Question 1190037: Convert the following decimal numbers to into binary format
a. 15
b. 210
c. 119
Can you please explain me this homework question step-by-step? Thank you!
Found 5 solutions by Boreal, MathLover1, greenestamps, Alan3354, math_tutor2020:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! 1 is 1
2 is 10
4 is 100
8 is 1000
15 is 1111, the first is 8, the second 4, the third 2, and the fourth 1--add them to get 15.
32 is 10000
64 is 100000
119 is 64+32+16 (112)+2+1, having to add 1,2,4,8
so for 119 it is 1110011 64+32+16+0+0+2+1
210 is 128 which is 1000000+64+0+16+0+0+2+0, or 11010010.
Read from right, no one+2+no four+no eight+16+no thirty-two+64+128=210
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instead of ones/tens/hundreds/thouands, there is 2^0=1, 2^1=2,2^2=4, etc. with the possible values being 0 or 1, 0 meaning blank, nothing, and 1 meaning present or the full value of that column.
Answer by MathLover1(20850)  (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
I'll do the second example, showing two different ways.
Place values, base 10:
1 = 10^0
10 = 10^1
100 = 10^2
1000 = 10^3
10000 = 10^4
etc.
Place values, base 2:
1(base 2) = 1 = 2^0
10(base 2) = 2 = 2^1
100(base 2) = 4 = 2^2
1000(base 2) = 8 = 2^3
10000(base 2) = 16 = 2^4
100000(base 2) = 32 = 2^5
1000000(base 2) = 64 = 2^6
10000000(base 2) = 128 = 2^7
100000000(base 2) = 256 = 2^8
etc.
You should understand how to convert from base 10 to base 2 using place values, because it helps you understand numbers in different bases. But it is a slow and inefficient process.
Here is how to convert 210 (base 10) to base 2 using place values.
The largest power of 2 less than 210 is 128, which is 10000000 (base 2); the number that remains is 210-128=82
The largest power of 2 less than 82 is 64, which is 1000000 (base 2); the partially converted number is 10000000+1000000 = 11000000; the number that remains is 82-64=18
The largest power of 2 less than 18 is 16, which is 10000 (base 2); the partially converted number is 11000000 + 10000 = 11010000; the number that remains is 18-16=2
2 is 10 in base 2
The conversion is 210 (base 10) = 11010000+10 = 11010010 (base 2).
The process involves repeated subtraction and then repeated referral to a list of powers of 2; that makes the process slow.
Here is a far more efficient way to perform the same conversion.
210/2 = 105 remainder 0
105 = 52 remainder 1
52 = 26 remainder 0
26 = 13 remainder 0
13 = 6 remainder 1
6 = 3 remainder 0
3 = 1 remainder 1
1 = 0 remainder 1
The converted number in base 2 is the remainders, written in reverse order: 11010010.
That is a much more efficient process.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! Convert the following decimal numbers to into binary format
a. 15
b. 210
c. 119
=============================
I would first convert them to Hexadecimal, then to binary.
---
15 = $F = 1111
210 = $D2 = 1101 0010
119 = $77 = 0111 0111
===========================
For bigger numbers, eg, 30975:
Divide by 256 --> 120.996 ...
120 = $78 = 0111 1000 binary
30975 - 256*120 = 255 = $FF = 1111 1111 binary
30975 = 0111 1000 1111 1111 binary
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Seems some of the other tutors think they're being paid by the word.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
The other tutors have great answers. Here's an alternative approach
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Part (a)
We'll need the powers of 2
1,2,4,8,16,32,...
Start with 15. Then subtract off the largest possible power of 2. In this case, we can subtract off 8
15-8 = 7
Then subtract off 4
7-4 = 3
Subtract off 2
3-2 = 1
Then subtract off 1
1-1 = 0
Work that process in reverse and we get
15 = 1+2+4+8
Showing that
15 base 10 = 1111 base 2
Each '1' is like a lightswitch that turns on the powers of 2 we're summing.
=============================================
Part (b)
We could use the previous idea but here's another way
15 base 10 = 1111 base 2
2 base 10 = 0010 base 2
13 base 10 = (15 base 10) - (2 base 10)
13 base 10 = (1111 base 2) - (0010 base 10)
13 base 10 = 1101 base 2
210 base 10 = 208 base 10 + 2 base 10
210 base 10 = (16*13) base 10 + 2 base 10
210 base 10 = 1101 0000 base 2 + 10 base 2 ... see note below
210 base 10 = 1101 0010 base 2
note: multiplying by 16 in base 2 is as easy as sticking four zeros at the far right end of the number
It's not a coincidence that 2^4 = 16.
=============================================
Part (c)
Like with part (b), I'll try to break this number into parts of 16. Or at least, try to get close to such multiples of 16.
As Alan alluded to, it's probably better to work with hexadecimal (base 16) since the numbers in binary are very long strings of 1's and 0's. It's easy to get lost.
7 base 10 = 0111 base 2
119 base 10 = 112 base 10 + 7 base 10
119 base 10 = (16*7) base 10 + 7 base 10
119 base 10 = 0111 0000 base 2 + 0111 base 2
119 base 10 = 0111 0111 base 2
If you are more familiar with the other methods presented by the other tutors, then ignore my post entirely.
I recommend the use of any online calculators that quickly convert from binary to decimal (and vice versa) to check your work. Keep in mind that it's not a good idea to rely entirely on the calculator because your teacher will want you to show your work (or thought process). The calculator is merely a tool to see if you have the right answer or not.
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