Is a - b always equal to √(a^2 - b^2)? A Closer Look at Mathematical Equivalence

Exploring the mathematical equivalence between the expressions (a - b) and (sqrt{a^2 - b^2}) can be intriguing, but it is not always the case that these two expressions are equivalent. Let's delve into various examples and logical steps to understand when and when not these expressions are equal.

Counterexample 1: Specific Values

Consider the values (a 0) and (b 1). Plugging these into the expressions:

(a - b 0 - 1 -1), which is a real number.

(sqrt{a^2 - b^2} sqrt{0^2 - 1^2} sqrt{0 - 1} sqrt{-1} i), which is an imaginary number.

Clearly, for these specific values, (a - b eq sqrt{a^2 - b^2}).

Expanding the Counterexample

Let's expand on the initial example to generalize and explore further:

Let (a 5) and (b 3).

(a - b 5 - 3 2)

(a^2 - b^2 25 - 9 16)

(sqrt{a^2 - b^2} sqrt{16} 4)

Here, (sqrt{a^2 - b^2} 4 eq 2 a - b).

Evaluating the Square Relationship

One might think that squaring both sides of the hypothetical equation (a - b sqrt{a^2 - b^2}) could help prove the equivalence. However, this approach does not work as seen in the following steps:

Square both sides:

((a - b)^2 (sqrt{a^2 - b^2})^2)

(a^2 - 2ab b^2 a^2 - b^2)

(-2ab b^2 -b^2)

(-2ab -2b^2)

(a b)

Therefore, (a - b sqrt{a^2 - b^2}) is only true when (a b). For any other values, this equality does not hold.

Further Breaking Down the Expression

Considering the expression (a - b^2) separately from (a^2 - b^2), it is clear that these are fundamentally different expressions:

(a - b^2 a - b^2)

(a^2 - b^2 (a b)(a - b))

Since (a^2 - b^2) can be factored into ((a b)(a - b)), it is not equal to (a - b^2).

Additional Counterexample

Consider the example (a 7) and (b 3).

(a^2 - b^2 49 - 9 40)

(sqrt{a^2 - b^2} sqrt{40} approx 6.32)

(a - b 7 - 3 4)

Thus, (sqrt{40} approx 6.32 eq 4 a - b).

Conclusion

From the examples and logical steps presented, it is evident that (a - b) is not always equal to (sqrt{a^2 - b^2}). The expressions are only equal when (a b), and for all other values, they are distinct.