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## Homework Statement

Let a and b be real numbers with a < b.

a. Derive a formula for the distance from a to b. Hint: Use 3 cases and a visual argument on the number line.

b. Use your work in part (a) to derive a formula for the distance between (a,c) and (b,c) in a plane.

c. Use the Pythagorean theorem to derive a formula for the distance between the points(a,c) and (b,d) in the adjacent sketch. The sketch is a line on x-y coordinate with two endpoints (a,c) and (b,d) where b > a and d > c.

d. Generalize the distance formula to R^n.

## Homework Equations

## The Attempt at a Solution

I tried doing (a):

Derive a formula for the distance from a to b. Hint: Use 3 cases and a visual argument on the number line.

Case 1: Assume that 0 ≤ a < b. Since a ≥ 0, the distance from a to 0 is a. Since b ≥ 0, the distance from b to 0 is b. The distance from a to b is the distance from b to 0 minus the distance from a to 0 which is b-a.

Case2: Assume that 0 ≥ b > a. Since 0 ≥ a, the distance from 0 to a is -a. Since 0 ≥ b, the distance from 0 to b is -b. The distance from a to b is the distance from 0 to a minus the distance from 0 to b which is b-a.

Case3: Assume that b ≥ 0 ≥ a. Since b ≥ 0, the distance from b to 0 is b. Since 0 ≥ a, the distance from 0 to a is -a. The distance from a to b is the distance from b to 0 plus the distance from 0 to a which is b-a.

Does it look correct?

Thanks.