There are faster ways to do exponentiation that require far fewer multiplies. In practice It's fun and educational to try writing these functions yourself, but in practice you will want to use an existing library that has been optimized and verified.
I think a part of the answer is in the question itself : To store these expressions, you can store the base or mantissa , and exponent separately, like scientific notation goes. Extending to that, you cannot possibly evaluate the expression completely and store such large numbers, although, you can theoretically predict certain properties of the consequent expression. I will take you through each of the properties you talked about:.
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How do computers evaluate huge numbers? Ask Question. Asked 6 years, 1 month ago. Active 5 years, 8 months ago. Viewed 3k times. Sam Sam 1 1 gold badge 14 14 silver badges 33 33 bronze badges. Dmitry Bychenko Dmitry Bychenko k 15 15 gold badges silver badges bronze badges.
Computer number format
Excellent answer! How is that explained? I've edited the post. So does this mean that there is no fancy shortcut to calculating massive exponentials? I assume Wolfram Alpha must have a huge distributed system used just for calculating large numbers? Sam Yes and no. There are some shortcuts in some specific cases, but in general you have to do full multiplications, and Wolfram Alpha probably has large data centers for answering queries. A single computer can do this in reasonable time, the "huge distributed" part only enters the picture because it's thousands of people running such queries concurrently.
A shortcut would be, for example, something that takes OP's expression and gives you the last few digits without calculating the millions of other digits. Hardly any measured quantity is known to anywhere near that much precision. For example, the constant in Newton's Law of Gravity is only known to four significant figures. The charge of an electron is known to 11 significant figures, much more precision than Newton's gravitational constant, but still less than a floating point number.
So when are 16 figures not enough? One problem area is subtraction. The other elementary operations -- addition, multiplication, division -- are very accurate. As long as you don't overflow or underflow, these operations often produce results that are correct to the last bit. But subtraction can be anywhere from exact to completely inaccurate. If two numbers agree to n figures, you can lose up to n figures of precision in their subtraction.
This problem can show up unexpectedly in the middle of other calculations. Number precision is a funny thing; did you know that an infinitely repeating sequence of 0. In mathematics, the repeating decimal 0. In other words: the notations 0. This equality has long been accepted by professional mathematicians and taught in textbooks. Proofs have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. Computers are awesome , yes, but they aren't infinite..
So any prospects of storing any infinitely repeating number on them are dim at best. The best we can do is work with approximations at varying levels of precision that are "good enough", where "good enough" depends on what you're doing, and how you're doing it. Octal number system has eight digits — 0, 1, 2, 3, 4, 5, 6 and 7. Decimal equivalent of any octal number is sum of product of each digit with its positional value.
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Octal number system has 16 symbols — 0 to 9 and A to F where A is equal to 10, B is equal to 11 and so on till F. Decimal equivalent of any hexadecimal number is sum of product of each digit with its positional value. The following table depicts the relationship between decimal, binary, octal and hexadecimal number systems. Besides numerical data, computer must be able to handle alphabets, punctuation marks, mathematical operators, special symbols, etc. The complete set of characters or symbols are called alphanumeric codes.
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Now a computer understands only numeric values, whatever the number system used. So all characters must have a numeric equivalent called the alphanumeric code. ASCII is a 7-bit code that has 27 possible codes. IISCI is mostly used by government departments and before it could catch on, a new universal encoding standard called Unicode was introduced.
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