How to convert the decimal 48.365 to binary?

Question

Please explain the part after the point carefully, I am weak in decimal-related divisions.....

Answer 1

First you need the base 2 table like this:

2^7 = 128

2^6 = 64

2^5 = 32

2^4 = 16

2^3 = 8

2^2 = 4

2^1 = 2 <-- 2nd to last least significant bit

2^0 = 1 <-- least significant bit

DECIMAL POINT

2^(-1) = 1/2 = 0.5 <-- 1st decimal

2^(-2) = 1/4 = 0.25 <-- 2nd decimal

2^(-3) = 1/8 = 0.125 <-- 3rd decimal

2^(-4) = 1/16 = 0.0625

2^(-5) = 1/32 = 0.03125

So the 48= 32+16 which is binary 00110000

Now for the decimal part that you are interested in...

1/2 is too big, so the first bit past the decimal is zero

Exponent -2: 0.365 - 0.25 = 0.115

Exponent -3: 0.125 is too big, so this bit is zero

Exponent -4: 0.115 - 0.0625 = 0.0525

Exponent -5: 0.03125 - 0.0525 = 0.02125

Exponent -6: 1/64 = 0.015625, so 0.02125-0.015625 = 0.005625

Exponent -7: 1/128 = 0.0078125 is to big so that bit is zero

Exponent -8: 1/256 = 0.00390625, so 0.005625 - 0.00390625=0.00171875

<next byte>

Exponent -9: 1/512 = 0.001953125 is too big , so that bit is zero

Exponent -10: 1/1024 = 0009765625, so 0.00171875-0.0009765625=

0.0007421875

Exponent -11: 1/2048 =.00048828125, so 0.0007421875-0.0007421875=

0.00025390625

Exponent -12: 1/4096 = 0.000244140625, so 0.00025390625- 0.000244140625=

0.000009765625

Exponent -13: 1/8192 = 0.0001220703125, is too big, so that bit is zero

Exponent -14: 1/16384 = 0.00006103515625 is too big, so that bit is zero

Exponent -15: 1/32786 = 0.00003050082352223510034770938815348

so 0.000009765625 - 0.00003050082352223510034770938815348=

0.00002073519852223510034770938815348

Exponent -16:

1/65572 = 0.00001525041176111755017385469407674,

so 0.00002073519852223510034770938815348-0.00001525041176111755017385469407674=

= 0.0000054847867611175501738546940732599

Hopefully by now you see the pattern...

the bit string is then

00110000.0101110101110011.....