8. CAMBRIDGE IGCSE (0478-0984) 1.1 Signed integers using two's complement

Summary

This video explores signed integers using two's complement in binary representation. It explains how positive numbers are represented with a leading zero and negative numbers with a leading one. The concept of two's complement is introduced through an analogy of a car's odometer, showing how negative values can be represented by flipping the digits. The video demonstrates how to convert positive 117 into its negative counterpart using two's complement and offers a trick to easily find the negative version of any binary number by inverting the bits after the first one encountered when reading from right to left.

Takeaways

• 🔢 The smallest binary number representable in eight bits is zero, denoted by a sequence of zeros.
• 🚗 Adding ones in any column increases the binary value, similar to a car's odometer.
• 🔄 Negative numbers in binary are represented using two's complement, a method analogous to a car's odometer running backward.
• 🌐 Positive numbers in two's complement start with a zero, and negative numbers start with a one.
• 💡 The most significant bit (MSB) in two's complement represents negative values, flipping the traditional binary representation.
• 🔑 To represent a positive number in two's complement, place a zero in the MSB and fill other bits as needed to achieve the value.
• 🔄 To represent a negative number in two's complement, start with a one in the MSB and fill other bits to achieve the negative value.
• 🎯 The process of converting a positive binary number to its negative two's complement involves flipping bits after the first one encountered from the right.
• 🔄 Flipping bits after the first one from the right in a binary number is a reliable trick to get the negative two's complement.
• 📚 The correctness of two's complement representation can be verified by adding a number to its negative and expecting zero as the result.

Q & A

What is the smallest number that can be represented in binary using eight bits?

The smallest number that can be represented in binary using eight bits is zero, which would be represented as a sequence of eight zeros.

How does adding ones to a binary number affect its value?

Adding ones to a binary number increases its value. For example, adding a one to the least significant bit (rightmost column) increases the value from zero to one.

What is the concept used by computers to represent negative numbers in binary?

Computers use a concept called two's complement to represent negative numbers in binary.

In two's complement, what is the significance of the most significant bit (MSB)?

In two's complement, the most significant bit (MSB) represents a negative value, with a one indicating a negative number and a zero indicating a non-negative number.

How are positive numbers represented in two's complement?

In two's complement, positive numbers start with a zero in the most significant bit, followed by the binary representation of the number.

What is the binary representation of the number 117 in two's complement?

The binary representation of the positive number 117 in two's complement would have a zero in the most significant bit, followed by the binary digits that sum up to 117.

How can you represent the negative version of 117 using two's complement?

To represent -117 in two's complement, you start with a one in the most significant bit and then place ones in the columns to bring the value up to -117, similar to how you would count up from -128.

What is the trick to convert a two's complement number into its negative version?

To convert a two's complement number into its negative version, write out the positive version, then starting from the right, copy each digit up to and including the first one, and after that point, swap every one for a zero and every zero for a one.

How can you verify the correctness of the negative version of a number in two's complement?

You can verify the correctness by adding the positive and negative versions of the number in two's complement, which should result in a row of zeros, indicating that the sum is zero.

What is the binary representation of negative 12 using the trick mentioned in the script?

To represent -12 in binary using the trick, you start with the positive binary representation of 12 (1100), copy from the right up to and including the first one, and then invert the remaining bits, resulting in 1001, which is the two's complement representation of -12.

Outlines

00:00

💡 Understanding Two's Complement for Signed Integers

This paragraph introduces the concept of two's complement, a method used in computing to represent signed integers in binary form. It explains that while adding ones to a binary number increases its value, representing negative numbers requires a different approach. The analogy of a car's odometer is used to explain how negative numbers can be represented by 'turning back' the count, similar to how two's complement works. The paragraph also notes that in two's complement, positive numbers start with a zero, zero is represented as a sequence of zeros, and negative numbers start with a one. The example of representing the number 117 in two's complement is given, showing how the most significant bit (MSB) indicates the sign of the number, with zeros for positive and ones for negative values. The process of representing -117 is also explained, where after setting the MSB to one, ones are added in other columns to adjust the value to -117.

05:02

🔄 Converting Positive to Negative Binary Numbers

This paragraph demonstrates a trick for converting a positive binary number into its negative counterpart using two's complement. The process involves writing out the positive binary number, copying each bit from right to left up to and including the first one encountered, and then inverting all subsequent bits (changing ones to zeros and zeros to ones). The example of converting the positive binary number for 12 into its negative form is provided. The paragraph concludes by suggesting that viewers will be able to verify the correctness of this method by adding the positive and negative binary numbers in an upcoming video, which should result in a zero if done correctly.

Keywords

💡Two's Complement

Two's complement is a method used in computer systems to represent signed integers in binary form. It allows for the representation of both positive and negative numbers using the same binary format. In the video, two's complement is introduced as a concept that enables computers to handle negative numbers alongside positive ones, with the most significant bit (MSB) indicating the sign of the number, where 0 represents positive and 1 represents negative. The video uses the example of representing -10 in binary to illustrate how two's complement works.

💡Binary

Binary is a numeral system that represents numeric values using two symbols, typically 0 and 1. It is the basis for all digital computing, as computers process and store data in binary form. The video discusses binary in the context of representing numbers, starting with the simplest binary representation of zero, which is a sequence of zeros, and then incrementing by adding ones to represent positive values.

💡Most Significant Bit (MSB)

The most significant bit, or MSB, is the leftmost bit in a binary number and has the greatest value in a positional numeral system. In the context of two's complement, the MSB is crucial as it determines whether the number is positive or negative. The video explains that in two's complement, a 0 in the MSB position indicates a positive number, while a 1 indicates a negative number.

💡Negative Numbers

Negative numbers are values that are less than zero. The video explores how negative numbers can be represented in binary using two's complement. It explains that all negative numbers in two's complement start with a 1, which is the MSB, and that the rest of the binary digits are used to specify the magnitude of the negative value.

💡Positive Numbers

Positive numbers are values that are greater than zero. The video clarifies that in two's complement, all positive numbers, including zero, start with a 0 in the MSB position. This is demonstrated when the video shows how to represent the positive number 117 in binary, where the MSB is 0, followed by a sequence of 1s in the columns that sum up to 117.

💡Mileometer Analogy

The mileometer analogy is used in the video to help explain the concept of two's complement by comparing it to a car's odometer. The analogy suggests that just as an odometer can represent both positive distances traveled (miles driven) and negative distances (miles driven back), two's complement allows a binary system to represent both positive and negative numbers.

💡Least Significant Bit

The least significant bit, or LSB, is the rightmost bit in a binary number and has the smallest value in a positional numeral system. The video mentions the LSB in the process of converting a positive binary number to its negative counterpart using two's complement, where the conversion starts from the LSB and the first one encountered marks the point where the rest of the bits are inverted.

💡Bit

A bit is the basic unit of information in computing and digital communications, representing a logical state with a value of 0 or 1. The video uses the term 'bit' when explaining binary representation and the two's complement system, where each bit in a sequence contributes to the overall value of the number being represented.

💡Magnitude

Magnitude in the context of the video refers to the size or absolute value of a number,不考虑正负号. It is used when discussing how to represent negative numbers in two's complement; the magnitude of the number is represented by the binary digits following the MSB, which indicates the sign.

💡Inversion

Inversion in the context of the video refers to the process of flipping binary digits from 0 to 1 or from 1 to 0. This is part of the trick mentioned for converting a positive two's complement number to its negative equivalent, where after copying the binary digits up to the first one, all subsequent digits are inverted.

Highlights

Binary representation of the smallest number is zero, with all bits set to zero.

Adding ones to any column in binary increases the value of a number.

Negative numbers in binary are represented using two's complement.

The car's milometer analogy explains the concept of negative values in binary.

In two's complement, positive numbers start with a zero, and negative numbers start with a one.

The most significant bit (MSB) in two's complement represents negative values.

The process of representing the positive number 117 in binary is explained.

The process of representing the negative version of 117 in binary using two's complement is detailed.

A neat trick is described for converting a two's complement number into its negative version.

The trick involves copying digits from the positive version up to and including the first one, then inverting the rest.

The conversion of positive 12 to negative 12 using the described trick is demonstrated.

The upcoming video series will teach how to add two binary numbers together.

The addition of +12 and -12 in binary should result in zero, proving the correctness of the representation.

The video concludes with a prompt to practice binary addition after watching the upcoming video.

Transcripts

in this video we take a look at signed

integers using two's complement

[Music]

so we know from our previous video that

the smallest or you could say lowest

number we can represent in binary is

zero and if we were using eight bits it

would simply be a sequence of zeros in

every column

we also know that if we add ones at all

in any column the number becomes

positive and increases in value so even

if we put a one in the smallest column

waiting on the right we still have a

value that's increased from zero

so this begs the question if adding any

ones at all increases the value of a

number

how can we possibly represent negative

numbers in binary for example -10

so let's just step back and think about

a little analogy

imagine a car's milometer so the car

leaves the factory and all the digits

are set to zero and we can think of this

as being

the value zero

drive the car for one mile

and the mile meter ends up at zero zero

zero zero zero one in this situation and

we could think of this as representing

positive one mile

imagine we could turn the meter back one

mile however from the starting position

it would now read 99999

and we could think of this or interpret

this as minus one mile

computers are able to use a very similar

concept called two's complement we're

going to take a look at that now

so before we dive into how it actually

works a couple of observations

here are the numbers minus 3

up to positive 3 in 2's complement

you will notice that when using the

two's complement method to represent

binary numbers

that all positive numbers start with

zero

this is also the case for the number

zero this is neither considered positive

nor negative in mathematics and that

also starts with a zero

likewise you'll notice that all negative

numbers always start with a one

so let's take a look at how this

actually works

so here's our standard binary waiting

line starting with a 1 on the right hand

side and then doubling as we move left 2

4 8 16 etc

you'll notice that when using the two's

complement method the most significant

bit the msb so that's the left most bit

now represents a negative value

so we've gone 16 32 64 and instead of

the left column representing 128 it now

represents

minus 128

okay so now we've changed our waiting

line let's try representing the positive

number 117

well it's a positive number therefore

the most significant bit must be a zero

so we'll pop that in

and then obviously like we've shown you

in the previous video we pop ones in

every column we need to add the values

to 117

so we have a 1 in the 64 column plus a

32 plus a 16 plus a four plus a one and

we have a positive one hundred and

seventeen

well now let's store the negative

version of 117

well as mentioned earlier

all negative numbers in binary start

with a 1 in the left-hand column if

we're using two's complement

because we have a value of a minus 128

because we have a one in that left

column we've got one lot of a minus 128

we now need to place ones in the columns

to bring the value

up to minus 117.

so if we put a 1 in the 8 column we're

saying

minus 128

plus

positive eight so that's bringing us up

to minus 120.

we then put a one in the two column and

now this brings our value up from minus

120 to -118

and finally we put a 1 in the 1 column

and that brings us up from -118 to the

target value we want of minus 117

now there is actually a really neat

trick that you can use that always works

which turns a two's complement number

into its negative version

so the process is as follows

start by writing out the positive

version of the number

then starting from the most right hand

digit so that's the least significant

bit

you copy out each digit exactly as they

appear up to and including the first one

you come across

after this point you continue but now

you swap every one for a zero and every

zero for one

so let's actually look at this in

practice we can convert the number

positive 12

into

negative 12.

so first we write out the positive

version of the number so here's the

number 12 written in binary we've got a

1 in the 8 column plus a 1 in the 4

column 8 plus 4 is 12.

we then start from the right hand side

the least significant bit and we simply

copy every value working left

up to

and including the first one so you can

see we've done that there zero becomes

zero zero becomes zero one becomes one

after this point we now switch

so where we had a one we instead write a

0 and where we had a 0 we then write a

1.

what you've ended up with is the

negative version of 12 and this works

every time now in a video coming up in

this series we're going to show you how

to add

two binary numbers together

once you've watched that video you'll be

able to prove to yourself that what

you've got on the screen here is correct

because as you know if you add plus 12

to -12 you should end up with zero

and indeed if you perform addition on

these two numbers you will end up with a

row of eight zeros give that a go once

you've watched the video

[Music]

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