Question

Can someone check my answer for this binary combo question?

Answer 1

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

Answer 2

@wio

Answer 3

@ganeshie8

Answer 4

Let \(n\) = 8 bits

whats the largest positive number in 1's complement system ?

Answer 5

um, 2^8?

Answer 6

nope

Answer 7

wait no, 2^8 - 1

Answer 8

no no, 2^7 -1

Answer 9

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

Answer 10

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

Answer 11

what exactly are you confused about ?

Answer 12

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

Answer 13

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

Answer 14

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?

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

you flip only when the left most bit is 1

Answer 19

oh beacuse it's positive?

Answer 20

yes

Answer 21

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

Answer 22

signed magnitude has nothing to do with 1's complement

Answer 23

well unsigned is always positive isn't it?

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

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`

Answer 28

so how is it different frmo signed magnitude system?

Answer 29

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

Answer 30

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 ?

Answer 31

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

Answer 32

tell me all the values in decimal form

Answer 33

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

Answer 34

you're correct about first one being 64 in both systems

second one is wrong, try again

Answer 35

wait positive 64 and 0.

Answer 36

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 \(\large -0\)

Answer 37

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

Answer 38

http://gyazo.com/ee939ee2e32939bec143ee78b1d6afd1

Answer 39

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

Answer 40

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.

Answer 41

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

Answer 42

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

Answer 43

do i have this right?

Answer 44

Exactly!

Answer 45

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

Answer 46

-127?

Answer 47

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

Answer 48

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

Answer 49

yes.

Answer 50

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

Answer 51

left most bit = 1, so thats a negative number

think a bit, how can a negative number be the biggest ?

Answer 52

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

Answer 53

1111 1110 = -1

Answer 54

how do you say -1 is the biggest number ?

Answer 55

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

Answer 56

sorry, 127.

Answer 57

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

Answer 58

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

Answer 59

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

Answer 60

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

what are you trying to say here

Answer 61

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

Answer 62

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

Answer 63

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

Answer 64

`1111 1111 ` = 0
right

Answer 65

oh sorry, i meant 1000 0000.

Answer 66

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

Answer 67

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

Answer 68

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 ?

Answer 69

no, i don't. why?