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Sladkaya [172]
3 years ago
13

How many bits are required to represent the decimal numbers in the range from 0 to 999 in straight binary code?

Mathematics
1 answer:
ad-work [718]3 years ago
8 0
Note that powers of 2 can be written in binary as

2^0=1_2
2^1=10_2
2^2=100_2

and so on. Observe that n+1 digits are required to represent the n-th power of 2 in binary.

Also observe that

\log_2(2^n)=n\log_22=n

so we need only add 1 to the logarithm to find the number of binary digits needed to represent powers of 2. For any other number (non-power-of-2), we would need to round down the logarithm to the nearest integer, since for example,

2_{10}=10_2\iff\log_2(2^1)=\log_22=1
3_{10}=11_2\iff\log_23=1+(\text{some number between 0 and 1})
4_{10}=100_2\iff\log_24=2

That is, both 2 and 3 require only two binary digits, so we don't care about the decimal part of \log_23. We only need the integer part, \lfloor\log_23\rfloor, then we add 1.

Now, 2^9=512, and 999 falls between these consecutive powers of 2. That means

\log_2999=9+\text{(some number between 0 and 1})

which means 999 requires \lfloor\log_2999\rfloor+1=9+1=10 binary digits.

Your question seems to ask how many binary digits in total you need to represent all of the numbers 0-999. That would depend on how you encode numbers that requires less than 10 digits, like 1. Do you simply write 1_2? Or do you pad this number with 0s to get 10 digits, i.e. 0000000001_2? In the latter case, the answer is obvious; 1000\times10=10^4 total binary digits are needed.

In the latter case, there's a bit more work involved, but really it's just a matter of finding how many number lie between successive powers of 2. For instance, 0 and 1 both require one digit, 2 and 3 require two, while 4-7 require three, while 8-15 require four, and so on.
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Step-by-step explanation:

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6 0
2 years ago
If 97 of 200 is 100% then how much of 200 is 112%
pickupchik [31]
To get the solution, we are looking for, we need to point out what we know.

1. We assume, that the number 112 is 100% - because it's the output value of the task.
2. We assume, that x is the value we are looking for.
3. If 112 is 100%, so we can write it down as 112=100%.
4. We know, that x is 200% of the output value, so we can write it down as x=200%.
5. Now we have two simple equations:
1) 112=100%
2) x=200%
where left sides of both of them have the same units, and both right sides have the same units, so we can do something like that:
112/x=100%/200%
6. Now we just have to solve the simple equation, and we will get the solution we are looking for.

7. Solution for what is 200% of 112

112/x=100/200
(112/x)*x=(100/200)*x - we multiply both sides of the equation by x
112=0.5*x - we divide both sides of the equation by (0.5) to get x
112/0.5=x
224=x
x=224

now we have:
200% of 112=224



●I tried my best I was never good with percentages my least favorite.... Please let me know if you got it wrong. If you do I'm sorry.
3 0
3 years ago
Three groups of students used different methods to estimate the diagonal length of a patio in feet. Their results were:
podryga [215]
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5 0
2 years ago
Let f be the function given by f(x) = x+4(x−1)(x+3) on the closed interval [−5,5]. On which closed interval is the function f gu
Vlad [161]

Answer:

Step-by-step explanation:

f(x)=x+4(x^2+2x-3)=4x^2+9x-12

f'(x)=8x+9

f'(x)=0,gives x=-9/8

f(-5)=-5+4(-5-1)(-5+3)=-5+4*-6*-2=43

f(-9/8)=-9/8+4(-9/8-1)(-9/8+3)

=-9/8+4*-17/8*15/8

=-9/8-255/16

=-273/16=-17 1/16

f(5)=4*5^2+9*5-12=100+45-12=133

absolute maximum=133

absolute minimum=-17 1/16

6 0
3 years ago
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Svetlanka [38]
94.1224489795918367 is the answer to your question
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3 years ago
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