79. OCR A Level (H046-H446) SLR13 - 1.4 Floating point binary part 1 - Overview

Summary

This video is the first in a series explaining floating point binary representation. It begins with an 8-bit binary number line, illustrating how integers and fractions are represented. The video then transitions to fixed point binary, where the binary point's position is fixed, limiting the range of numbers that can be stored. To overcome this, floating point binary is introduced, where the binary point 'floats', allowing for a trade-off between the size and accuracy of numbers. The video explains the division of bits into the mantissa (value) and exponent (binary point position), crucial for storing real numbers in binary. Examples are provided to demonstrate how numbers are converted from floating point binary to decimal.

Takeaways

• 📏 In floating point binary, the position of the binary point can change, allowing for the representation of both large and small numbers with varying degrees of precision.
• 🔢 The binary number line's place values double as you move from right to left, which is different from unsigned binary or two's complement.
• 👉 The most significant bit (leftmost bit) in floating point binary represents whether a number is negative or positive.
• 📉 To represent fractions, the number line is extended to the right, with place values halving as you move from left to right.
• 🚫 Fixed point binary has a limited range of numbers it can store accurately, as some bits are used for the fractional part of the number.
• 🔄 The method of changing the position of the binary point to increase accuracy or range is known as floating point binary.
• 💾 Storing a number in floating point binary requires splitting the bits into the mantissa (the actual value) and the exponent (the position of the binary point).
• 📉 The number of bits used for the mantissa and exponent is determined by the data type, such as 24 bits for the mantissa and 8 for the exponent in a 32-bit single precision number.
• ➕ The exponent in floating point binary is stored in two's complement, which indicates how many places to move the binary point.
• 🔄 Moving the binary point to the right increases the size of the number, while moving it to the left decreases the size but can increase precision.

Q & A

What is the significance of the most significant bit in an 8-bit binary number representing a negative number?

In an 8-bit binary number representing a negative number, the most significant bit, which is the leftmost bit, must be a 1. This bit indicates that the number is negative.

How is the number 6.5 represented in fixed point binary format?

The number 6.5 is represented in fixed point binary format as 0011 0100. This representation includes a one in the four and two column (which equals six), plus a one in the half column, resulting in 6.5.

What is the largest positive number that can be stored in an 8-bit format using fixed point binary?

The largest positive number that can be stored in an 8-bit format using fixed point binary is 01111111, which equals 15.875 when calculated as 8 + 4 + 2 + 1 + 0.5 + 0.25 + 0.125.

Why can't the fraction 'a third' be accurately stored in the given fixed point binary format?

The fraction 'a third' cannot be accurately stored in the given fixed point binary format because the binary representation of a third (0.333...) would repeat infinitely, and the format does not allow for infinite precision.

How does the floating point binary system allow for a trade-off between the size and accuracy of numbers?

The floating point binary system allows for a trade-off between the size and accuracy of numbers by adjusting the position of the binary point. More bits can be allocated to the fractional part for higher accuracy, or to the whole number part for a larger range of values.

What are the two parts that the bits in a floating point binary number are split into, and what do they represent?

The bits in a floating point binary number are split into two parts: the mantissa, which represents the actual value of the number, and the exponent, which represents the position of the binary point within the number.

How many bits are used for the mantissa and exponent in a 32-bit single precision floating point number?

In a 32-bit single precision floating point number, 24 bits are used for the mantissa and 8 bits for the exponent.

What does the exponent in a floating point binary number indicate?

The exponent in a floating point binary number indicates how many places to move the binary point within the mantissa to determine the actual value of the number.

How is the position of the binary point determined in a floating point binary number?

The position of the binary point in a floating point binary number is determined by the exponent. The binary point starts after the most significant bit in the mantissa and is moved to the right or left based on the value of the exponent.

What is the significance of backfilling zeros in the mantissa of a floating point binary number?

Backfilling zeros in the mantissa of a floating point binary number is significant because it allows for the adjustment of the binary point's position without changing the value of the number. It is permissible as numerically, zeros are not significant.

Outlines

00:00

💡 Introduction to Floating Point Binary

This segment introduces the concept of floating point binary representation in a three-part series. It explains the binary number line, highlighting the difference between unsigned binary and two's complement systems. The significance of the most significant bit in representing negative numbers is discussed. The video uses an 8-bit example to demonstrate how to represent whole numbers and fractions, such as 6.5 and 3.75, in binary. The concept of fixed point binary is introduced, where the position of the binary point is fixed, and the limitations in the range of numbers that can be represented are discussed. The example of representing a third (0.333) is used to show the inaccuracy that can occur with fixed point representation. The video concludes by discussing the need for a more flexible system to increase accuracy, hinting at the floating point binary system.

05:00

🔍 Exploring Floating Point Binary

This part of the video delves deeper into floating point binary representation. It explains how the binary point can 'float' to increase the size or accuracy of the number being represented. The trade-off between the range of numbers and their precision is discussed, using examples to show how the binary point's position affects the number's size and accuracy. The segment introduces the concept of the mantissa and exponent in floating point binary, explaining how the bits are split to represent the number's value and the position of the binary point. The video uses examples to demonstrate how to convert floating point binary numbers into decimal, emphasizing the importance of the exponent in determining the binary point's final position. The examples illustrate how the mantissa and exponent work together to represent numbers like 6, 1.75, and 0.125 in floating point binary.

10:02

📚 Understanding the Storage of Fractions in Computers

The final segment of the video focuses on how computers store fractions or real numbers using floating point binary. It explains the process of moving the binary point based on the exponent and how to disregard the exponent after determining the point's position. The video uses a negative exponent example to show how the binary point is moved to the left, and how zeros are backfilled to complete the number. The segment concludes by posing a key question to the viewer: how does a computer store fractions or real numbers? This question is meant to reinforce the understanding of floating point binary representation and its importance in computer science.

Keywords

💡Floating Point Binary

Floating Point Binary is a method used by computers to represent real numbers, which includes both whole numbers and fractions. Unlike fixed-point binary, where the position of the binary point is fixed, floating point binary allows the point to 'float' or move. This flexibility enables the representation of a wide range of numbers with varying degrees of precision. In the video, the concept is introduced as a way to store numbers with fractional components, and it's explained that the size of the number that can be stored can be increased or decreased depending on how many bits are used for the fractional component.

💡Binary Number Line

The Binary Number Line is a visual representation used in the video to explain how numbers are represented in binary form. It shows how place values double as you move from right to left, which is different from the decimal system. This concept is fundamental to understanding how binary numbers work and is used to demonstrate the representation of both positive and negative numbers.

💡Mantissa

The Mantissa in floating point binary represents the significant digits of a number. It is the part of the number that is stored after the binary point has been moved to its correct position. The video explains that the mantissa is one of the two parts into which the bits are split, with the other part being the exponent. The number of bits used for the mantissa can vary depending on the data type, and it determines the precision of the stored number.

💡Exponent

In the context of floating point binary, the Exponent indicates the position of the binary point relative to the most significant bit of the mantissa. The video explains that the exponent is stored in two's complement form, and it tells us how many places to move the binary point to the right (for positive exponents) or to the left (for negative exponents). The exponent is crucial for adjusting the size of the number that can be represented.

💡Two's Complement

Two's Complement is a method for representing negative numbers in binary. The video mentions that the most significant bit of a number is used to indicate whether it is negative or positive. In two's complement, a negative number is represented by inverting all the bits of its positive counterpart and then adding one. This system is used for both the mantissa and the exponent in floating point binary.

💡Fixed Point Binary

Fixed Point Binary is a system where the position of the binary point is fixed, and it is used to represent numbers with a fractional component. The video contrasts this with floating point binary, explaining that in fixed point, the range of numbers that can be stored is more limited because some positions on the number line are used to store the fractional part of the number.

💡Place Values

Place Values in binary, as in any number system, refer to the value of a digit based on its position. In binary, place values double as you move from right to left, which is different from the decimal system where they increase by a factor of ten. The video uses the concept of place values to explain how binary numbers are constructed and how they represent both positive and negative values.

💡Most Significant Bit (MSB)

The Most Significant Bit (MSB) is the leftmost bit in a binary number and plays a crucial role in determining the sign of the number in two's complement representation. The video explains that if the MSB is a 1, the number is negative, and if it's a 0, the number is positive. This is a fundamental concept in understanding how binary numbers represent both magnitude and sign.

💡Precision

Precision in the context of floating point binary refers to the accuracy or the number of significant digits that can be represented. The video discusses how the precision of a number can be increased by using more bits for the fractional component, but this comes at the cost of reducing the range of numbers that can be represented.

💡Data Type

A Data Type in computing defines the kind of data a variable can hold and the operations that can be performed on it. The video mentions different data types, such as 32-bit single precision, which uses a specific number of bits for the mantissa and the exponent. Understanding data types is essential for knowing how numbers are stored and processed in a computer system.

Highlights

Introduction to floating point binary in a three-part series.

Explanation of 8-bit binary number line with place values doubling to the left.

Differentiation between unsigned binary and two's complement for representing negative numbers.

Representation of the number six in binary as 00000110.

Extension of the number line for fractional components, halving place values to the right.

Binary representation of 6.5 as 001101000.

Binary representation of 3.75 as 00011110.

Storing negative numbers with fractional components using the most significant bit.

Binary representation of -6.5 as 110001100.

Definition and example of fixed point binary with a fixed position for the binary point.

Limitation in the range of numbers storable in fixed point binary due to fractional part usage.

Challenge of accurately representing fractions like a third in fixed point binary.

Introduction to floating point binary with a floating binary point for increased flexibility.

Trade-off between number size and accuracy in floating point binary representation.

Explanation of mantissa and exponent in floating point binary to store the number and binary point position.

Example of storing the number 110001 in floating point binary with 5 bits for mantissa and 3 for exponent.

Conversion process from floating point binary to base 10 using the mantissa and exponent.

Handling of negative exponents in floating point binary and their effect on binary point movement.

Final example demonstrating the storage of 0.125 in floating point binary with negative exponent.

Summary of how floating point binary allows for dynamic adjustment of number size and accuracy.

Transcripts

this video is part one in a three-part

series on floating point binary

in this video we provide an overview of

the topic

[Music]

look at this 8-bit binary number line

note how the place values double as we

move right to left

unlike unsigned binary with two's

complement the leftmost bit the most

significant bit

always represents a negative number

so the number six would be represented

as zero zero zero zero

zero one one zero that's a one in the

four column

plus a one in the two column four plus

two is six

to store numbers with a fractional

component such as six point five

we extend the number line from left to

right

note how the place values now half as we

move from left to right

we place a binary point between the 1

and the half

using this format 6.5 will be

represented as 0 0

1 1 zero one zero zero

so that's a one in the four and two

column four plus two is six

plus a one and a half column for six

point five

the number three point seven would be

represented as zero zero zero one

one one one zero so a two plus a one

is three plus a half is three point five

plus a quarter is three point seven five

we can also easily store negative

numbers with a fractional component

using this format

remember if this is a negative number

the most significant bit that's the

leftmost bit must be a 1.

so using this format minus 6.5 will be

represented as 1

1 0 0 1 1 0 0 that's minus 16

plus an 8 bring it up to 8 plus a 1

bringing us up to 7 plus a half bringing

us up to

minus 6.5

this method of storing is known as fixed

point binary

as the position of the point is fixed on

the number line

note that the range of numbers we can

now store is more limited

as some positions on the number line are

now being used to store

the fractional part of the number

biggest positive number we could now

store in this eight bit format

is zero one one one one one

one an eight plus a four plus a two plus

a one

plus a half quarter and an eighth or

15.875

now you may have spotted that some

numbers

can't be stored accurately at all

so looking at the format we've got on

the screen now how would you store

a third

well the best we could manage in this

format would be zero

zero zero zero zero zero one zero

which represents a quarter or 0.25

not a third however it's the closest we

can get

in this format

we could extend the number line to allow

for more fractional parts

and therefore more accuracy but the

pattern just repeats

one zero one zero one zero and it

actually repeats forever

we can't actually ever represent a third

precisely

to increase accuracy we can change

the way we use the binary point

while we're still using eight bits to

store the number here the eight bits on

the number line

change

to put it another way the binary point

is floating

up and down the number line

in this example we're using 8 bits from

-8 to 1 16 4 bits for the whole number

and 4 bits for the fractional component

in this example we're still using 8 bits

but this time the numbers from -64

down to a half so we're using seven bits

for the whole number

and only one bit for the fractional part

note how the size of the number we can

store has increased

but that's at the cost of reduced

accuracy

and in this example we're still using

eight bits

from minus four but this time all the

way down to one thirty second

that's three bits for the whole number

and a whole five bits for the fractional

part

so the size of the number we can store

has now really decreased

but we can now store it with a much

higher degree of accuracy

or precision

now the important takeaway here is that

the binary point is

moving it's floating up and down the

number line

as it does we can use more or fewer bits

to store the fractional component

and this is known as floating point

binary

the big advantage of this approach with

just eight bits is we can either choose

to increase the size of the number

or the accuracy of the number

now it may not have escaped or noticed

that with floating point binary

we've actually introduced a new problem

along with the actual number which we

need to store in binary

we also need to store the position of

the binary point

on the number line

we do this by splitting the bits into

two parts

the mantissa which is going to represent

the actual value of the number itself

and an exponent the position of the

binary point

within that number the number of bits

used for mantis for an exponent is

determined by the data type

a 32-bit single precision number uses 24

bits for the mantissa

and 8 for the exponent giving 24 bits of

precision

now in the exam the number of bits used

for mantissa for an exponent will always

be stated

in this example we're going to have five

bits for mantissa

and three for the excellent

so let's look at an example of a number

being stored in floating point binary

we're going to go with zero one one zero

zero

zero one one with the first five bits

for mantissa

and the last three bits for the exponent

to convert this into base 10 dna we

first have to look at the exponent as

this is going to tell us

where the binary point is going to need

to end up

note that the binary point always starts

after

the most significant bit in the mantissa

between the first two digits but that's

not

where it's going to end up we're going

to move it

based on this exponent

so we start by converting the exponent

now note

the most significant bit is representing

negative four because we store it in

two's complement

so our exponent is three we've got a one

in the two column

and a one in the one column two plus one

is three

so our exponent of three is telling us

we need to move the binary point in our

mantissa

three places to the right we move it to

the right

because the exponent is positive

now we're finished with the exponent now

all it was doing was storing how many

places to move the binary point

now we've moved it we can ignore it so

we're now left with the number

not one one naught point naught

we apply the normal binary waiting line

and add up

any columns that have a one in them

again note the most significant bit is

representing minus eight not positive

because we're using two's complement so

we've got a one in the four column and a

one in the two

four plus two is six so that's what this

actual number

was representing the dna value six

okay let's have another look at another

example again eight bits

five for the mantissa and three for the

exponent this number is zero one one one

zero

zero zero one

now our exponent is one that's all we've

got here nothing in the two column

nothing in the minus four

so it tells us to move the binary point

in the mantissa

one place to the right positive exponent

moves the binary point to the right

finished with the exponent so we

disregard it and we end up with the

number zero one

point one one zero we apply the normal

binary waiting line

and add up any columns with a one in we

have a one

plus a half plus a quarter so this

number

is 1.75

okay in this final example we're going

to use a negative exponent so again 8

bits

5 for the mantissa three for the

exponents the full number

zero one zero zero zero one one zero

so we start by working out the exponent

the exponent is

minus two minus four plus two is minus

-2

so we need to move the binary point in

our mantissa

two places to the left this time because

it's a negative

exponent now we can't easily move the

binary point off the left of this screen

visually so we're just going to slide

the bits across two places which serves

the same purpose

we can now disregard the exponent and we

end up with

zero point zero zero one zero zero zero

we apply the normal binary waiting line

and add up the columns with a one

all we've got is a one in the eighth

column so this number was naught

point one two five note how we had to

backfill for a couple of zeros to make

the number work

this is allowed as numerically zeros are

not significant

if the number is positive we can always

backfill with zeros as needed

if the number is negative we backfill

with ones as needed instead

having watched this video you should be

able to answer the following key

question

how does a computer store fractions or

real numbers

[Music]

you