max negative value value in 8 bit

Question

in 8 bit representation max +ve value is +127  which is understandable as 01111111 but max - ve value is -128 how it can be stored     as left most bit will always be 1 and rest 7 bits can only store 111111 i.e max of 127 so how -128 is stored

Comments

In 8 bit representation max +ve value is $255$ (not +127) i,e. $11111111$, $+127$ is max. value in $2's$ complement representation.

Answer 1

Computer generally uses $2's$ complement representation for storing numbers. So $+127$ is stored as $\color{red}{01111111}$ with MSB i.e. $8^{th}$ bit being $0$ indicating that it is a positive number and $-128$ is represented as $10000000$, i.e., $8^{th}$ bit indicating that number represented by this bit pattern is a negative number and magnitude is represented as $-2^7*1 + 0*2^6 + 0*2^5 + \dots + 0.2^0 = -128$

So, $-128$ is stored as $\color{red}{10000000}$

Comments

so left most bit works as both sign bit and magnitude bit ? and can +ve numbers also be represented in 2's complement form

---

Yes, for example $0110 = +6$ and $1100 = -4$.

Answer 2

-128 = 10000000

1's complement of 10000000  is,  01111111

2'complement is,  01111111 + 1 = 10000000