course Mth 151 ??g?????????????
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08:03:11 `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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RESPONSE --> 3*4^2 +2*4^1 + 1*4^0 = 3*16 + 2*4 + 1*1 = 48 +8 + 1= 57 confidence assessment: 3
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08:04:40 In base 10, 3 * 4^2 + 2 * 4^1 + 1 * 4^0 = 3 * 16 + 2 * 4 + 1 * 1 = 48 + 8 + 1 = 57.
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RESPONSE --> to change it to base ten we first need to worked out the powers, tthen change it to base ten self critique assessment: 3
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08:08:51 `q002. What would the number 213{base 4} be in base 10 notation?
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RESPONSE --> 2*4^2 + 1*4^1 + 3*4^0 = 2*16 + 1*4+ 3*1 = 32+ 4+ 3 =39 confidence assessment: 3
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08:11:47 213{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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RESPONSE --> by simplifying the powers it is changed to base 10 self critique assessment: 3
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08:17:57 `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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RESPONSE --> first we simplify the powers 6*4^2 + 7*4^1 + 3*4^0 then it can be express in base 4 = 6*4^4 + 7*4^1 + 3*4^0 = 6*4^4 + 7*4 + 3*4 = 96+ 28+ 12 = 136 confidence assessment: 3
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08:35:08 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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RESPONSE --> first we simplified 6 and 7, then can simplify to the power numbers and add self critique assessment: 3
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08:36:48 `q004. What would happen to the number 1333{base 4} if we added 1?
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RESPONSE --> we would have to adjust the entire operation confidence assessment: 3
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08:47:48 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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RESPONSE --> ok self critique assessment: 3
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09:04:48 `q005. How would the decimal number 659 be expressed in base 4?
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RESPONSE --> .00659 = 0*4^-1 + 0*4^-2 + 6*4^-3 + 5*4^-4 + 9*4^-5 confidence assessment: 3
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09:12:56 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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RESPONSE --> we first look for the power of 4 that will not exceed 659, then we continue working with each keepeing track of not exceeding 659 self critique assessment: 3