It’s A Math, Math World (The Substitution Principle)


This week’s blog post is a little different than the past few weeks. It is a little less math-intensive, but still very relevant. I will introduce a concept in mathematics that extends from Algebra and Geometry into higher math. It is a concept that eludes some of my best tutoring students. It is the concept of The Substitution Principle.

Basically, the substitution principal states in plain English that:

If quantities have the same value then they are interchangeable.

Sometimes the wording is different or the concept is clouded by the details of the problem, but it merely states that if two quantities are equal, in numeric or algebraic value, then one can take the place of the other in your calculation or mathematical proof.

Think about it in terms of an equation:

Example 1:

Given:   x+ y = z


Conclusion:         substitute 5 in for z into the top equation and get


Example 2:         

a=b and a=f

We can conclude that b=f

Example 3 (from Geometry):

A = πr2


We can conclude that πr2 =18πy and we can solve for r or y.

Example 4 (from a geometric proof):

Given:   <1 complementary to <3 (i.e. m<1 + m<3 = 90)

                <2 complementary to <3 (i.e. m<2 + m<3 = 90)

Conclusion:         m<1 + m<3 = m<2 + m<3

                                m<1 = m<2

As you can see, this principal has many useful applications.

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  1. There is one huge caveat with the substitution principle — identifying “quantity” in the textual form of an expression. Basically, a substring represents a quantity if and only if you can surround it with parentheses without changing the meaning of the whole expression.

  2. Dijkstra called this Leibniz’s rule in “Program Semantics and Predicate Calculus” (which I highly recommend, by the way). It’s never given enough explicit mention, since to an experienced practitioner it seems blindingly obvious. Good on you for bringing it up.

  3. Thank you . Sometimes the obvious can be overlooked at times. It is amazing the number of HS math students who find the principle foreign.

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  5. Thank you for providing such an awesome refresher for me. It has been a little while since my graduate course in biostatistics and your blog is priceless for reviewing the fundamentals.