Q: Why does every rational number have a additive inverse?

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The additive inverse of EVERY positive rational number is a negative number.

It is a fundamental requirement of algebraic structures called groups.

For every number, a,there exists a number called the additive inverse, -a, such that a + -a = 0.

Yes. Just put a minus sign in front of it. Note that except for the zero, the additive inverse is no longer a natural number.

Yes.

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The additive inverse of EVERY positive rational number is a negative number.

yes

Yes.

It is a fundamental requirement of algebraic structures called groups.

For every number, a,there exists a number called the additive inverse, -a, such that a + -a = 0.

By definition, every rational number x can be expressed as a ratio p/q where p and q are integers and q is not zero. Consider -p/q. Then by the properties of integers, -p is an integer and is the additive inverse of p. Therefore p + (-p) = 0Then p/q + (-p/q) = [p + (-p)] /q = 0/q.Also, -p/q is a ratio of two integers, with q non-zero and so -p/q is also a rational number. That is, -p/q is the additive inverse of x, expressed as a ratio.

The additive inverse states that a number added to its opposite will equal zero. A + (-A) = 0. The "opposite" number here is the "negative" of the number. For any number n, the additive inverse is (-1)n. So therefore yes.

Yes. Just put a minus sign in front of it. Note that except for the zero, the additive inverse is no longer a natural number.

The additive inverse of a number is the value that, when added to the original number, results in a sum of zero. In this case, the additive inverse of 22 would be -22, as 22 + (-22) equals zero. The additive inverse is essentially the negative version of the original number.

Yes.

One example would be a Galois Field size 4 (ie GF(4)). Here, the elements are {0,1,2,3} and every element is its own additive inverse.

To form the additive inverse, negate all parts of the complex number → 8 + 3i → -8 - 3i The sum of a number and its additive inverse is 0: (8 + 3i) + (-8 - 3i) = (8 + -8) + (3 + -3)i = (8 - 8) + (3 - 3)i = 0 + 0i = 0.