There are sixteen Sutras, (Main rules are called Sutras) and thirteen sub Sutras (corollaries) in Vedic Mathematics. Jagadguru Swami Bharathi Krishna Thirtha (1884-1960) of Sri Govardhana Mutt, Puri , after extensive research rediscovered these sutras. Jagadguru writes (Vedic Mathematics XXXV)

“The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including arithmetic, algebra, geometry- plane and solid, trigonometry- plane and spherical, conics-geometrical and analytical, astronomy, calculus-differential and integral etc.). In fact, there is no part of mathematics, pure or applied, which is beyond their jurisdiction.”

The following are the Sutras

Sutra	Translation
एकाधिकेन पूर्वेन	By one more than the one before
निखिलं नवतश्चरमं दशतः	All from 9 and the last from 10
ऊर्ध्वतिर्यग्भ्याम्	Vertically and Cross-wise
परावर्त्य योजयेत्	Transpose and Apply
शून्यं साम्यसमुच्चये	If the Samuccaya is the same it is zero
आनुरूप्ये शून्यं अन्यत्	If one is in Ratio the other is zero
संकलन व्यवकलनाभ्यां	By Addition and by subtraction
पूरणापूरणाभ्यां	By the completion or Non-completion
चलनकलनाभ्याम्	Differential Calculus
यावदूनं	By the Deficiency
व्यष्टिसमष्टिः	Specific and general
शेषाण्यडेन चरमेण	The Remainders by the last digit
सोपान्त्यदयमन्त्यं	The ultimate and twice the penultimate
एकन्यूनेन पूर्वेन	By one less than the one before
गुणितसमुच्चयः	The Product of the Sum
गुणकसमुच्चयः	All the Multipliers

The Trachtenberg speed system of Basic Mathematics is somewhat similar to vedic Mathematics. Jackaow Trachtenberg, a Russian Born Engineer discovered this method. He derived the whole system while being detained in a Nazi concentration camp.

Let us start examining the vedic sutras one by one. There is no special meaning attached to the order in which the sutras are discussed.

The sutra एकाधिकेन पूर्वेन meaning “By one more than the previous one ” is applied in practice as under:

To find the answer for the mathematical sum 1/19 to an accuracy of 18 digits. Answer = 0.052631578947368421

Step 1 : The number which is previous to the last digit 9 is 1 (1 9) and therefore one more than that is 2

Step 2 : Divide the numerator 1 by 2. The quotient is 0 and reminder is 1. Therefore the first decimal digit of the answer should be 0

Step 3 : Take the quotient 0 and Prefix the reminder 1 to it to get the number 10. Divide that by to get quotient 5 and reminder as 0. The second digit of the answer is therefore 5

Step 4 : The reminder in previous step is 0 and when prefixed to get quotient 5 we get the digit 5. Divide that by 2 to get Quotient 2 and reminder 1. The third digit of the answer is therefore 2

Step 5 : Divide 12 ( prefix reminder 1 to quotient 2) by 2 to get 6 as quotient and reminder 0. The fourth decimal of the answer is 6.

Step 6 : Divide 6 (reminder 0 prefixed to quotient 6 is 6) by 2 to get 3 as quotient and 0 as reminder. Therefore the fifth decimal of the answer is 3.

Repeat the same procedure to get the answer, 0.052631578947368421.

It can be seen that the digits after the first 9 digits are complements of 9.

Therefore one need to workout only the first 9 decimal digits and then subtract each from 9, starting with first decimal digit and write down the result.

That is simple !!!

Exit pupil:
The circular image or beam of light formed by the eyepiece of a telescope. To take full advantage of a scope’s light-gathering capacity, the diameter of an eyepiece exit pupil should be no larger than the 7mm diameter of your eye’s dark-adapted pupil, so that all of the light collected by the telescope enters your eye. (The eyepiece exit pupil diameter is found by dividing the eyepiece focal length by the telescope focal ratio.) Your eye’s ability to dilate declines with increasing age (to a dark-adapted pupil of about 5mm by age 50 or so). For those in this age group, eyepieces with exit pupils larger than their eyes can dilate to simply waste their telescope’s light-gathering capacity, as some of the scope’s light will fall on their iris instead of entering their eye.

Celestron NexStar 5 SE
Aperture: 5 inches (125mm)
Focal length: 1250 mm
Focal ratio: 10
Limiting Stellar Magnitude: 13

With a 25mm eyepiece Magnification: 50x
Highest useful magnification: 295x (4mm eyepiece)
Lowest useful magnification: 17.86x (70mm eyepiece)

Useful links:
Astronomy Terms
Scope Math
New Yorker article on light pollution and the Bortle Scale