How Do You Convert the Square Root of Two to Binary with Six Decimal Places?

[StonedPanda]

So I'm taking a freshan analysis class. I've never covered converting things to binary and other number systems before, and in a chapter about sequences the book mentions binary and has an exercise to convert the square root of two to six decimal places. Can someone tell me what binary is and to do a problem like that?

 

[comwiz72]

well...

you know the way our numbers go:

0
1
2
3
4
5
6
7
8
9
10
?

Well, binary goes different.
Binary is a base-2 system (as opposed to our base-10 system), and is useful because there are only two digits. It forms the basis for every digital system, as 0 and 1 can be compared to "off" and "on" respectively.
Reading a binary system is a bit different. You read it from right to left. A few examples:

00001 = 1
00010 = 2
00011 = 3
00100 = 4
01000 = 8
10000 = 16

Do you see how this works? Each digit from the right is the next power of 2, starting with 2^0 (= 1) at the extreme right.

As for fractional values, binary has none as such. So the closest you can get is whole numbers. (Unless I'm wrong of course :->)

 

[lurflurf]

As for fractional values, binary has none as such. So the closest you can get is whole numbers. (Unless I'm wrong of course :->)

wrong
binary-decimal
1.0=1
0.1=1/2
0.11=3/4
0.01=1/4
0.111=7/8
0.101=5/8
0.011=3/8

a binary number is an expansion of the form
$$x=\sum_{n=-\infty}^\infty b_n \ 2^n$$
each bn is either 0 or 1
for real numbers their is an n such that bm=0 for all m>n
for integers bn=0 for all n<0 (in standard form)

 

[comwiz72]

wrong
binary-decimal
1.0=1
0.1=1/2
0.11=3/4
0.01=1/4
0.111=7/8
0.101=5/8
0.011=3/8

a binary number is an expansion of the form
$$x=\sum_{n=-\infty}^\infty b_n \ 2^n$$
each bn is either 0 or 1
for real numbers their is an n such that bm=0 for all m>n
for integers bn=0 for all n<0 (in standard form)

i never knew that! learn something every day!

 

[mezarashi]

Can someone tell me what binary is and to do a problem like that?

The binary system tells us about how arbitrary our base 10 system is. We have a total of 10 symbols in our number system:

0,1,2,3,4,5,6,7,8,9

The question is, what would have happened if we only somehow invented two numbers instead of 10, namely, 0 and 1. We would then have binary, and we would have to relearn all our arithmetic, and multiplication tables again (sigh*).

Or what if we invented 16 symbols instead (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) also known as hex? How would our math be different now, having numbers like 41E, and B7A90.

 

[CarlB]

I've never covered converting things to binary and other number systems before, and in a chapter about sequences the book mentions binary and has an exercise to convert the square root of two to six decimal places.

sqrt(2) = 1.4142135623730950488016887242097

First convert the integer part to binary. That's easy. 1 converts to 1.

Now take the remainder, and multiply it by 2 a bunch of times, each time keeping the integer part (which will be zero or one) as the binary bit:

1.4142135623730950488016887242097
.
0.8284271247461900976033774484194
1.6568542494923801952067548968388
1.3137084989847603904135097936776
0.62741699796952078082701958735517
1.2548339959390415616540391747103
0.50966799187808312330807834942068
1.0193359837561662466161566988414
0.038671967512332493232313397682708
0.077343935024664986464626795365416
0.15468787004932997292925359073083
0.30937574009865994585850718146166
0.61875148019731989171701436292333
1.2375029603946397834340287258467
0.47500592078927956686805745169332
0.95001184157855913373611490338664
1.9000236831571182674722298067733
1.8000473663142365349444596135466
1.6000947326284730698889192270931
1.2001894652569461397778384541862
0.40037893051389227955567690837245
0.8007578610277845591113538167449
...

That should be enough for 6 decimal places. The answer is read off the left hand column, namely (and rounding the last bit):

sqrt(10) = 1.011010100000100111101

To check this, just convert it into decimal and square it. Yep, it works.

Another fun thing to do: Figure out what sqrt(2) is to six decimal places written out in base "pi".

Carl