Natural exponents covered the positive integers. Negative exponents extended through zero and below. Rational exponents filled in every fraction. But the real number line still has gaps — irrational numbers like 2 and π sit between the rationals, and a complete definition of ax must account for them.
The Problem
The expression 23 means multiply 2 by itself three times. The expression 2−1 means take the reciprocal. The expression 23/4 means a combination of root and power. Each of these follows from a clear definition.
But what is 2π? The exponent π is not a natural number, not a negative integer, and not a fraction. It cannot be expressed as a ratio of two integers. The repeated multiplication idea is meaningless, and the root interpretation does not apply — there is no "πth root" in the sense that a1/n defines an nth root.
Yet 2π must have a value if the laws of exponents are to work for all real numbers. The product rule requires 2π⋅21−π=21=2. The power rule requires (2π)1/π=2. The algebraic framework assumes this number exists. The question is how to define it.
Intuition Through Approximation
The number π is irrational, but it can be trapped between rational numbers to any desired precision.
π lies between 3 and 4. So 2π lies between 23=8 and 24=16.
Narrowing: π lies between 3.1 and 3.2. So 2π lies between 23.1≈8.574 and 23.2≈9.190.
Narrowing further: π lies between 3.14 and 3.15. So 2π lies between 23.14≈8.815 and 23.15≈8.876.
Each rational approximation of π gives a value of 2x that we already know how to compute — using rational exponents. As the rational bounds squeeze closer to π, the corresponding powers squeeze closer to a single value.
That limiting value is 2π≈8.825. It is not reached by any single rational exponent, but it is pinned down uniquely by the sequence of rational approximations closing in from both sides.
Why This Works
The approximation method succeeds because the function ax, when restricted to rational values of x, behaves smoothly. Plot the points (x,2x) for rational x and the result is a curve with no gaps, no jumps, and no sudden direction changes.
Between any two rational numbers there are infinitely many more. The rational values of 2x fill the curve so densely that only isolated points — corresponding to irrational exponents — are missing.
Defining 2π as the limit of 2rn for a sequence of rationals rn→π fills in exactly those missing points. The process is called continuous extension — the smooth behavior of ax at rational points guarantees that the extension to irrational points is unique and well-defined.
The result is a complete curve: ax is now defined for every real number x, with no holes remaining on the number line.
Domain Restriction: a>0
Every previous extension tightened the restriction on the base. Natural exponents allowed any real base. Negative exponents required a=0. Rational exponents with even roots required a≥0. Irrational exponents tighten it once more: the base must be strictly positive.
The reason is that negative bases produce erratic behavior under dense rational exponents. Consider (−1)x at rationals near 21. The value (−1)1/3=−1, but (−1)1/2 is undefined in the reals. Nearby rational exponents alternate between real and undefined, making it impossible to define a smooth limiting value.
For a>0, no such problem arises. The function ax is positive for all rational x, changes smoothly, and extends without ambiguity to every irrational x. The restriction a>0 is the price of completeness — it is what allows ax to be defined for the entire real number line.
Zero is excluded as well. The expression 0x equals 0 for positive x and is undefined for negative x (since 0−n=0n1), so no continuous extension to all real exponents is possible.
The Laws Still Hold
By now the pattern is familiar: each extension of the exponent definition preserves the laws of exponents, and the irrational case is no different.
The product rule am⋅an=am+n holds when m and n are irrational. The proof relies on the fact that the rule holds for all rational approximations, and the limiting process preserves algebraic identities.
The same continuity argument extends every other law — quotient rule, power of a power, power of a product, power of a quotient. If a rule works for every rational exponent, and irrational exponents are defined as limits of rational ones, then the rule works for irrational exponents as well.
No separate derivation is needed. Three rounds of verification — natural, negative, rational — established the laws for all rational exponents. The continuous extension to irrationals carries those laws across the finish line automatically.
Completing the Picture
The definition of ax has now been built in four stages, each extending the previous one while preserving the same set of rules.
Natural exponents defined an as repeated multiplication for n=1,2,3,… — covering the positive integers on the number line.
Negative exponents defined a0=1 and a−n=an1, extending coverage to zero and all negative integers.
Rational exponents defined am/n=nam, filling in every fraction between and beyond the integers.
Irrational exponents defined ax for the remaining points through continuous extension, completing the real number line.
The result: for any base a>0 and any real exponent x, the expression ax is defined, positive, and obeys every law of exponents. This is the foundation on which exponential functions are built — functions where the base is fixed and x becomes a variable free to range over all real numbers.