What's the additive inverse of -7?

Answer 1

We will answers the two questions:
1. What is the additive inverse of -7

2. What's an additive identity.

The additive inverse of a number is the number you have to add to the number in order to get 0. (Or more generically speaking, to get the additive identity element of the group or field.) So the additive inverse of -7 is +7. For any real number a, the additive inverse is -a. If z is a complex number, a+bi, then the additive inverse is (-a-bi) since (a+bi)+(-a-bi)=0.

The case becomes a little more interesting in fields other than the real or the complex numbers. The integers mod p, where p is a prime, form a finite field. So if we look at integers mod 7, the additive inverse of 5, for example, would be 2 since 5+2=7 which is congruent to 0 in this field.

The additive identity in the field of real or complex numbers is 0.
"Additive identity" means the number you can add to any other number in order to get the same number back. Since -7 + 0 = -7, the additive identity of -7 is 0.

In the case of a+bi where i^2=-1, the additive identity is still 0. If it helps you to think of it as 0+0i, that is fine. In the finite field of integers mod p, where p is a prime, we have p as the additive identity. For example, 2 mod 7 is just 2, and if we add 7 it is 9 but that is still 2 mod 7.

All of these ideas can be extended to fields of invertible matrices and many other exciting algebraic structures!