Many mathematical discoveries are the result of taking what is already known and extending it a step further. If a line segment is a point stretched in one dimension, a square is a segment stretched in a direction perpendicular to the first, and a cube is a square stretched in a direction perpendicular to the last two, what comes next?
I am not going to write about hypercubes here, but a concept which was discovered by very similar means - hyperoperators.
Consider the addition of two positive integers, a and b. This is done by repeating the action of taking a+1, a's successor, denoted by S(a), b times. For example, 2+4 = S(S(S(S(2)))) = S(S(S(3)) = S(S(4)) = S(5) = 6. Addition is repeated application of the successor function.
Multiplication is defined in a similar fashion. To calculate a*b, start with 0 and add a b times. 2*3 = 2+2+2 = 6. Multiplication is repeated addition. (This will become interesting soon.)
Exponentiation continues this pattern. 2^3, or 2↑3 (which I am writing for a reason which you will soon see), = 2*2*2 = 8. Exponentiation is repeated multiplication.
The question becomes, what is the logical continuation of this? The answer is repeated exponentiation, or tetration. It gets its name from the fact that it is the fourth operator after addition, multiplication, and exponentiation, and it is represented by two up arrows. 2↑↑3 = 2↑(2↑2) = 2↑4 = 16.
Tetration is a way to generate large numbers very fast. 3↑↑3 = 3↑(3↑3) = 3↑27 = 7625597484987, or about 7.6 trillion!
Tetration is the fourth of an infinite series of hyperoperators, each defined as repeated application of the last, each represented by one more arrow than the last, and each climbing to higher numbers than the last. After tetration comes pentation, hexation, heptation, and beyond. 3↑↑↑3 (3 pentated to 3) = 3↑↑(3↑↑3) = 3↑↑7625597484987, which is
a power tower of 3^(3^(3^(3^(... 7.6 trillion threes high -
truly a gargantuan number.
65536 is my favorite number because of hyperoperators. First, it's 2↑16, which is 2↑(2↑4), or 2↑(2↑(2↑2)), which, by the definition of tetration, is equal to 2↑↑4. But 4 = 2↑↑2, so it's 2↑↑(2↑↑2). Since this is repeated tetration, it can be expressed in terms of pentation: 2↑↑↑3. 65536 is my favorite number because it's the largest number that isn't as large as something like 7 trillion, but can still be expressed in terms of a high-level hyperoperator like pentation.
How would one go about formally defining hyperoperators, building one's way up from the successor function to pentation and beyond? For that, we will have to discuss recursion, which will be done in Part 2!
There are functions that grow much, much faster than any hyperoperator, but as I don't understand them fully yet, you can find them on the Googology Wiki, an extensive list of extremely large numbers.
I am not going to write about hypercubes here, but a concept which was discovered by very similar means - hyperoperators.
Consider the addition of two positive integers, a and b. This is done by repeating the action of taking a+1, a's successor, denoted by S(a), b times. For example, 2+4 = S(S(S(S(2)))) = S(S(S(3)) = S(S(4)) = S(5) = 6. Addition is repeated application of the successor function.
Multiplication is defined in a similar fashion. To calculate a*b, start with 0 and add a b times. 2*3 = 2+2+2 = 6. Multiplication is repeated addition. (This will become interesting soon.)
Exponentiation continues this pattern. 2^3, or 2↑3 (which I am writing for a reason which you will soon see), = 2*2*2 = 8. Exponentiation is repeated multiplication.
The question becomes, what is the logical continuation of this? The answer is repeated exponentiation, or tetration. It gets its name from the fact that it is the fourth operator after addition, multiplication, and exponentiation, and it is represented by two up arrows. 2↑↑3 = 2↑(2↑2) = 2↑4 = 16.
Tetration is a way to generate large numbers very fast. 3↑↑3 = 3↑(3↑3) = 3↑27 = 7625597484987, or about 7.6 trillion!
Tetration is the fourth of an infinite series of hyperoperators, each defined as repeated application of the last, each represented by one more arrow than the last, and each climbing to higher numbers than the last. After tetration comes pentation, hexation, heptation, and beyond. 3↑↑↑3 (3 pentated to 3) = 3↑↑(3↑↑3) = 3↑↑7625597484987, which is
a power tower of 3^(3^(3^(3^(... 7.6 trillion threes high -
truly a gargantuan number.
65536 is my favorite number because of hyperoperators. First, it's 2↑16, which is 2↑(2↑4), or 2↑(2↑(2↑2)), which, by the definition of tetration, is equal to 2↑↑4. But 4 = 2↑↑2, so it's 2↑↑(2↑↑2). Since this is repeated tetration, it can be expressed in terms of pentation: 2↑↑↑3. 65536 is my favorite number because it's the largest number that isn't as large as something like 7 trillion, but can still be expressed in terms of a high-level hyperoperator like pentation.
How would one go about formally defining hyperoperators, building one's way up from the successor function to pentation and beyond? For that, we will have to discuss recursion, which will be done in Part 2!
There are functions that grow much, much faster than any hyperoperator, but as I don't understand them fully yet, you can find them on the Googology Wiki, an extensive list of extremely large numbers.