How many decimal places are required by scientists for π?
NASA uses 15 π digits for interplanetary travel. 40 digits allow precise universe circumference calculation, deviating less than a hydrogen atom's diameter.
How many digits of π do most scientists and engineers typically employ for calculations? Is the computation solely based on π≈3.14, or do they incorporate a larger number of digits beyond the decimal point?
The actual situation might leave you slightly disappointed. Contrary to popular belief, most scientists, physicists, and astronomers – even those working on orbit and interplanetary navigation calculations for deep space exploration missions – do not employ the extensively multi-digit constant π that is often envisioned. The utmost precision utilized stands at 15 decimal places, namely π=3.141592653589793. Why is this the case? Pi is defined as the ratio of a circle’s circumference to its diameter.
Let’s take Voyager 1, the spacecraft currently farthest from Earth, as an example for calculation. With its present distance of about 23 billion kilometers, we consider this as the radius of a circle. Accordingly, the circumference of said circle amounts to 2×π×23,000,000,000. When employing π=3.14, the calculated circumference turns out to be 144,440,000,000 kilometers. However, if we utilize π to 15 decimal places (π=3.141592653589793), the computed circumference becomes 144,513,262,065.130478 km. Upon comparison, the two values exhibit a difference exceeding 73 million kilometers, highlighting a substantial disparity.
Now, let’s take the precision of π a step further by extending it by an additional 5 orders of magnitude – how about 20 digits beyond the decimal point? Assuming π=3.14159265358979323846, the resulting calculated circumference stands at 144,513,262,065.13048896916 kilometers. Comparing 144,513,262,065.130478 km and 144,513,262,065.13048896916 km, you’ll observe that these two numbers only diverge from the fifth decimal place, indicating a difference of merely 0.00001 kilometers, which is equivalent to 1 centimeter! In simpler terms, even with a dedicated effort to enhance precision by a factor of 100,000, the error remains no wider than a finger’s breadth on a grand circle with a diameter of 23 billion kilometers. Consequently, this error can be safely disregarded, rendering the deliberate pursuit of ultimate precision unreasonable.
If we restrict our circle’s diameter to be that of Earth, utilizing π accurate to 15 decimal places results in a circumference error so minuscule it’s akin to the scale of a single molecule or a few, rendering it entirely inconsequential for the precision of your driving or walking navigation. Naturally, one might raise the question of its application across the entire cosmos. Well...
The observable universe’s radius spans about 46 billion light-years, roughly equivalent to 4.35×10^26 meters. Therefore, the discrepancy between the ‘circumference’ of the universe computed using π to 15 decimal places and π to 20 decimal places amounts to a mere 200 meters—comparable to a mere playground track found in some elementary schools.
Comparison: 2,734,379,413,831,484,031.34 meters
To achieve precision comparable to the diameter of a hydrogen atom in the 'circumference' measurement of the universe, one would likely need to extend π to 39 or 40 decimal places (given that atomic diameters are not a critical factor). Yet, how does an atom measure up against the vastness of the cosmos?
In reality, contemporary scientists adhere to a usage of π accurate to 15 decimal places for another significant reason. Within electronic computers, numerical storage is not arbitrary. Generally, decimal numbers (real numbers) are represented through 'floating-point numbers.' The term 'floating point' implies that the position of the decimal point is not fixed. For instance: 1.2345 or 123.456—here, the location of the decimal point differs.
In 1985, the Institute of Electrical and Electronics Engineers (IEEE) introduced the floating-point standard, which standardized the representation of floating-point numbers and established two formats: single-precision floating-point numbers (single) and double-precision floating-point numbers (double). Each floating-point number is stored within the computer’s memory in scientific notation, comprised of three components: sign bit, exponent, and mantissa. The single-precision floating-point number occupies 32 bits (4 bytes) of storage space, with the sign using 1 bit, the exponent using 8 bits, and the mantissa using 23 bits. The double-precision floating-point number occupies 64 bits (8 bytes) of storage, with the sign using 1 bit, the exponent using 11 bits, and the mantissa containing 52 bits. The term ‘bit’ here represents a binary digit, capable of storing either 1 or 0.
Storage structure of double-precision floating-point numbers | Sourced from the Internet
Evidently, it is the 52-bit mantissa (or significant digits) that dictates decimal precision (i.e., the number of digits following the decimal point). When translated to a decimal number, this corresponds to roughly 15 or 16 significant digits. This elucidates why the scientific community commonly employs π accurate to 15 decimal places.
Reference: NASA uses 15 Digits only.
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