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Prove that if a and b are real numbers such that a² + b² = 0, then a = 0 or b = 0?

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Final answer:

To show that a² + b² = 0 implies a = 0 and/or b = 0, we use the fact that a squared number in the real numbers is always non-negative. Since both a² and b² must be non-negative for their sum to be zero, it follows that a and b must both be zero.

Step-by-step explanation:

To prove that if a and b are real numbers such that a² + b² = 0, then a = 0 and/or b = 0, we can use the properties of real numbers and the nature of squared terms.

In the realm of real numbers, a number squared is always non-negative, meaning it is either positive or zero. This is because the product of two positive numbers or two negative numbers is positive, and zero squared is zero. For the equation a² + b² = 0, both terms a² and b² are non-negative and their sum is zero. This implies that both a² and b² must separately equal zero. Therefore, a = 0 and b = 0.

Conclusively, the only possibility for the sum of two squared real numbers to equal zero is if both of those numbers themselves equal zero.

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