Numbers are free creations of the human mind that serve as a medium for the easier and clearer understanding of the diversity of thought.
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From our early childhood we know that having two chocolates at disposal is a better proposition than having one! This indeed, tells about our intuitive acceptance of the order relation among positive integers (here 2 is more than 1). A similar ordering among all real numbers enables us to make statements about one real number being greater (or smaller) than the other.
For any two real numbers x and y, if x is less than y (written x < y) then y – x is positive. Geometrically, x < y if and only if x lies to the left of y on the real line. Likewise, y > x means x < y.
The case, either x = y and x < y can be described as x ≤ y. Similarly, y ≥ x means x ≤ y. We further have x > 0 if and only if x is positive and if x < 0, we say x is negative. However, for x ≥ 0 we say x to be non-negative.
For any two real numbers x and y, exactly one of the three relations x > y, x = y, x < y holds. It is the great trichotomy law of real numbers. It follows that, for any real number x, either x > 0 or x = 0 or x < 0.
If x ≠ 0, then x2 > 0 and x2 = 0 only when x = 0. It follows that x2 ≥ 0 for any real number x. Using this, we can prove that 1 > 0. In fact, 1 = 12 and 12 > 0. Hence 1 > 0. If xy > 0, then either both x and y are positive or, both x and y are negative. Finally, what if x2 < 0? In that case x is a complex number.
Theorem 1. If a2 + b2 = 0 for any two real numbers a and b, then a = 0 = b and conversely.
If a = 0 = b, then certainly a2 + b2 = 0 and we are done.
There are many ways to prove the other part, namely a2 + b2 = 0 means a = 0 = b. The above facts and results are important in proving this. The first proof is based on the law of trichotomy of real numbers.
By the law of trichotomy, either a > 0 or a = 0 or a < 0. In a similar fashion, we can have either b > 0 or b = 0 or b < 0. Collecting all the possibilities in the tabular form, we get the following table, from where we conclude that a = 0 = b is the only possibility for the present case.
Recall that, x2 = 0 means x = 0. When a ≠ 0 and b ≠ 0, we have a2 > 0 and b2 > 0. It follows that a2 + b2 > 0, an impossibility. Next suppose, a = 0 but b ≠ 0. This means, a2 = 0 and b2 > 0. It follows that a2 + b2 > 0, a contradiction. Finally, assume that b = 0 but a ≠ 0. Hence, b2 = 0 and a2 > 0, whence a2 + b2 > 0, a contradiction. The only possibility a = 0 = b is evident now.
Now a2 + b2 = 0 means a2 = − b2. Since a, b are real numbers, we have a2, b2 ≥ 0. If both of them are not equal to 0, let a2 = m > 0. Then, b2 = − a2 = − m < 0, contradicting the fact that b2 ≥ 0. Only possibility a = 0 = b follows.
We can write, from a2 + b2 = 0, that (a − b)2 + 2ab = 0 and (a + b)2 − 2ab = 0. Using the fact x2 ≥ 0 for all real number x, it follows that 2ab ≥ 0 and − 2ab ≥ 0. Hence ab ≥ 0 and ab ≤ 0. For both to hold simultaneously, only possibility is a = 0 = b.
Consider the points Q(a, b) and P(0, 0) in the plane. The distance between them is PQ = √( a2 + b2). Now a2 + b2 =0 means PQ = 0. This is possible only when Q coincides with P, the origin. As a result (a, b) = (0, 0), whence a = 0 = b as required.
We now use complex numbers to establish the result. Recall that the product of two complex numbers is zero if and only if each one of them is zero. Now a2 + b2 = 0 means a2 − (−1)b2 = 0 or a2 − i2b2 = 0. This in turn implies that (a + ib)(a − ib) = 0, the product of two complex numbers. Hence, either a + ib = 0 or a – ib =0. Equating real and imaginary parts on both sides, we get a = 0 = b in both cases. Therefore, only possibility for equality to hold is a = 0 = b.
Few applications of the above result
Of course, there is a similar generalization to the above theorem, which is given below.
Theorem 2. If a2 + b2 + c2 = 0 for any three real numbers a, b and c, then a = b = c = 0 and conversely.
Below, we will consider two examples which can be solved using the above theorem.
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