Multiplication of Equations

As we know, multiplication of integers may be defined in terms of addition. This is why Euclid did not mention multiplication among his Common Notions. Properties of multiplication are derivable from those of addition and, therefore, if we can add equations, we may multiply them as well. It's interesting that Euclid stipulates among Common Notions not only validity of addition but also that it's permissible to subtract equal quantities from equal quantities without disturbing their equality. The modern math would derive one property from another; however, the fact remains that both addition and its inverse operation (subtraction) may apply to equations. A legitimate question is whether the same is true of multiplication and the operations inverse to multiplication.

Division is one operation that reverses the result of multiplication. However, division has limitations: it's impossible to divide by 0. Other than that, if a = b then a / r = b / r. However, concealed division by 0 results in errors and what's commonly known as arithmetic paradoxes. Here is a well known one: assume a = b and that both are positive numbers. Write

 
1.a, b > 0that's given
2.a = bthat's also given
3.ab = b²this is step 2 times b
4.ab - a² = b² - a²Euclid's Common Notion (3)
5.a(b - a) = (b + a)(b - a)distributive law
6.a = b + adividing by (b - a)
7.0 = bEuclid's Common Notion (3)
8.b = 2bEuclid's Common Notion (2)
9.1 = 2Nice, right?

What went wrong?


Another case of reversing the result of multiplication is squaring. Let a × a = b. Given b, what is a? As we know, a = b. As in the case of division, squaring should be used cautiously.

If two numbers are equal, their squares are also equal. Is the reverse true? That is, is it true that squares of two numbers are equal provided the numbers are equal, to start with? In short, does u = v imply u = v? Or simpler yet: is the square root of a number unique?

Assume the answer is unqualified "yes" and let's see where this will lead us.

 
1.(n + 1)² = n² + 2n + 1 Do not you know it?
2.(n + 1)² - (2n + 1) = n² Euclid's Common Notion (3)
3.(n + 1)² - (2n + 1) - n(2n + 1) = n²-n(2n + 1) ditto
4.(n + 1)² - (n + 1)(2n + 1) = n² - n(2n + 1) plain factoring
5.(n + 1)² - (n + 1)(2n + 1) + (2n + 1)²/4 = n²-n(2n + 1) + (2n + 1)²/4 Euclid's Common Notion (2)
6.[(n + 1) - (2n + 1)/2]² = [n - (2n + 1)/2]² a little algebra
7.(n + 1) - (2n + 1)/2 = n - (2n + 1)/2 taking square roots of both sides
8.n + 1 = n Euclid's Common Notion (3)
9.1 = 0 That's also nice, right?

What went wrong?

References

  1. T. Pappas, The Joy of Mathematics, Wide World Publishing, 1989
  2. J. R. Newman, The World of Mathematics, v3, Dover, 2003

What Can Be Multiplied?


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We assumed at the outset that a = b. But then, on step #6, divided the equation by b - a which is, by our assumption, zero. Dividing by 0 makes no sense and may lead to unpredictable results.


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The problem here is with a careless use of the term "square root" and the corresponding symbol . Squaring is a many-to-one operation: 2² = (-2)², implying that the converse operation is not defined uniquely. If a non-zero real number has a real square root, it has two square roots, i.e., there are two numbers whose square equals the given one. For a given R, one of such numbers is denoted R while the other -R. In short, ±R. One of this is positive, the other negative. As a matter of definition, R stands for the positive number, e.g., 25 = 5; writing or assuming 25 = -5 is a common mistake.

The above derivation played out on that difference between the uniqueness of and a lack of uniqueness in interpreting what stands behind the verbal "square root". The opening questions were designed to conceal this difference and hopefully confuse the reader.

If two numbers are equal, their squares are also equal. Is the reverse true? That is, is it true that square roots of two numbers are equal provided the numbers are equal, to start with? (Innocuously this is where the confusion starts: the square root is not unique. So, which one is the question about?) In short, does u = v imply u = v? (The answer to this question is an unqualified "Yes". By the definition of this is true. This question and the implied answer serve as a diversion that sets a trap for the next question.) Or simpler yet: is the square root of a number unique? (After the previous question, there may be an impulse to answer "Yes" to that question too. We go on and make that assumption.) Assume the answer is unqualified "yes" and let's see where this will lead us.

The correct result on step 7 would be (n + 1) - (2n + 1)/2 = -(n - (2n + 1)/2). Ignoring the fact that the number on the right is negative (and, therefore, cannot serve as of another number) leads to a contradiction.


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