Why Tetration Never Made It Into the Arithmetic Textbooks

Tetration is the next iterated operator after exponentiation. Just as exponentiation extends multiplication through repetitive iteration, tetration expands exponentiation through further iterated operation. I have already explored this topic in several previous posts—for example, see my post on the Dunning-Kruger distribution for comparison.

Without going into excessive detail, let us build up the arithmetic operators step by step, beginning with the successor operator. The successor operation was likely one of the earliest concepts a primitive human mind grappled with when working with numbers. It simply returns the element that immediately follows a given input element within an ordered set. The successor operator can thus act on any well-ordered set. Interestingly, it aligns closely with the construction of the natural (counting) numbers, and considering its inverse naturally extends the number line to include zero and the negative integers.

Addition is the iterated application of the successor function. It can be understood as repeatedly applying the successor operation a specified number of times. Since the number of iterations and the starting value need not be identical, addition is a binary operator. Considering its inverse (subtraction) does not introduce an entirely new class of numbers—all values were already reachable, just more laboriously expressed.

Multiplication is the iterated application of addition. It amounts to applying addition to itself a certain number of times. Once again, the number of iterations need not equal the starting value of the product. Considering multiplication by itself does not introduce new numbers. However, its inverse—division—does create an entirely new class: the rational numbers. These arise whenever division of two integers does not yield another integer (i.e., when the dividend is not evenly divisible by the divisor). Once we have the rationals, the earlier operations (successor, addition, and subtraction) must be rigorously extended and defined on this new domain to maintain consistency.

Exponentiation is the iterated application of multiplication. It can be conceived of as applying multiplication to itself a number of times. The number of iterations again need not match the base, so exponentiation is also a binary operator. When we allow rational exponents whose denominators are even, we encounter a new class of values: the complex numbers (via even roots of negative quantities). When we fully explore all possible inverses of exponentiation (and apply prior inverses in every combination), and when we impose the axiom of completeness, an entire continuum of real numbers emerges.

Given this natural progression, it seems intuitive that we would next consider tetration—the iterated application of exponentiation. Yet our arithmetic textbooks and handbooks make it plain that tetration is not treated as a fundamental or everyday operation for modeling the natural world. Why is this the case?

In this short article, I offer three primary reasons that, taken together, largely explain humanity’s relative indifference to operators beyond exponentiation.

First, nothing we observe in the physical world grows faster than exponentially. Although tetration is a mathematically elegant operator that extends the iterative pattern of the earlier operations, we have not identified any natural phenomenon whose growth is best described by tetration. Because we encounter nothing that expands at such an extreme rate, we have little practical need to develop symbolic tools for representing or analyzing it. Consequently, tetration remains largely outside our everyday mathematical vocabulary.

Second, while the narrative above presents the operations as though each is built by iterating the previous one, that is not how arithmetic actually developed historically. The appearance of a clean iterative hierarchy is largely a retrospective reconstruction. The operations we use today were not constructed by deliberately iterating their predecessors in a systematic way. It is therefore a significant conceptual leap to treat tetration as the “obvious next step” after exponentiation in the same seamless manner that exponentiation follows from multiplication. Hyperoperations are not a brand-new invention, but extending the sequence in this particular direction does not follow automatically from the way earlier operations entered mathematical practice.

Third, it is genuinely difficult to develop clear intuitions about what the higher hyperoperations actually mean—especially once we begin asking about their inverses. Consider tetration (sometimes called hyperoperation level 4). We can visualize it via power towers. Tetration of 3 to height 3, written (3 \uparrow\uparrow 3) or (^{3}3), is the power tower (3^{3^{3}}). Because exponentiation is right-associative, this evaluates as (3^{(3^{3})} = 3^{27}), already an enormous number that illustrates the explosive growth. But what does it mean to tetrate 3 to a height of ½? What new classes of numbers (if any) does a fractional height unlock? Can the process be continued indefinitely upward or downward? Is there a canonical, well-behaved generalization to real or complex heights? These questions quickly become subtle and technically demanding; their inverses (super-roots and super-logarithms) are even less intuitive.

I believe these three considerations—together with several related factors not discussed here—explain why tetration has not achieved the same foundational status in our mathematics as addition, multiplication, or even exponentiation.

Thank you for reading. If you would like to discuss these ideas further, please feel free to reach out.