QA 19

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course Mth 151

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

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Question: `q001. There are 5 questions in this set.

The preceding calculations have been 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?

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Your solution:

It would be 3 x 4^2 + 2 x 4^1 + 1 x 4^0 = 57

confidence rating #$&*: 2

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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?

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Self-critique (if necessary): OK

Self-critique Rating: OK

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Question: `q002. What would the number 213{base 4} be in base 10 notation?

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Your solution:

I think it would be 2 x 4^2 + 1 x 4^1 + 3 x 4^0, which ends up = 39

confidence rating #$&*:: 2

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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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.

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Self-critique (if necessary):

I got confused by this!

&#Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the parts of the given solution on which your solution didn't agree, and if necessary asking specific questions (to which I will respond).

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self-critique rating #$&*: 2

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Question: `q003. Suppose we had a number expressed in the form 6 * 4^2 + 7 * 4^1 + 3 * 4^0. This number isn't quite in the form needs to be if it is to be expressed in base 4. This is because we have the numbers 6 and 5, which exceed 4. How would this number be expressed without using any numbers 4 or greater?

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Your solution:

I understand the answer, I think, but I am not sure how you originally got there. I see that the answer is 1 x 4^3 + 3 x 4^2 + 3 x 4^1 + 3 x 4^0, but don’t understand the first paragraph.

confidence rating #$&*: 1

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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 * 4^1 =

(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}.

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Self-critique (if necessary): See above.

Self-critique should go here, not in your solution.

How is it that you see that the answer is as given?

Specifically what do you understand, and what do you not understand, in the first paragraph of the given solution?

Here is an expanded version of the first sentence, which might resolve your confusion:

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.

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self-critique rating #$&*:

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Question: `q004. What would happ en to the number 1333{base 4} if we added 1?

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Your solution:

You would have 1 x 4^3 + 3 x 4^2 + 4 x 4^1 + 3 x 4^0 which ends up giving you 4^1.

So, 1 x 4^3 + 3 x 4^2 + 3 x 4^1 + 1 x 4^1 + 0 x 4^0 = 4^2.

So, 1 x 4^3 + 1 x 4^3 + 1 x 4^2 + 0 x 4^1 + 0 x 4^0 = 4^3

So, 1 x 4^3 + 1 x 4^3 + 0 x 4^2 + 0 x 4^1 + 0 x 4^0 = 2000 in base 4

confidence rating #$&*:: 2

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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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}.

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Self-critique (if necessary): OK

Self-critique Rating: OK

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Question: `q005. How would the decimal number 659 be expressed in base 4?

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Your solution:

4^4 = 256, which is the last power that wont end up being higher than 659.

So, 2 x $6$ + 2 x 4^3 + 1 x 4^2 +0 x 4^1 + 4 x 4^0

confidence rating #$&*: 2

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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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.

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Self-critique (if necessary): OK

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self-critique rating #$&*: OK

&#Good work. See my notes and let me know if you have questions. &#

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