1047: Approximations - Revision history

82.132.237.40: /* Explanation */ Correct "more" to "less" (and a couple of missed characters added

2026-04-02T15:12:37Z

‎Explanation: Correct "more" to "less" (and a couple of missed characters added

174.34.4.47: Various improvements, updated populations

2026-04-02T14:45:26Z

Various improvements, updated populations

DollarStoreBa'al: cat

2026-03-31T17:02:21Z

cat

B for brain: /* Explanation */ It's been 3 months since January 1st.

2026-03-29T16:14:01Z

‎Explanation: It's been 3 months since January 1st.

2001:4C4E:1C01:F800:3D5A:8F0:A0B9:43A4: /* Equations */ the error has increased a little bit on May 20, 2019

2025-09-29T10:30:59Z

‎Equations: the error has increased a little bit on May 20, 2019

CalibansCreations: /* Equations */

2025-04-17T08:26:00Z

‎Equations

108.162.245.19: /* Explanation */

2025-03-30T04:04:09Z

‎Explanation

Firestar233: not everyone here has a math degree

2025-02-16T06:33:22Z

not everyone here has a math degree

172.70.210.235: /* Proof */

2025-02-15T22:08:59Z

‎Proof

172.68.54.209: /* Explanation */ 2025 update

2025-02-03T17:55:48Z

‎Explanation: 2025 update

@@ Line 183: / Line 183: @@
 — 385<br>
 |align="center"|
-|align="left"|Grows by 3 million each year. As of April 2026 the predicted number is 358 million, while the actual count is 342 million([https://www.census.gov/popclock source]), or 16 milion more than predicted.
+|align="left"|Grows by 3 million each year. As of April 2026 the predicted number is 358 million, while the actual count is 342 million ([https://www.census.gov/popclock source]), or 16 million less than predicted.
 |-
 |align="center"|Electron rest energy (joules)

@@ Line 61: / Line 61: @@
 |align="center"|31,640,625
 |align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
-|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he did add this statement to the top of the comic page:
+|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he added this statement to the top of the comic page:
 "Lots of emails mention the physicist favorite, 1 year = pi × 10<sup>7</sup> seconds. 75<sup>4</sup> is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 10<sup>7</sup> is nearly equal to 31,415,926.536, and 75<sup>4</sup> is exactly 31,640,625. Randall's elegance belongs to the number π, but it should be multiplied by the factor of ten.
@@ Line 142: / Line 142: @@
 — 8.8<br>
 |align="center"|
-|align="left"|Grows by 75 million every year on average (100 million every year, except for a pause every leap-year). As of 2025, still more or less accurate. (8.0B)
+|align="left"|Grows by 75 million every year on average (100 million every year, except for a pause every leap-year). As of April 2026, a little under the real number (the predicted population is 8.1 billion, while the real number is ~8.18 billion([https://www.census.gov/popclock source]))
 |-
 |align="center"|U.S. population estimate (millions)
@@ Line 183: / Line 183: @@
 — 385<br>
 |align="center"|
-|align="left"|Grows by 3 million each year. As of 2024 the actual number is ~10 million smaller.
+|align="left"|Grows by 3 million each year. As of April 2026 the predicted number is 358 million, while the actual count is 342 million([https://www.census.gov/popclock source]), or 16 milion more than predicted.
 |-
 |align="center"|Electron rest energy (joules)

← Older revision		Revision as of 17:02, 31 March 2026
Line 537:		Line 537:
	[[Category:Protip]]		[[Category:Protip]]
	[[Category:Large drawings]]		[[Category:Large drawings]]
		+	[[Category:Comics with custom header texts]]

@@ Line 18: / Line 18: @@
 Randall says he compiled this table through "a mix of trial-and-error, ''{{w|Mathematica}}'', and Robert Munafo's [http://mrob.com/pub/ries/ Ries] tool."  "Ries" is a "{{w|Closed-form expression#Conversion from numerical forms|reverse calculator}}" that forms equations matching a given number.
-The {{w|world population}} estimate for 2025 is still somewhat accurate. The estimate is 8.0 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: [https://www.census.gov/popclock/ United States Census Bureau - U.S. and World Population Clock]. Nevertheless, there are other numbers listed by different sources.
+The {{w|world population}} estimate for 2026 is still roughly accurate. The estimate is 8.1 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: [https://www.census.gov/popclock/ United States Census Bureau - U.S. and World Population Clock]. Nevertheless, there are other numbers listed by different sources.
 The first part of the title text notes that "Jenny's constant," which is actually a telephone number referenced in Tommy Tutone's 1982 song {{w|867-5309/Jenny}}, is not only prime but a {{w|twin prime}} because 8675311 is also a prime. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them.  Twin primes were also referenced in [[1310: Goldbach Conjectures]].

@@ Line 93: / Line 93: @@
 |align="center"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
 |align="center"|1.59895121062716 × 10<sup>−19</sup>
-|align="center"|1.602176565 × 10<sup>−19</sup>
+|align="center"|1.602176634 × 10<sup>−19</sup>
 |align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567: Urgent Mission|for that]].)
 |-

@@ Line 70: / Line 70: @@
 |align="center"|31,536,000
 |align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
-|align="left"|"''Rent'' Method" refers to the song "{{w|Seasons of Love}}" from the musical ''{{w|Rent (musical)|Rent}}''. The song asks, "How do you measure a year?" One line says "525,600 minutes" while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one. This method for remembering how many seconds are in a year was also referenced in [https://what-if.xkcd.com/23/ What If? 23].
+|align="left"|"''Rent'' Method" refers to the song "{{w|Seasons of Love}}" from the musical ''{{w|Rent (musical)|Rent}}''. The song asks, "How do you measure a year?" One line says "525,600 minutes" while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one. This method for remembering how many seconds are in a year was also referenced in ''{{what if|23|Short Answer Section II}}'', where [[Cueball]] gets the song stuck in his head whilst calculating how many counterfeit bills he can print in a year.
 |-
 |align="center"|Age of the universe (seconds)
@@ Line 94: / Line 94: @@
 |align="center"|1.59895121062716 × 10<sup>−19</sup>
 |align="center"|1.602176565 × 10<sup>−19</sup>
-|align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567|for that]].)
+|align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567: Urgent Mission|for that]].)
 |-
 |align="center"|Telephone number for the {{w|White House}} switchboard

@@ Line 350: / Line 350: @@
 \end{align}</math>
-To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7) <!--<math>\textstyle{ \cos \frac{2k\pi}{7} }</math>--> is the real part of the k-th 7-th root of unity, exp(2 k i pi/7)<!--<math>\textstyle{ \exp \frac{2 k i\pi}{7} }</math>-->. The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:
+To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7) <!--<math>\textstyle{ \cos \frac{2k\pi}{7} }</math>--> is the real part of the k-th 7-th {{w|root of unity}}, exp(2 k i pi/7)<!--<math>\textstyle{ \exp \frac{2 k i\pi}{7} }</math>-->. The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:
 <!--:<math>0 = \cos \frac{0\pi}{7} + \cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} + \cos \frac{8\pi}{7} + \cos \frac{10\pi}{7} + \cos \frac{12\pi}{7} </math>-->

@@ Line 365: / Line 365: @@
 :cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2
-which, because cos(x) = cos(pi - x),<!--<math>\cos (x) = - \cos(\pi - x)</math>,--> can be rewritten as
+which, because cos(x) = -cos(pi - x),<!--<math>\cos (x) = - \cos(\pi - x)</math>,--> can be rewritten as
 <!--:<math>\cos \frac{5\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{pi}{7} = 1/2 </math>-->