SOLUTION: Convert sqrt312.117 base 10 to binary number

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Question 1132706: Convert sqrt312.117 base 10 to binary number
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

sqrt%28312.117+%29=17.666833=17.667
Decimal number is converted into binary by dividing the number successively by 2 and printing the remainder in reverse order.
Divide the number by 2.
17.667%2F2=8.8335.....remainder 1
8.8335%2F2=4.41675.....remainder 0

4.41675%2F2=2.208375.....remainder 0
2.208375%2F2=1.1041875.....remainder 0
1.1041875%2F2=0.55209375.....remainder 1
so far binary number is 10001, here comes decimal point
after that you have
55209375%2F2=276046875.....remainder 1
276046875%2F2=1380234375.....remainder 0
1380234375%2F2=6901171875.....remainder 1
6901171875%2F2=34505859375.....remainder 0
34505859375%2F2=17252929687.....remainder 1
17252929687%2F2=8626464843.....remainder 1
8626464843%2F2=4313232421.....remainder 0
4313232421%2F2=215661621.....remainder 0
215661621%2F2=107830810.....remainder 0
107830810%2F2=+53915405.....remainder 1
53915405%2F2=26957702.....remainder 1
26957702%2F2=13478851.....remainder 0
13478851%2F2=6739425.....remainder 1
6739425%2F2=3369712.....remainder 0
3369712%2F2=1684856.....remainder 1
:
so on up to reminder is 0
Binary number:+10001.101011000110101
when I convert it to decimal, I got
17.673492431640625 which is approximately 17.667

Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


sqrt(312.117) = 17.66683333; or 17.667 to 3 decimal places.

Convert the whole number and decimal parts separately.

To convert the whole number part, repeatedly divide by the new base (2) and pick off the remainders.

17/2 = 8 R 1
8/2 = 4 R 0
4/2 = 2 R 0
2/2 = 1 R 0
1/2 = 0 R 1

The binary equivalent of the decimal whole number 17 is the remainders in the REVERSE order: 17 (decimal) = 10001 (binary)

Convert the fraction part to binary by repeatedly multiplying by 2, picking off the whole number parts.

.667 * 2 = 1 + .334
.334 * 2 = 0 + .668
.668 * 2 = 1 + .336
.336 * 2 = 0 + .672
.672 * 2 = 1 + .344
.344 * 2 = 0 + .688
.688 * 2 = 1 + .376
.376 * 2 = 0 + .752
.752 * 2 = 1 + .504
.504 * 2 = 1 + .008

The digits in the binary representation of the fraction part of the number is the whole number parts, in the SAME order they are obtained.

To 10 places after the decimal, .667 decimal is .1010101011 binary.

ANSWER: sqrt(312.117) (decimal) = (approximately) 10001.1010101011 (binary)

You can verify this answer using the following online binary converter:

https://www.mathsisfun.com/binary-decimal-hexadecimal-converter.html

Note if the 312.117 was an approximation and the number being converted is actually 17.66666... = 17 2/3, the binary representation can be found exactly. The whole number part is still 10001 binary; to find the binary representation of the fraction 2/3...

2/3 * 2 = 1 + 1/3
1/3 * 2 = 0 + 2/3
2/3 * 2 = 1 + 1/3
1/3 * 2 = 0 + 2/3
2/3 * 2 = 1 + 1/3
...
etc., etc.

The binary representation of the decimal fraction 2/3 is 0.101010... repeating.

In that case the binary representation of the given number would be

10001.101010... repeating