3 is the coefficient of 4^0.
*@</h3>
<h3>@&
7 * 4^1 is not in base-4 notation, precisely because 7 is greater than 3.
So we write it as
4 * 4^1 + 3 * 4^1.
Now the 3 * 4^1 is OK, at least for the moment, but the 4 * 4^1 is still
problematic.
So we just multiply 4 by 4^1, which gives us
4 * 4^1 = 4 * 4 = 4^2.
That is the plain unadorened 4^2 term in the step
4^3 + 2 * 4^2 + 4^2 + 3 * 4^1 + 3 * 4^0,
the one without a coefficient (which is understood to be 1):
We have two 4^2 terms, 2 * 4^2 and 4^2, which add up to 3 * 4^2. We see this
term in the following step
1 * 4^3 + 3 * 4^2 + 3 * 4^1 + 3 * 4^0.
*@</h3>
Self-critique (if necessary):
Self-critique Rating:
Question: `q004. What would happen to the number 1333{base 4} if we added 1?
Your solution:
Confidence Assessment:
Given Solution:
Since 1 = 1 * 4^0, Adding one to 1 * 4^3 + 3 * 4^2 + 3 * 4^1 + 3 * 4^0 would give us
1 * 4^3 + 3 * 4^2 + 3 * 4^1 + 3 * 4^0 + 1 * 4^0 =
1 * 4^3 + 3 * 4^2 + 3 * 4^1 + 4 * 4^0.
But 4 * 4^0 = 4^1, so we would have
1 * 4^3 + 3 * 4^2 + 3 * 4^1 + 1 * 4^1 + 0 * 4^0 =
1 * 4^3 + 3 * 4^2 + 4 * 4^1 + 0 * 4^0 .
But 4 * 4^1 = 4^2, so we would have
1 * 4^3 + 3 * 4^2 + 1 * 4^2 + 0 * 4^1 + 0 * 4^0 =
1 * 4^3 + 4 * 4^2 + 0 * 4^1 + 0 * 4^0 .
But 4 * 4^2 = 4^3, so we would have
1 * 4^3 + 1 * 4^3 + 0 * 4^2 + 0 * 4^1 + 0 * 4^0 =
2 * 4^3 + 0 * 4^2 + 0 * 4^1 + 0 * 4^0.
We thus have the number 2000{base 4}.
Self-critique (if necessary):
Self-critique Rating:
Question: `q005. How would the decimal number 659 be expressed in base 4?
Your solution:
Confidence Assessment:
Given Solution:
We need to express 659 in terms of multiples powers of 4, with the multiple not exceeding 3. The powers of 4 are 4^0 = 4, 4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024. We could continue to higher powers of 4, but since 4^5 = 1024 already exceeds 659 we need not do any further.
The highest power of 4 that doesn't exceed 659 is 4^4 = 256. So we will use the highest multiple of 256 that doesn't exceed 659. 2 * 256 = 512, and 3 * 256 exceeds 659, so we will use 2 * 256 = 2 * 4^4.
This takes care of 512 of the 659, leaving us 147 to account for using lower powers of 4.
We then account for as much of the remaining 147 using the next-lower power 4^3 = 64. Since 2 * 64 = 128 is less than 147 while 3 * 64 is greater than 147, we use 2 * 64 = 2 * 4^3.
This accounts for 128 of the remaining 147, which now leaves us 19.
The next-lower power of 4 is 4^2 = 16. We can use one 16 but not more, so we use 1 * 16 = 1 * 4^2.
This will account for 16 of the remaining 19, leaving us 3. This 3 is accounted for by 3 * 4^0 = 3 * 1. Note that we didn't need 4^1 at all.
So we see that 659 = 2 * 4^4 + 2 * 4^3 + 1 * 4^2 + 0 * 4^1 + 3 * 4^0.
Self-critique (if necessary):
Self-critique Rating:
Question: `q006. Find the base-10 equivalent of the number 322{base 4}.
Your solution:
Confidence Rating:
Question: `q007. Find the base-4 equivalent of the number 487.
Your solution:
Confidence Rating:
Self-critique Rating: