What is the best approximation to the value one-tenth that can be represented using the 8 -bit f

What is the best approximation to the value one-tenth that...

Key Concepts

Floating-Point Representation

This concept involves representing real numbers using a fixed number of bits that are typically divided into a sign, exponent, and significand (or mantissa). Understanding how these parts work together to encode a wide range of values is critical, as it explains why some numbers cannot be represented exactly and must be approximated.

Normalization and Bias in Exponent Encoding

Floating-point formats usually represent numbers in normalized form, where the significand is adjusted to have a single nonzero digit before the binary point. An exponent bias is applied to allow for both positive and negative exponents. These ideas are essential to understanding how the range and precision of representable numbers are determined in any floating-point format.

Binary Representation of Decimal Fractions

Many decimal fractions, such as one-tenth, have repeating representations in binary. This is a fundamental concept because it means that such numbers cannot be exactly represented in a binary floating-point format, which leads to the need for approximation and introduces rounding errors.

Precision and Rounding in Limited Bit-Width Formats

When representing numbers like one-tenth using a format with a limited number of bits (such as an 8-bit floating-point format), the concept of rounding becomes crucial. The finite precision forces the representation to be rounded to the nearest representable value, which makes it important to understand rounding rules and the resulting quantization errors.

Transcript

00:01 All right, so in this question, we're asked to approximate one -tenth into a floating point representation.

00:08 We've got a couple options like 1 -8th and 1 -16th, and i've written a question mark down here at the bottom to show that there are actually a few more options.

00:17 First, i want to talk about how floating point is going to work for this problem.

00:22 So 1 8th is going to look something like this with 0 .001.

00:29 116th is going to be 0 .0001.

00:32 But there are ways to get much, much smaller numbers.

00:36 For example, if we have our radix point starting out right here, we have 1 .010.

00:43 But then in floating point representation, we tell the radix point to move three spaces to the left.

00:49 That's going to be a pretty big difference.

00:54 That's a 16th place and a 64th place.

01:00 So it's possible that this is a closer approximation to one tenths.

01:04 So the first thing we're going to do is find one of these small numbers, maybe something in the 64th place or the 128th place that is actually a lot closer to one tenth.

01:16 So if you're interested in doing this part, go ahead and pause the video, get out a calculator, and try to see which number in 64th or 128th.

01:23 28th place will be closest to match 1 tenths and we'll come back.

01:32 Okay, here's what we've got so far.

01:35 In the top right, first i wanted to look at our, well, our smallest number would be 1 over 256.

01:41 I wrote 15 over 256 and that's still too small.

01:46 So next i went to 15 over 128, which is getting a lot closer.

01:51 And from there, i used a calculator to find 13 over 128 and 12.

01:55 I approximated those, and it's looking to me, like out of all of these options, this one is the closest to one -tenth...