Fixed Point: Taming Those Tricky Fractions!

So, you thought binary was just for whole numbers, did you? Wrong! It turns out computers can also get their digital teeth into fractions – those bits of numbers that aren't quite whole. This is where something called fixed point binary comes into play. It's a way of drawing a line in the digital sand (or rather, a point in the binary string) to separate the whole number part from the fractional bit. Clever, eh?

Think of it like this: you've got your usual binary numbers marching along, and then, suddenly, there's a binary point. Everything to the left of that point is the normal whole number bit. But everything to the right? That's where the fractions live. The first digit after the point represents 2^-1 (which is a half in our regular numbers), the next one is 2^-2 (a quarter), then 2^-3 (an eighth), and so on, getting smaller and smaller as you go further to the right. It's like a digital shrinking ray for fractions!

Fixed Point to Denary: Decoding the Fractional Fun

Now, if you've got a fixed point binary number and you want to know what it means in our regular denary (or decimal) system, there's a little process to follow. They've even given us a handy way to organise it: a table! Let's have a squint at that:

See how that point separates the powers of 2? On the left, you've got the usual positive powers (like 2⁰ which is 1, 2¹ which is 2, and so on). But on the right, after the dot, the powers become negative. That 2^-1 is like saying ½, and 2^-2 is like 1/2² = ¼, and so on. It's all about those inverse powers!

Denary to Fixed Point: Wrangling Whole Numbers and Naughty Fractions!

So, the digital overlords now want you to go the other way – take our perfectly sensible denary numbers and cram them into the world of fixed point binary. It's like trying to fit a particularly awkward octopus into a very specific jam jar.

If you're dealing with the nice, well-behaved whole number part of your denary number, the process is pretty much what you'd expect. You just keep dividing by 2 and noting down the remainders (those 0s and 1s) until you hit zero. Then you read the remainders in reverse, and Bob's your uncle – the whole number bit is sorted.

But then you've got those pesky fractions after the decimal point. They don't behave so nicely. To convert them, you have to get into a multiplying mood. You take the fractional part and keep multiplying it by 2. Each time you do this, you look at the bit that pops up *before* the decimal point (it will be either a 0 or a 1). You write these down, from top to bottom, and that's your fractional part in binary. You keep going until you hopefully end up with exactly 1.0. Sometimes, though, these fractions can be a bit stubborn and never quite reach 1.0, meaning you might have to stop after a reasonable number of steps. It's a bit like chasing a digital greased pig!