Monday, 25 September 2023

Misuse of Mathematical Symbols

I often see arithmetical challenges like:$$ \begin{align} 2+5&=14\\3+6&=27\\5+1&=30\\7+2&=\text{ ?} \end{align}$$What's annoying about this is the abuse of the addition and equal signs. Clearly the numbers shown on the left hand side of the equation do NOT add up the number shown on the right hand side and so the equal sign should NOT be used.

Instead some operation, not addition, is being performed on the two numbers on the left in order to generate the outcome shown. In other words, we have a function of two variables. Let's call the variables \(x\) and \(y\) and the function \(f\). This can be represented then as \(f(x,y)\) and we knows that:$$ \begin{align} f(2,5)&=14\\f(3,6)&=27\\f(5,1)&=30\\f(7,2)&=\text{ ?} \end{align}$$This is the correct way to present the problem and then by a little bit of experimentation, it can be seen that:$$ \begin{align} f(x,y)&=x \times (x+y)\\&=x^2+xy \end{align}$$The solution to the original problem is then:$$ \begin{align} f(7,2)&=49+14\\&=63 \end{align}$$Using this method, it's easy to generate new challenges. For example let's suppose that:$$ \begin{align} f(x,y)&=x \times (x-y)\\&=x^2-xy \end{align} $$ We could then formulate the challenge: $$ \begin{align} f(3,1)&=6\\f(5,3)&=10\\f(7,6)&=7\\f(8,5)&=\text{ ?} \end{align}$$The correct answer is then:$$ \begin{align} f(8,5)&= 64-40\\&=24 \end{align} $$Of course, the function could be of three variables. For example:$$f(x,y,z)=(x+y-z)^2$$A challenge could then be formulated as follows:$$ \begin{align} f(2,3,1)&=16\\f(1,4,2)&=9\\f(3,5,1)&=49\\f(5,2,3)&= \text{ ?} \end{align}$$The answer is then:$$ \begin{align} f(5,2,3)&=(5+2-3)^2\\&=4^2\\&=16 \end{align} $$And so it goes.

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