Hype for the Future 94C: Important “Off-by-One” Numbers in Computing

Introduction

In the world of computing, the numbers formed by the powers of two subtracted by one, with the exception of the number zero, are of significant interest and relevance in the world of computer science and cybersecurity. Even zero (0) is of significant importance even though the binary representation is also 0 because of the binary nature of computing and related features.

Significant Numbers by Number of Bits Required in Binary

1. 1
2. 3
3. 7
4. 15
5. 31
6. 63
7. 127
8. 255
9. 511
10. 1023
11. 2047
12. 4095
13. 8191
14. 16383
15. 32767
16. 65535
17. 131071
18. 262143
19. 524287
20. 1048575
21. 2097151
22. 4194303
23. 8388607
24. 16777215

Usually, for numbers higher than 16777215, which most commonly refers to data storage sizes such as 16 MiB (mebibytes) minus one byte, the notation is written out in a condensed form. For instance, 65535 KiB (kibibytes) would likely be written out as 64 MiB (mebibytes) minus one kibibyte.

Hype for the Future 94A: Syllables Required to Pronounce Numbers in American English

Introduction

Throughout the number line, counting only the natural numbers and excluding the whole number of zero and the negative integers, the minimum numbers required to be spoken in a certain number of syllables are, as follows, numbers often containing the digits one, two, and/or seven.

Numbers by Number of Syllables

1. 1 (one)
2. 7 (se•ven)
3. 11 (e•le•ven)
4. 27 (twen•ty•se•ven)
5. 77 (se•ven•ty•se•ven)
6. 111 (one•hun•dred•e•le•ven)
7. 127 (one•hun•dred•twen•ty•se•ven)
8. 177
9. 777
10. 1127
11. 1177
12. 1777
13. 7777
14. 11777
15. 27777
16. 77777
17. 111777
18. 127777
19. 177777
20. 777777

As a general pattern, the largest place values in American English are, in increasing numbers of syllables, “one hundred eleven,” “one hundred twenty-seven,” “one hundred seventy-seven,” and “seven hundred seventy-seven,” where applicable.

Hype for the Future 68J: The Derivatives of the Different Trigonometric Functions?

Overview

While novaTopFlex would like to explain the derivatives and integrals of the different trigonometric functions, LaTeX would be required in order to correctly interpret the underlying situation.

Hype for the Future 68F: YouTube Channel Plans

novaTopFlex

For the channel at https://www.youtube.com/@novaTopFlex:

• Existing Shorts
• Gummi for LaTeX Programming and Notation

geoTopFlex

For the channel at https://www.youtube.com/@geoTopFlex:

• No programming anticipated at this time.

futoTopFlex

For the channel at https://www.youtube.com/@futoTopFlex:

• Trigonometry, Geometry, Algebra, Units

Hype for the Future 44D: Calculated Saturdays for 2026

The Saturdays of the year 2026 are of the utmost importance for novaTopFlex, since every Saturday has a unique story to tell in particular. The first Saturday associated with the new year is January 3, since the new year begins on a Thursday in the case of 2026. In order, the Saturdays of the new year are:

1. January 3
2. January 10
3. January 17
4. January 24
5. January 31
6. February 7
7. February 14
8. February 21
9. February 28
10. March 7
11. March 14
12. March 21
13. March 28
14. April 4
15. April 11
16. April 18
17. April 25
18. May 2
19. May 9
20. May 16
21. May 23
22. May 30
23. June 6
24. June 13
25. June 20
26. June 27
27. July 4
28. July 11
29. July 18
30. July 25
31. August 1
32. August 8
33. August 15
34. August 22
35. August 29
36. September 5
37. September 12
38. September 19
39. September 26
40. October 3
41. October 10
42. October 17
43. October 24
44. October 31
45. November 7
46. November 14
47. November 21
48. November 28
49. December 5
50. December 12
51. December 19
52. December 26

Hype for the Future 9B: Verfunctions in Trigonometry

The graph attached below showcases eight distinct and antiquated trigonometric functions regarding “versed” functions. Haversine in particular has historically been of importance in navigation, sailing, and perhaps even in communities such as fishing and whaling as well.

From top to bottom, the functions listed in the legend correspond with the versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, and hacovercosine, respectively.

Hype for the Future 9A: What to Expect from the YouTube Channels

On Days 7 and 8 of the 284-day “Hype for the Future” series, novaTopFlex has exported numerous mathematical graphs in PNG format as follows (as futoTopFlex):

Hype for the Future 8/284: Mathematics Edition

novaTopFlex takes on the ultimate mathematics competition as well as the test of success as a member of the novaTop community. Throughout the ideals of the novaTop Identity, Hype for the Future signifies a guaranteed sense of success, and over the next two-hundred-seventy-six (276) days, novaTopFlex shall maintain upkeep for the Three Identities with additional context, including from journalism.

Hype for the Future 7A: Mathematics Edition

The hyperbolic functions of the sine and of the cosine are identified as follows:

sinh(x)=(e^x-e^(-x))/2

cosh(x)=(e^x+e^(-x))/2

The remaining hyperbolic trigonometric functions can be constructed with the hyperbolic versions of the appropriate functions. While tan(x)=sin(x)/cos(x), tanh(x)=sinh(x)/cosh(x) and can be constructed as (e^x-e^(-x))/(e^x+e^(-x)).

Hype for the Future 7/284: The novaTop Experience, Mathematics Edition

novaTopFlex is preparing mathematics knowledge for greater mathematics competitions through the end of the line at the end of May 2026. As of November 7, 2025, novaTopFlex has silently been interested in trigonometry and in trigonometric functions at a particularly deep level.

Computer Science: Is it really a subject?

Throughout my age and schooling, computer science has often been crowned as my favorite school subject in my mind and worldview. As far as I have known throughout my past years, even all the way back to 2015, I had been on an active and increasing trajectory into technical geekery, though I have had personal experience with technology since effectively the end of 2007.

The Ultimate November 2025 Experience by novaTopFlex

novaTopFlex is going to be on multiple breaks for special events that may interfere with the regular streaming and recording schedules. Dates that may be less likely to be available for the Three Identities are November 4 (for reasons not to be disclosed), November 6 (for a cybersecurity awareness and event competition), November 7-8 (for a statewide mathematics contest), November 15 (for the first swim meet), and November 26-30 (reduced capacity due to Thanksgiving accommodations).

Why is arctan(1/2) equal to arctan(1/3) + arctan(1/7)?

The truth behind the signature arctangent question is simply that the arctangent addition formula, in direct resemblance to the tangent addition formula, is directly related to the fact that 1/3 + 1/7 equals 7/21 + 3/21 and the fact that 7 + 3 = 10, resulting in a final solution of 10/21 just in the numerator. In the denominator, however, is 1 - (1/3)(1/7), which results in 1 - 1/21, which in return results in (21-1)/21 equals 20/21. If the numerator is 10/21 and the denominator is 20/21, the ratio is exactly identical to the ratio with numerator 10 and denominator 20. Furthermore, 10/20 is equivalent to the simplified expression 1/2.

Related Question: Why can Pi be represented by arctan(1)+arctan(2)+arctan(3)?

The arctangent of one plus the arctangent of two is equivalent to the arctangent of (1+2)/(1-1*2), which is the arctangent of 3/-1 equals -3. With two and three, the formula becomes (2+3)/(1-2*3), which becomes 5/-5 and thus becomes -1. For one and three, the formula is (1+3)/(1-1*3), which equates to 4/-2 equals -2. Multiplying the arctangents of the negatives by the multiplicative inverses effectively cancels out the numerators, as in x+(-x) equals x-x equals 0 (zero). Even though the arctangent of zero is, in fact, zero, the tangent function is notable for having a period of π, and with the positive arctangents, arctan(1)+arctan(2)+arctan(3) can, in fact, equal π.

The novaTopFlex Video Topics for Halloween

In anticipation of a large-scale mathematics competition on Saturday, November 8, 2025, and the ultimate competition following my penultimate loss from November 2, 2024, novaTopFlex shall enter the futoTop configuration with extensive mathematics topics. Expect upcoming videos to cover topics in algebra, geometry, trigonometry, and numerous associated topics and subtopics.

What is Binary?

According to novaTopFlex, the sole reason to mention binary is if and when mentioning binary digits. For instance, 1000 is 8 in binary, so 8 in binary is 1000. What is 6 in binary? What is 7 in binary? The correct answers, of course, would be 110 and 111, respectively.

Allyship Targets for novaTopFlex

When referring to discussions of social roles, the novaTop Community shall ally with the State of Vermont, as well as with Western Massachusetts, specifically the Berkshires and the Pioneer Valley. When referring to tourist and local interest attractions, the Community shall focus primarily on New England, then on the Mid-Atlantic, then on the Great Lakes region, then on Mid-Appalachia and the Lower Midwest.

For novaTopFlex and geoTopFlex interests, the technology and geography sub-communities shall communicate—and ally—with the Boston Metropolitan Area. With economic concerns, novaTop shall ally with the Berkshires and into the Taconics across the New York state line.

Historical Context of the Dual Contest Qualifiers

The historical context of both novaTopFlex and the math powerhouse often overlaps, though not always so. With both novaTopFlex and the math powerhouse, individuals can thank Susan B. Anthony, Lucretia Mott, Elizabeth Cady Stanton, W. E. B DuBois, Mark Twain, Nathan Hale, Ada Lovelace, and Maria Mitchell as historical figures of interest and influence. Both are relatively nonconforming individuals, albeit in different ways, and novaTopFlex is actually three separate identities to an individual.

Mathematics Performance

novaTopFlex is glad to announce a high confidence in performance in the subjects of mathematics in general as well as a plan to continue within the mathematical journey toward a more positive outlook in the future! Nearly all mathematics topics are covered in the performance, and new horizons and new chapters of the life story and of general public-facing personal and community journalism are coming soon as well.

Interesting Mathematics Facts

Preamble

novaTopFlex returns to a mathematic conquest of the journey to the top, the gateway to pure mathematic joy and excellence, and all the way to the purest forms of excitement! The novaTop community shall anticipate the highest performance levels possible in every respect in terms of what is considered ideal, desirable, and acceptable, with very high standards to ensure accuracy, fairness, and correctness. Starting with phase one, where a calculator is allowed, recommended, and encouraged, but going into phases two and three, as novaTop approaches the state and national conquest stages, it gets very important to be prepared for every possible next step.

Curious Facts about the Journey

Individuals and groups may understand trigonometry and related mathematics topics at a basic level, but the common people may not know, for instance, that adding the arctangent of one-third to the arctangent of one-seventh results in the arctangent of one-half. Additional quirks also include the identities which occur when cosines of angles are multiplied by cosines of sixty degrees minus the respective angles followed by cosines of sixty degrees plus the respective angles. With both sine and cosine relationships, the identities are identified as triple-angle identities thence divided by four.

Additional formulae can be derived from existing trigonometric properties, including from a baseline understanding of the sine and cosine functions and relationships. But any trigonometric function can be expressed in terms of the remaining functions, depending on the specific context. For instance, sine is the square root of the difference of one and cosine squared, while cosine is the square root of the difference of one and sine squared.

With algebraic formulae intact, cubic functions can be solved with Cardano’s formula for the determinants at a minimum, but the solutions are often difficult if not impossible to achieve. Quartic functions can also be solved using radicals; however, per the Abel-Ruffini theorem, quintic and higher-order functions almost never can be solved in such terms, with the simplest example being “x to the fifth minus x minus one equals zero.”

The ‘novaTop Authority

novaTopFlex is an authority in the subject matters of technology, geography, and mathematics. Depending on the specific context, language, and/or application, abilities may vary dramatically. For instance, while I may be easily capable of utilizing my most commonly used commands or my applications, I may often utilize documentation sets provided to my system(s) by Dash by Kapeli through Zeal for video content on select platforms.

Part of the novaTop authority is the separation of identities into novaTop, geoTop, and futoTop standards. novaTop is for technology, geoTop is for geography, and futoTop is for mathematics. Video content, especially on mainstream platforms such as YouTube, may be further subdivided as follows:

• Documentation-based Topics
  • Content provided through Dash/Zeal and intended for educational, documentary, scientific, and artistic purposes
• Content-based Topics
  • Content provided by commands and features on the host system
  • Parameters of the content should also be provided within the content-based category.
• Context-based Topics
  • Content provided within the specific content of commands
  • A successive string of commands related to a singular stream or novaTop project are the core context-based topics.

The Identity

The novaTop Identity has just become official! Significant influences to the identity are prevalent from Rossmann, Stallman, and, of course, even from the Voldemort of Tech! When you access my YouTube channels, there may occasionally be a dichotomy between formal and informal influences, so if you are looking for more professional content, please check the video content longer than five minutes in real time or videos where the video stream has been slowed down. Or for slightly more informal tech suggestions, the videos that had been sped up or are in real time less than five minutes may be of better help. There will also be the occasional video that requires a mix of speeds, but videos that include a slowdown portion should be more suitable for practical and professional settings.

Categories of videos from the three identities may be categorized based on length, speed, type, and topic. Turning points in length occur at five minutes and again at fifteen minutes, while speeds can be faster, normal, slower, or even mixed. Types may include documentation-, content-, and context-based topics. Topics will be included at the beginning of every novaTop video. Technologies will be under the novaTop category, while geospatial progress and recommendations will be under geoTop. futoTop does not have an official identity, so the identity may become more prevalent in select novaTop and geoTop content streams, if any given topic relates to mathematics, or possibly science, in any way.

Documentation-based topics refer to topics present on YouTube videos on the novaTopFlex channel that are entirely based on documentation on Dash or Zeal. Topics with additional influences beyond the Zeal documentation may be content- or context-based. All videos in the geoTopFlex channel are considered context-based channels. Content-based topics specifically refer to the output of specific command(s), regardless of the requirement to view additional internal and/or external documentation. Context-based topics are specifically based on a specific situation occurring within novaTopFlex.

If there are interviews or external forces involved in a novaTopFlex video or stream, then these meetings must break through. Expect novaTopFlex to speak about the future and issues in the technology industry during these times of elevated risk. Questions may also be focused on the context, but videos with third parties may not be considered “context-based.”

Two plus One: Exploring the ‘topFlex Identities

May 31, 2025, and beyond

novaTopFlex is an identity, a community, and an ethical channel on YouTube, all focusing predominantly on advocacy for open contributions and the elimination of closed contributions. Although the topics of interest may vary substantially through the three identities, the ideologies remain fairly similar.

novaTopFlex

novaTopFlex is the technological identity, acting as the main identity for the novaTop personality.

novaTopFlex Examples

Examples of content containing features of the novaTop identity may include:

• Documentation-based topics (from system documentation sets, including, but not limited to:)
  • Ada
  • Apache HTTP Server
  • Bash
  • Blender
  • Bootstrap
  • C
  • CSS
  • CUDA
  • Crystal
  • Clojure
  • GLib
  • GLibC
  • GNU make
  • D
  • Django
  • Docker
  • Elvish
  • Fish
  • FFmpeg
  • FLTK
  • Kotlin
  • Kubernetes
  • J
  • Jekyll
  • Jinja
  • Joomla
  • Julia
  • Java
  • JavaScript
  • Lua
  • LaTeX
  • Linux Man Pages
  • Nginx
  • Sphinx
  • Tcl
  • Tk
  • Debian Policy
  • Debian History
  • Rust
  • Ruby
  • Go
  • Odin
  • Drupal Console (decommissioned due to drush)
  • WP-CLI
  • WordPress
  • Zsh
  • PEPs
  • RFCs
  • Racket
  • R
  • Perl
  • Python
  • PHP
  • MySQL
  • MariaDB
  • PostgreSQL
  • gi-docgen
  • Haskell
  • Haml
  • HTML
  • HTTP
  • Qt
  • UnderscoreJS
  • MomentJS
  • Node.js
• Content-based topics
  • Videos based on system manual “man” pages
  • Videos requiring access to software not preinstalled on an Ubuntu system
  • Videos showcasing output from commands and pagers such as “less”
  • The output of a single command
• Context-based topics
  • The output of multiple commands
  • The output of a chain of commands
  • Successful how-to guides for videos (rare but feasible, particularly for smaller projects and videos)

geoTopFlex

geoTopFlex is the broad and obvious geographical identity, and acts as a side-identity and geospatial channel to standard novaTop content. Often independent of the primary identity, sometimes novaTop content must be contained within geoTop context for reliability.

geoTopFlex Examples

Examples of content containing features of the geoTop identity may include:

• The “Advocacy for OpenStreetMap Contribution” playlist, originally from novaTopFlex prior to the identity split
• Active live streams for polygons from OpenStreetMap data for the “novaTop QGIS Project”
• Videos of ideal “control cities” for US and state highways across the nation
• Additional videos containing data from OpenStreetMap contributors

futoTopFlex

futoTopFlex is the secret mathematics identity, and acts as a back-identity to the standard novaTop and geoTop personalities. Unless combined with novaTop aspects on video, the futoTop personality will be respected as such. Otherwise, this is the futaTop personality to avoid confusion. Normally, content combinations will not occur; however, within a novaTopFlex or geoTopFlex video, combinations may be required to occur for an extended understanding of the context.

futoTopFlex Examples

Examples of content containing features of the futoTop identity may include:

• C and C++ Mathematics Libraries
• The J Programming Language, which is mostly comprised of mathematics commands
• Mathematics features in additional languages
• System libraries requiring mathematics
• Content from OpenStreetMap containing polygons
  • Delaunay Triangulation
  • Voronoi Polygons
  • Calculating centroids