8. CAMBRIDGE IGCSE (0478-0984) 1.1 Signed integers using two's complement
Summary
TLDRThis 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
💡 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.
🔄 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.
Mindmap
Keywords
💡Two's Complement
💡Binary
💡Most Significant Bit (MSB)
💡Negative Numbers
💡Positive Numbers
💡Mileometer Analogy
💡Least Significant Bit
💡Bit
💡Magnitude
💡Inversion
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]
you
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