First, ∞ is *not* number. It’s symbol for infinity. For example in the natural numbers set **N** (the set of all positive integers), there are infinite number of numbers, but none of them is ∞.

But now to the example…

What is 1^{∞}? Layman probably would say that that’s 1, but according to mathematicians 1^{∞} is not defined; we can’t say (at least by our understanding on infinity we have at the moment) what it is.

But what if we get ”close” to 1^{∞}? Let’s examine following:

From arithmetic operations in infinity we remember, that

So what we have at the first limit sentence is ”1^{∞} ” and for

an exact value can in fact be determined! It’s the Napier’s number *e*, that’s an irrational number i.e. its decimal representation is infinite. It’s approximately 2.718281828459.

What I presented here, I came familiar with on my first university year.

Mathematically the number of digits in given integer > 0 can be determined with logarithm.

The number of digits in an integer > 0 is

Trunc(Log_{k}(number)) + 1

Trunc comes from the Pascal programming language and means truncation. ”number” is 10 base system number of *k* base system number.

The idea is to examine how many powers of base system *k* there is in the number, that is how many times the given number can be divided with the base number until we get number < 1. With this our friend logarithm helps us to determine that. As a reminder, logarithm is reverse operation to power.

Examples:

In 10 base system in number 12345678 there is 7 powers of 10: Trunc(Log_{10}12345678) = 7, the number of digits is 7 + 1 = 8.

The hexdecimal system number 999_{16} = 2457_{10}, Log_{16} 2457 = 2.8156…, the number of digits is 2 + 1 = 3.

Hex number FF_{16} = 255_{10}, Log_{16} 255 = 1.9985…, the number of digits is 1 + 1 = 2.

Binary number (*k *= 2) 10000000_{2} = 128, Log_{2} 128 = 7, the number of digits is 7 + 1 = 8.

Binary number 10100001_{2} = 161_{10}, Trunc(Log_{2 }161) = Trunc(7.33091…) = 7, the number of digits is 7 + 1 = 8.

*Image courtesy of Stuart Miles at FreeDigitalPhotos.net*

Via programming the number of digits can be determined by dividing the number by the base system number *k* until the number being divided is < 1. Now the number of digits is the number of divisions.

Below is a C programming language program that determines the number digits of 10 base system number that can be positive or negative decimal number:

#include <stdio.h> #include <math.h> int detDigits(double); void main(void) { double decimalnumber; int number_of_digits; printf("Give a decimal number: "); scanf("%lf", &decimalnumber); number_of_digits = detDigits(decimalnumber); printf("The number of digits is %d.",number_of_digits); } int detDigits(double dn) { int digits_integer_part, digits_decimal_part; int worknumber; if (dn == 0) return 1; if (dn < 0) dn = -dn; digits_integer_part = ((int)log10(dn)) + 1; dn = dn – (int)dn; worknumber = (int) dn; while (worknumber % 10 != 0) { dn = dn * 10; worknumber = (int)dn; } /* Variable "worknumber" now holds the number of digits of decimal part of the given number + 1 */ digits_decimal_part = worknumber – 1; return digits_integer_part + digits_decimal_part; }

Another way of determining the number of the digits is of course converting the decimal number to string and by examining the number of characters.

I remember the time when I was still studying at *lukio *(equivalent school is high school in the US, *gymnasium* in Sweden, *Gymnasium* in Germany) when there were exercises to determine positive integer powers of imaginary unit *i.* I invented at that time a method to easily determine the power without using the taught method of grouping the exponent. Later on my university times I invented a formula to determine arbitrary real number power of imaginary unit *i.*

**The method**

Complex number *z* is of form *z = a + bi*, where the previous is the real part and the latter the imaginary part. Now we’re interested only in imaginary unit *i*. So we examine complex number *z = i*, the magnitude of *i *= 1.

The argument (arg) of imaginary unit *i *is 90°. From this fact it follows that multiplying with *i* corresponds 90° rotation on the complex plane. Thus, if the exponent is a positive integer, there’s exactly 4 possibilities for the power of *i*: 1, -1, *i*, *-i*.

The power can be determined by dividing the exponent by 4 and examining the decimal part of the division. There are now 4 possibilities: .0; .25; .5; .75. These tell how many percent of unit circle on the complex plane has been rotated.

Based on this the following table can be presented:

The first line is the decimal part of division by 4 of the exponent, the second line tells place in the unit circle of the complex plane, the third line tells the the power of imaginary unit *i*.

For example to determine *i*^{12345678}* *, we first divide the exponent 12345678 by 4, the result is 3086419.5. The decimal part is .5, so we see from the table above, that the power is -1.

**The formula for arbitrary real number exponent**

I invented this formula myself some time in the year 1997 when I was studying for the second year at the university of Jyväskylä. I thought my formula was too simple, so I didn’t show it to the maths department staff. Anyway, the formula is as follows:

Many years ago, I asked on local science magazine’s net forum, that does this kind of formula already exist. The existing formula was a bit different. In the same forum one reader presented a proof to this formula.