Electronics World articles Popular Electronics articles QST articles Radio & TV News articles Radio-Craft articles Radio-Electronics articles Short Wave Craft articles Wireless World articles Google Search of RF Cafe website Sitemap Electronics Equations Mathematics Equations Equations physics Manufacturers & distributors LinkedIn Crosswords Engineering Humor Kirt's Cogitations RF Engineering Quizzes Notable Quotes Calculators Education Engineering Magazine Articles Engineering software RF Cafe Archives Magazine Sponsor RF Cafe Sponsor Links Saturday Evening Post NEETS EW Radar Handbook Microwave Museum About RF Cafe Aegis Power Systems Alliance Test Equipment Centric RF Empower RF ISOTEC Reactel RF Connector Technology San Francisco Circuits Anritsu Amplifier Solutions Anatech Electronics Axiom Test Equipment Conduct RF Copper Mountain Technologies Exodus Advanced Communications Innovative Power Products KR Filters LadyBug Technologies Rigol TotalTemp Technologies Werbel Microwave Windfreak Technologies Wireless Telecom Group Withwave RF Cafe Software Resources Vintage Magazines RF Cafe Software WhoIs entry for RF Cafe.com Thank you for visiting RF Cafe!

Temwell Filters

PCB Directory (Manufacturers)

Please Support RF Cafe by purchasing my  ridiculously low−priced products, all of which I created.

RF Cascade Workbook for Excel

RF & Electronics Symbols for Visio

RF & Electronics Symbols for Office

RF & Electronics Stencils for Visio

RF Workbench

T-Shirts, Mugs, Cups, Ball Caps, Mouse Pads

These Are Available for Free

Espresso Engineering Workbook™

Smith Chart™ for Excel

Copper Mountain Technologies (VNA) - RF Cafe

Pi (π) - a Closer Look

There are so many pages on the Internet with information on π (pi) that I will not even attempt to outdo them. However, if you happened upon this website and are looking for series expansions for calculating π, then you have come to the right place. Here are a few of the most popular.

The earliest renditions of π resulted from estimating the relationship of measurements of the circumference of a circle to its diameter. Here are a few early values, which were all ratios of whole numbers because at the time it was inconceivable that something "irrational" could exist:

  • 3 = value implied in the Bible in I Kings 7:23
  •  - RF Cafe = upper bound by Archimedes

  •  - RF Cafe = lower bound by Adriaan

  •  - RF Cafe = Otho's value

  •  - RF Cafe = Ptolemy's value (he used 3.14167 in his calculations).

  • A few others:  - RF Cafe

Here's an interesting property I just noticed myself about Ptolemy's value (even though it is of no consequence): the numerator is the impedance of free space, and the denominator times π yields the numerator, so, rearranged, 120π=377!

π was calculated to 200 places in 1844 by Johann Martin Zacharias Dase (1824-1861).

π was calculated to 200 places in 1844 by Johann Martin Zacharias Dase - RF Cafe and since  - RF Cafe, then pi approximatino - RF Cafe,

which is the first infinite series ever found for π, by James Gregory (1638-1675). The problem is that this series (called the Gregory-Leibniz series) converges very slowly on π: 300 terms are needed to be accurate to only two decimal places!

Newton (1656-1742) had a better idea (many, actually) and discovered the following:

Newton (1656-1742) had a better idea - RF Cafe, and since  - RF Cafe, then Newton (1656-1742) had a better idea - RF Cafe, which converges much more quickly.

He then determined that π to 16 decimal places using only 22 terms - RF Cafe, which produces π to 16 decimal places using only 22 terms.

Another series by Gregory was produced using two arctangents which resulted in:

                           Another series by Gregory was produced using two arctangents - RF Cafe

Euler (1707-1783) determined that  - RF Cafe, exactly. Of course, the arctangents are irrational. Euler calculated π to 20 decimal places in one hour using this formula.

This series by David and Gregory Chudnovsky of Columbia University produced over 4 billion digits in 1994, and each term gives an additional 14 digits:

David and Gregory Chudnovsky of Columbia University - RF Cafe

The incredibly simple series here was recently discovered: pi series - RF Cafe

...and was used as the basis for a handy algorithm that produces any digit of π one might desire, albeit in hexadecimal - see Reference 2, below, for the formula (too much involved to repeat it here).

 (thanks to Hugo K. for these last three items and for Reference 2, below)

1. For a very extensive treatise on π, please read the source of this material, "A History of Pi" by Petr Beckmann

2. "The Quest for Pi" by David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe
June 25, 1996 Ref: Mathematical Intelligencer, vol. 19, no. 1 (Jan. 1997), pg. 50–57

About RF Cafe

Kirt Blattenberger - RF Cafe Webmaster

Copyright: 1996 - 2024


    Kirt Blattenberger,


RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The World Wide Web (Internet) was largely an unknown entity at the time and bandwidth was a scarce commodity. Dial-up modems blazed along at 14.4 kbps while tying up your telephone line, and a nice lady's voice announced "You've Got Mail" when a new message arrived...

Copyright  1996 - 2026

All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged.

All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged.

My Hobby Website: AirplanesAndRockets.com

My Daughter's Website: EquineKingdom

Exodus Advanced Communications Best in Class RF Amplifier SSPAs

TotalTemp Technologies (Thermal Platforms) - RF Cafe

Innovative Power Products Passive RF Products - RF Cafe