Express 473.5629 in base 10 as a binary number?

1 Answer
Jul 3, 2018

#473.5625 = 111011001.1001_2#

#473.5629 ~~ 111011001.100100000001101000110110111001011_2#

Explanation:

I wonder whether the number given should have been #473.5625# since that has a terminating binary representation, but let us assume that the number given is correct.

First work on the integer part:
The first few powers of #2# (which can be found by taking #1# and doubling it repeatedly) are:

#1, 2, 4, 8, 16, 32, 64, 128, 256, 512#

We find:

#473 = 256+128+64+16+8+1 = 111011001_2#

For the fractional part, we can repeatedly double and note down the integer part before discarding it to give the next binary digit:

#0.5629 * 2 = color(red)(1).1258#

#0.1258 * 2 = color(red)(0).2516#

#0.2516 * 2 = color(red)(0).5032#

#0.5032 * 2 = color(red)(1).0064#

#0.0064 * 2 = color(red)(0).0128#

#0.0128 * 2 = color(red)(0).0256#

#0.0256 * 2 = color(red)(0).0512#

#0.0512 * 2 = color(red)(0).1024#

#0.1024 * 2 = color(red)(0).2048#

#0.2048 * 2 = color(red)(0).4096#

#0.4096 * 2 = color(red)(0).8192#

#0.8192 * 2 = color(red)(1).6384#

#0.6384 * 2 = color(red)(1).2768#

#0.2768 * 2 = color(red)(0).5536#

#0.5536 * 2 = color(red)(1).1072#

#0.1072 * 2 = color(red)(0).2144#

#0.2144 * 2 = color(red)(0).4288#

#0.4288 * 2 = color(red)(0).8576#

#0.8576 * 2 = color(red)(1).7152#

#0.7152 * 2 = color(red)(1).4304#

#0.4304 * 2 = color(red)(0).8608#

#0.8608 * 2 = color(red)(1).7216#

#0.7216 * 2 = color(red)(1).4432#

#0.4432 * 2 = color(red)(0).8864#

#0.8864 * 2 = color(red)(1).7728#

#0.7728 * 2 = color(red)(1).5456#

#0.5456 * 2 = color(red)(1).0912#

#0.0912 * 2 = color(red)(0).1824#

#0.1824 * 2 = color(red)(0).3648#

#0.3648 * 2 = color(red)(0).7296#

#0.7296 * 2 = color(red)(1).4592#

#0.4592 * 2 = color(red)(0).9184#

#0.9184 * 2 = color(red)(1).8368#

#0.8368 * 2 = color(red)(1).6736#

...

Eventually the fractional part will repeat, and so will the binary expansion, but I will leave that to you to find.

For now, we can combine our integer and fractional parts to find:

#473.5625 = 111011001.1001_2#

#473.5629 ~~ 111011001.100100000001101000110110111001011_2#