Can someone check my answer for this binary combo question?

- Curry

Can someone check my answer for this binary combo question?

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- Curry

##### 1 Attachment

- Curry

1. [-2^(n-1)+1, 2^n-1]
2. [-2^(n-1)+1, 2^n-1]

- Curry

@wio

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## More answers

- Curry

@ganeshie8

- ganeshie8

Let \(n\) = 8 bits
whats the largest positive number in 1's complement system ?

- Curry

um, 2^8?

- ganeshie8

nope

- Curry

wait no, 2^8 - 1

- Curry

no no, 2^7 -1

- ganeshie8

Yes, \(\large 2^7-1\) is the largest number in 1's complement system using 8 bits :
\[\large 2^7-1~~=~~0111 ~1111\]

- Curry

wait, actually i'm kinda confused about that to begin with.

- ganeshie8

what exactly are you confused about ?

- Curry

so, if i have 0000 0000, in one's compliment, cant i write 1111 1111?

- ganeshie8

thats the reason 1's complement system sucks, we have two different representations for 0 in this system :
```
0000 0000
```
and
```
1111 1111
```
both represent the same number 0 in this system

- Curry

oh ok, so 1 is the smallest number i can write that can be converted to one's complement so that it bcomes the largest?

- ganeshie8

I don't get you, could you elaborate a bit

- Curry

since i'm flipping the bits, the smallest number i write in binary would give me the greatest number in one's complement right?

- ganeshie8

you don't flip bits when the left most bit is 0

- ganeshie8

you flip only when the left most bit is 1

- Curry

oh beacuse it's positive?

- ganeshie8

yes

- Curry

so one's complement only works when we're dealing with signed magnitude numbers?

- ganeshie8

signed magnitude has nothing to do with 1's complement

- Curry

well unsigned is always positive isn't it?

- Curry

so it has to be a negative number for one's complement to have an effect right?

- ganeshie8

You maybe correct but I'm not getting you
First notice that "one's complement of a number" and "ones complement system" are two different things

- ganeshie8

In "one's complement system",
binary strings with 0 as left most bit are treated as positive
binary strings with 1 as left most bit are treated as negative

- ganeshie8

for example,
`010 0000` is a positive number because the left most bit is `0`
`1000 0000` is a negative number because the left most bit is `1`

- Curry

so how is it different frmo signed magnitude system?

- Curry

isnn't signed magnitude the same thing? where 0 is positive and 1 is negative?

- ganeshie8

before answering that, let me ask you two simple questions :
what are the values of `0100 0000` in 1's complement and signed magnitude ?
what are the values of `1000 0000` in 1's complement and signed magnitude ?

- Curry

for the first one: one complement = 0100 0000 signed magnitude = 64
second one: one complement = 1111 1111 signed magnitude = 0 ?

- ganeshie8

tell me all the values in decimal form

- Curry

for the first one it'll be 64 both times.
second: -64, 0

- ganeshie8

you're correct about first one being 64 in both systems
second one is wrong, try again

- Curry

wait positive 64 and 0.

- ganeshie8

what are the values of `1000 0000` in 1's complement and signed magnitude ?
#1's complement system :
since the left most bit is `1`, we need to take 1's complement of this for the value and put minus sign.
1's complement = 0111 1111 = 127
so the value is \(\large -127\)
#signed magnitude system :
since the left most bit is `1`, the value is simply \(\large -0\)

- ganeshie8

so what do you notice ? what exactly is the difference between '"ones complement system" and "signed magnitude system" ?

- ganeshie8

http://gyazo.com/ee939ee2e32939bec143ee78b1d6afd1

- Curry

well just with respect to the left most bit, it's the same for positive numbers.

- Curry

OOO! so if i have a binary number, and if it's positive, for one/two complement and signed magnitude, it'll be the same.

- Curry

if there is a one in front of the binary, then i flip all the bits, calculate new decimal value, and add negative number.

- Curry

and for two's compliment, i write teh number as positive first, flip all the bits, and add 1.

- Curry

do i have this right?

- ganeshie8

Exactly!

- ganeshie8

coming back the original question, whats the smallest number in 1's complement system using 8 bits ?

- Curry

-127?

- ganeshie8

Yes
\[-127 = 1000~0000\]
this is the smallest number in 1's complement system using 8 bits

- ganeshie8

using \(8\) bits, it seems we can express the numbers from \(-(2^{7}-1)\) to \(2^7-1\) ?

- Curry

yes.

- Curry

so for the biggest number would it be, 1111 1110?

- ganeshie8

left most bit = 1, so thats a negative number
think a bit, how can a negative number be the biggest ?

- Curry

no 1111 1110 is in one's complement. so in just binary, it'd be 0000 0001.

- ganeshie8

1111 1110 = -1

- ganeshie8

how do you say -1 is the biggest number ?

- Curry

so if it doesn't matter when it's a positive number, would it be 0111 1111, which is 128?

- Curry

sorry, 127.

- ganeshie8

thats right, 127 is the biggest number using 8 bits in 1's complement system

- Curry

oh so it's just that simple? 1111 1111 , 0111 1111?

- Curry

How would I do it for tw's complement? or should i post this as a new question?

- ganeshie8

wait,
```
oh so it's just that simple? 1111 1111 , 0111 1111?
```
what are you trying to say here

- Curry

so in one's complement, to find the greatest and smallest number given n bits, it's always
2^(n-1) -1

- Curry

wait positive and negative 2^(n-1) -1.

- ganeshie8

thats right, but what has that do with `1111 1111 ` ?

- ganeshie8

`1111 1111 ` = 0
right

- Curry

oh sorry, i meant 1000 0000.

- ganeshie8

now that makes sense, yeah `1000 0000` is the least number and `0111 1111` is the max number in 1's complement system using 8 bits

- Curry

oh ok, this makes a lot of sense now, thanks!!

- ganeshie8

good, sry to ask too many questions but do you happen to know why signed magnitude is not so good for representing numbers in a computer ?

- Curry

no, i don't. why?

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