>>Brushing up

Significant digits 1.414213562373095048801688724209698078569671875 AKA: Pythagoras’ number Stats: First irrational discovered; decimal expansion is infinite and nonrepeating.

Where it comes from: 2 1guess at

= 1.5; 2

1guess + 1.5 = 1.25; 2

2 = 1.6; 1.25

1.25 + 1.6 = 1.425; 2

Courtesy John A. Jaszczak

The science of numbers called mathematics was described by Poincaré as the art of giving the same name to different things. This is true whether counting bushels of barley or grouping dimensionless parameters for fluid modeling. Numbers are a tool, our human tool for exploring, comprehending, and altering our surroundings. But there is evidence that numbers would exist even if we had never named and used them. Says G.H. Hardy in A Mathematician’s Apology: “In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.” So even though we may play with and use numbers, they seem to follow their own laws — the laws of the universe.

√2 2 = 1.4035; 1.425

1.425 + 1.4035 = 1.4142543... 2

Algorithm from ancient Babylonia

Natural occurrences: Natural rhombic dodecahedron crystals are as symmetrical as any cube. The ratio of the two diagonals’ lengths in each rhombus is the square root of two.

3.141592653589793238462643383279502884197169399 AKA: Archmedes’ or Ludolph’s number; Pi Stats: Irrational, transcendental; it might be logically assumed that all ten numbers appear with the same frequency in pi’s first 4 billion decimal digits, but not so. Six appears 0.0016% more than 2, a statistically significant difference.

π Where it = comes from: 2

b

π

1 1 1 1 1 1 1 1 1 1 + + + ... 2 2 2 2 2 2 2 2 2

a = b

Natural occurrences: In spherical celestial bodies. The number of ways to get a chosen integer by squaring and summing two others (c.i.=m2+n2) approaches π as the chosen integer gets bigger.

0.5772156649015328606065120900824024310421593359 AKA: Euler-Mascheroni constant Stats: Rational and transcendental status unknown.



Where it comes from:  n 1  γ = lim  ∑ − ln( n)  n → ∞  k =1 k Relationship to fellow famous figures: Surprisingly, the difference of the harmonic series minus ln(x) converges to g. Occurs in integrals; shadowed only by π and e in significance. Natural occurrences: In asymptotic formulas used to predict the probability of severe 36 MSD ● Motion System Design ● September 2002

Courtesy NASA

Courtesy NASA

Equation derived by François Viète

a

2.718281828459045235360287471352662497757247093 AKA: Napier’s number; the natural logarithm base Stats: Derivative of loge(x) is 1/x. Transcendental: not the root of any rational number coefficient

e

polynomial. Though transcendental numbers such as e occur infinitely more frequently than algebraic numbers, strangely only a few have been found. Relationship to fellow famous figures: eπ - π ≈ 20 Natural occurrences: In growth, decay, continuously compounding interest, and alternating current models.

Where it comes from: 0! 1 1! 1 2! 1 3! 1 4! 1 5! 1 6! 1 7! 1 8! 1 9!

= = =

1 1 1

= 0.0416666666666666666666667

24 1

= = =

1

= 0.0013888888888888888888889

720 1

= 0.0001984126984126984126984

5040 1 40320 1 362880 1

= 0.0000248015873015873015873

AKA: Golden Mean Stats: Algebraic, irrational. Where it comes from:

= 0.0000002755731922398589065

3628800

ϕ=

1± 5 2



= 0.0000000250521083854417188 = 0.0000000020876756987868099 = 0.0000000001605904383682161

13! 1

1.618033988749894848204586834365638117720309179

= 0.0000027557319223985890653

12! 1

Shown here is the decay of a manmade subatomic boson particle into four muons, which are identical to electrons, but bigger. The natural logarithm can model this decay.

= 0.0083333333333333333333333

120

=

11! 1

= 0.1666666666666666666666667

6 1

=

10! 1

= 0.5000000000000000000000000

2 1

=

1

= 1.0000000000000000000000000

1 1

=

=

= 1.0000000000000000000000000

Courtesy Lucas Taylor, Northeastern Univ.

1

Relationship to fellow famous figures: The ratios of successive number pairs in the Fibonacci sequence approaches the Golden Mean. Natural occurrences: In seashell and human fist spiral geometry; in early architecture.

= 0.0000000000114707455977297

14! 1

= 0.0000000000007647163731820

15! 1

= 0.0000000000000477947733239

16! 1

= 0.0000000000000028114572543

17! 1 18! 1

+

19! 1

= 0.0000000000000001561920697 = 0.0000000000000000082206352 = 0.0000000000000000004110318

20!

≈ 2.718281828459045235360 September 2002 ● Motion System Design ● MSD 37

Significant digits: Numbers in nature

The science of numbers called mathematics was described by Poincaré as the art of giving the same name to different things.

Significant digits: Numbers in nature

The science of numbers called mathematics was described by Poincaré as the art of giving the same name to different things.