Is the map $y=2x+3$ linear?

"Of course it is." -- a high school teacher will answer.

"Nope; it's affine, but not linear." -- a college student will contradict.

This difference terminology that basic ought to have historical roots. Could you point toward them?

  • 1
    $\begingroup$ Related: hsm.stackexchange.com/questions/2490/… $\endgroup$
    – Michael E2
    Jun 5, 2019 at 23:08
  • 4
    $\begingroup$ This question is more suitable for Math Ed SE. The original "linear" refers to the graph being a line. In more advanced fields (linear algebra, functional analysis) it makes sense to make the finer linear/affine distinction, but there is little point to it in school/freshman math. $\endgroup$
    – Conifold
    Jun 6, 2019 at 1:46

1 Answer 1


Yes, it has historic roots. The term "linear" is much older than "affine" and the function $ax+b$ is "linear" because its graph is a straight line. With the invention of linear algebra, the meaning of the world linear changed. But educators, especially on the lower level are very conservative, and also reluctant to introduce extra Greek terms (which on their opinion intimidate students). So the terminology in lower levels of mathematical education lags behind the development of mathematics.

Similar situation we have with the words "equal" and "congruent" in geometry. Euclid and later educators called two figures "equal" if one can be moved to coincide with another. With the spread of set theory, the word "equal" was reserved for equality in the set-theoretic sense. But school education (in most countries) continues to use the word "equal" in Euclid's sense.

  • $\begingroup$ Your 2nd paragraph reminded me of the Soviet middle-high school geometry textbook by Kolmogorov and Semenovich that stressed the difference between "equal" and "congruent" and caused much grief among middle-high school students. $\endgroup$
    – Michael
    Jun 7, 2019 at 15:43

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