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
Another case of reversing the result of multiplication is squaring. Let
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
Assume the answer is unqualified "yes" and let's see where this will lead us.
- T. Pappas, The Joy of Mathematics, Wide World Publishing, 1989
- J. R. Newman, The World of Mathematics, v3, Dover, 2003
What Can Be Multiplied?
- What Is Multiplication?
- Multiplication of Equations
- Multiplication of Functions
- Multiplication of Matrices
- Multiplication of Numbers
- Peg Solitaire and Group Theory
- Multiplication of Permutations
- Multiplication of Sets
- Multiplication of Vectors
- Multiplication of a Vector by a Matrix
- Vector Space and Spaces with the Scalar Product
- Addition and Multiplication Tables in Various Bases
- Multiplication of Points on a Circle
- Multiplication of Points on an Ellipse
We assumed at the outset that a = b. But then, on step #6, divided the equation by
The problem here is with a careless use of the term "square root" and the corresponding symbol
The above derivation played out on that difference between the uniqueness of
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
The correct result on step 7 would be