Why Do We Introduce Constants in Proportional Relationships?

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SUMMARY

The discussion centers on the necessity of introducing a constant in proportional relationships to convert proportionality into equality. A constant of proportionality quantifies the relationship between two variables, such as in the equations y = 3x and y = 2x, where the constants 3 and 2 determine the rate of change in y relative to x. This constant allows for the selection of a specific curve from a family of curves defined by the proportionality, thus enabling precise mathematical modeling.

PREREQUISITES
  • Understanding of proportional relationships in mathematics
  • Familiarity with linear equations
  • Knowledge of constants in mathematical expressions
  • Basic algebra skills
NEXT STEPS
  • Study the concept of the constant of proportionality in detail
  • Explore the implications of proportional relationships in real-world applications
  • Learn about different types of mathematical curves and their properties
  • Investigate the role of constants in other mathematical contexts, such as calculus
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Students of mathematics, educators teaching proportional relationships, and anyone interested in understanding the foundations of mathematical modeling.

StupidGenius
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My proff mentioned something about proportionality:
"To make an a proportionality into an equality, we must introduce a constant"
Something along those words: my question is why? Can show prove this to me??
 
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StupidGenius said:
My proff mentioned something about proportionality:
"To make an a proportionality into an equality, we must introduce a constant"
Something along those words: my question is why? Can show prove this to me??
I am not aware of any proof of the concept, but I can offer a reason. Consider the statement "y is proportional to x". This means that a finite change in x induces a finite change in y. The size of the change in y depends on the size of the change in x. Now to quantify it, we need to introduce the so-called constant of proportionality. It is this constant that determines how big the change in y is for a given change in x.

For example, take the equation y = 3x. Clearly y is proportional to x and if x changes by +1, then y changes by +3. Equally, consider the equation y = 2x. Once again, y is clearly proportional to x, but this time for each integer step of x, y changes by 2.

This is why we need to introduce such constants. More technically, a proportionality defines a family of curves, by fixing the constant of proportionality (as well as the additive constant) you are effectively choosing a single curve from the family.
 

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