The ONLY April day where this function is undefined

Wednesday, April 13, 2016
 

Today is Wednesday, April 13, 2016, and the cotangent of 4,130 degrees, equivalent to cot(170 degrees) and tan(100 degrees), equals −5.67128182. However, tomorrow, Thursday, April 14, is the only day in April where cot(10*N) is undefined, where N is the date in MM/DD format, disregarding the “slash” to form an integer. Four hundred and thirteen (413) represents April 13, since April 13 is written as “4/13,” while “414” and “415” represent April 14 and April 15, respectively. For 29 of the 30 days of April, the function is defined. The only exception is April 14, since cot(4140) is undefined. Go ahead, try this on Windows Calculator. Type “4140,” then select the “Trigonometry” button, then select “cot,” which stands for “cotangent.” Since tan(4140) = 0, and since cotangent is defined as the reciprocal of the tangent, or one divided by tan, this is undefined and returns the error message, “Cannot divide by zero.” Like with other error messages such as the “Invalid input” message, all calculator buttons are disabled except the 0-9 buttons and the “Back” and “Clear” buttons.

Why is cot(4140) undefined? That’s because tan(4140) = 0, and to find cot(4140), you have to take the reciprocal of zero, which, as you know, is not a valid operation in mathematics. You cannot divide by zero, so you cannot take its reciprocal. Zero does not have a reciprocal. Zero also does not have a logarithm in any base; this is negative infinity for any positive base such as base 10 and base ${\displaystyle e}$ (approximately 2.7182818), since ${\displaystyle e^{x}}$ and ${\displaystyle 10^{x}}$ infinitely approach, but never reach, zero as ${\displaystyle x}$ tends to negative infinity. For example, 10⁻⁹ is the reciprocal of 1,000,000,000, or one divided by one billion. This is extremely close to zero but not mathematically equal to zero. In fact, no finite exponent will cause the value of 10^−x to equal zero.

Although most people in real life would write April 1st as “4/1,” the zero-padded form “4/01” works better for consistency in mathematical contexts. April 1 through April 9 would be rendered as the numbers 401 through 409, not 41 through 49. This is consistent because 410 is 9 more than 401, just like April 10 is nine days after April 1. Similarly, April 11 is three days after April 8, just like 408 + 3 = 411. Although consistency is still lost when moving from one month to another (i.e. 202 is much larger than 131 even though February 2 is only two days after January 31), consistency is preserved within a month. The numbers 401 through 430 represent the dates April 1 through April 30. In computing, dates are often zero-padded for similar reasons of clarity and consistency; dates, when translated to integers, use the YYYY-MM-DD format, so “20160405” represents April 5, 2016, or 2016-April-05. Note that both the month number (04) and the day number (05) must be zero-padded. Therefore, for a date such as August 31, 2015, you should write “20150831,” not “2015831.” Every date in our lifetime should be written with exactly eight digits; no more, no less.

Example 1: 20110405 + 20 = 20110425, so April 25, 2011, was 20 days after April 5, 2011.

Example 2: 20160411 + 2 = 20160413, so April 13, 2016, today, is two days after April 11, 2016.

Example 3: 20160411 + 5 = 20160416, so the coming Saturday, April 16, 2016, will be five days after April 11, 2016.

Anyway, cot(4010), cot(4020), cot(4030), all the way through cot(4300), are all defined EXCEPT cot(4140), which is infinity.

• [ ] —  e.g. December 31, 1999
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