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Problem:

Find the points of trisection of the line segment AB, where $A\equiv(4,0), B\equiv(0,3)$

Process 1:

Let the P & Q be the points of trisection. Let $x_1$ & $y_1$ be the abscissa and ordinate of P, and let $x_2$ & $y_2$ be the abscissa and ordinate of Q. Now,

$$(x_1,y_1)=\left(\frac{(2)(4)+(1)(0)}{3},\frac{(2)(0)+(1)(3)}{3}\right)$$

$$=\left(\frac{8}{3}, 1\right)$$

$$...$$

Process 2:

Let the P & Q be the points of trisection. Let $x_1$ & $y_1$ be the abscissa and ordinate of P, and let $x_2$ & $y_2$ be the abscissa and ordinate of Q. Now,

$$(x_1,y_1)\equiv\left(\frac{(2)(4)+(1)(0)}{3},\frac{(2)(0)+(1)(3)}{3}\right)$$

$$\equiv\left(\frac{8}{3}, 1\right)\tag{1}$$

$$...$$

Question:

  1. Should I use the $=$ sign or the $\equiv$ sign?
  2. If process 2 is correct, then in line $(1)$ should I use $=$ or $\equiv$ before $\left(\frac{8}{3}, 1\right)$? Currently, there is an $\equiv$ before $\left(\frac{8}{3}, 1\right)$ in line $(1)$.

Comments:

The usage of $=$ in process 1 seems completely fine to me. However, in my book, it has been written like process 2. So, am I correct to write in process 1?

ryang
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tryingtobeastoic
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1 Answers1

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The symbol $≡$ has multiple meanings in mathematics: to indicate an equivalence relation (like the congruence of geometric figures or in modular arithmetic), an identity (like the equation $x^2-y^2≡(x+y)(x-y),$ which is true for every allowable combination of variable values), a definition, a mathematical equivalence between two statements, a logical equivalence between two sentences, etc.

In your quoted example, there is no compelling reason to switch $=$ and $≡.$ The latter is just confusing here, unless the author has prefaced it with a note on their particular usage of $≡.$

ryang
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  • Thanks for your answer! I had another question: why are points denoted using the identity sign always? Example: $A\equiv(0,10)$ – tryingtobeastoic Dec 14 '21 at 11:11
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    @tryingtobeastoic They're not, typically. But if your text does this, it is probably defining point $A$ as $(0,10).$ You could also write $A:=(0,10)$ to indicate a definition/declaration. But all this may be a little pompous. Check out this Answer in which, for differerent emphases, I'd used all three symbols. – ryang Dec 14 '21 at 11:47