Decimal Points and Binary Points

How can you take 100.1 (binary) and get 4.5 (decimal)

I understand that:
100(binary) = 4.0(dec) because that is the same as saying there are no ones, no twos, and one four.

I also understand 1101 to mean 13 because there it is 1 + (0 * 2) + (1 * 4) + (1 * 8).

I don't understand how 100.1(bin) can equal 4.5(dec). I would have thought it to mean 4.1

Can you have a binary number be something like 2021 which in decimal would be (1 * 1) + (2*2) + (0 * 4) + (2 * 8). It is still in binary powers, it is just saying how many of each power exists.

EDIT
and btw this is from a book, in case u cared!

You seem to mixing binary and decimal. Just because you go to the right of the 'decimal' point, doesn't mean you switch back to decimal again.

100.1 is 4.5 because 1 is half of 10(b), in the same way 5 is half 10(d).

Here's a question for you. How do you express 4.2 in binary terms?

>Can you have a binary number be something like 2021

I don't follow you. Are saying can 2021 be a binary number? The answer is no, there is no digit 2 in binary.

Assume this is a number in binary:

abc.def

Where each letter is a digit, 1 or 0.
To get the decimal number you do this:

(a * 2^2) + (b * 2^1) + (c * 2^0) + (d * 2^-1) + (e * 2^-2) + (f * 2^-3)
a*4 + b*2 + c*1 + d*(1/2) + e*(1/4) + f*(1/8)

The power is positive on the left side and negative on the right side, and incrementing by 1 in both directions from the decimal point.

Quote:

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Originally posted by Shadow12345
100.01?

I don't really get it

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100.01

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v

(1 * 4) + (0 * 2) + (0 * 1) + (0 * (1/2)) + (1 * (1/4)) =
4 + (1/4) =
4.25

HOw do i take 4.2 and turn that into binary?
I'm still confused lol

4.2 = 100.01

I get 4.2 to be 100.0011

To get the fractional value you keep multiplying the fractional value until you get the same value started with or you have reached the required precision.

e.g 0.2 to binary.

0.2 * 16 = 3.2
0.2 * 16 = 3.2

As you keep multiplying you take the fractional value from the result to the next multiplication. You could multiply the fractional value by 2, but it would take a bit longer. Once you have finished multiplying you take each significant digit, and convert to binary. Since we have a 3, 3 in binary is 0011 (4 binary bits to one hex). Notice how we got the same answer as what we started with! You then stop.

Keep in mind that not all numbers can be represented exactly. For example 1/3 = 0.3333333.... in decimal. Binary numbers have the same characteristics for certain numbers also.

In decimal numbers, the first digit (in the decimal part) represents 1/10, the second 1/100, the third 1/1000 and so on...

In other words:
10^-1, 10^-2, 10^-3 and so on...

So it isn't so hard to realize that, in binary numbers, the first digit after (in the decimal part) represents 1/2, the second 1/4, the third 1/8 and so on...

In other words:
2^-1, 2^-2, 2^-3 and so on...