You've seen how this problem arises in GEOMETRY. But I may be the only one who can, and wishes to, tell you that IT IS A PROBLEM OF ARITHMETIC.

Briefly, ARITHMETIC HAS PRIMARY OPERATIONS:

1. ADDING
2. MULTIPLYING
3. EXPONENTIATING.

But the Danish philosopher and theologian, Sören Kierkegaard told us: "Life can only be understood backward, but must be lived forward." MATH should help us in this.

In fact, ALGEBRA IS ARITHMETIC BACKWARD. WE START FROM A KNOWN ANSWER AND UNRAVEL THE ALGEBRA TO SEE HOW THAT ANSWER RESULTS. It's no accident that the first important achievements in ALGEBRA and in ASTRONOMY occured among a single PEO0LE, the ancient BABYLONIANS. Babylonian priests used ALGEBRA to trace ASTRONOMICAL EVENTS BACKWARD, so they could later PREDICT THEM, giving some degree of CONTROL.

But this means that each of the PRIMARY ARITHMETIC OPERATIONS must have an INVERSE. And they do:

1. ADDING → SUBTRACTION. Making SUBTRACTION TOTAL (ALWAYS WORKS!) LEADS TO INTEGER NUMBER SYSTEM.
2. MULTIPLYING → (NONZERO) DIVIDING. Making (NONZERO) DIVISION TOTAL LEADS TO RATIONAL NUMBER SYSTEM.
3. EXPONENTIATING → LOGARITHM and
   EXPONENTIATING → ROOT EXTRACTION.

Why TWO INVERSES for EXPONENTIATING, and only ONE INVERSE for the other PRIMARY ARITHMETIC OPERATIONS?

Each of them has TWO INVERSES, BUT COMMUATIVITY OF THE OPERATION CAUSE THESE TWO TO MERGE INTO ONE. However, EXPONENTIATION IS NOT COMMUTATIVE!

WritE b for BASE; e for EXPONENT; p for POWER. Then we have, b^e = p, which is NOT COMMUTATIVE. Thus, 2³ = 8, but 3² = 9.

So we have the INVERSES: b^e = p → b = (p)^1/3
and b^e = p → e = log_bp.

Making LOGARITHM TOTAL YIELDS THE REAL NUMBER SYSTEM. Making ROOT EXTRACTION YIELDS THE COMPLEX NUMBER SYSTEM.

But the REAL NUMBER SYSTEM INCLUDES √2. Hence, making LOGARITHM TOTAL (as one INVERSE of EXPONENTIATION) solves the √2-problem WITHOUT REFERENCE TO GEOMETRY.

But this is not taught. And logarithm seems like something the cat drug in!