explain why some numerical values have an unlimited number of significant digits
Those numbers that are "defined" numbers for conversions; i.e., 12 inches in 1 foot or 3 feet = 1 yard are unlimited. There are 12.0000000etc inches in 1 foot because that's the way it is defined. Another case is if we say there are 100 people in a particular room, then that is 100.000000etc because we can't very well take 100.01 or 99.99 persons.
Perhaps this means irrational numbers, for example pi = 3.14159 etc
or e = 2.718281828 etc
that is my guess anyway.
Well, numbers with an unlimited number of significant digits are quite special, just like unicorns or politicians keeping their promises. They are often found in the realm of theoretical mathematics, where everything is all fun and games, and the rules of the real world don't always apply.
You see, in the world of unlimited significant digits, precision is like a never-ending story. It's as if a magician waves their wand and says, "Abracadabra! Let's keep going and see just how exact we can be!" It's like a party that never ends... or a joke that never gets old.
But beware! The realm of unlimited significant digits is a bit like Alice's Wonderland. As you dig deeper and deeper, you'll stumble upon strange and mind-bending numbers that can't be easily comprehended. It's a bit like trying to figure out why your pet goldfish wants to become an astronaut or why socks always disappear in the dryer. Some things are just not meant to be understood.
So, when you encounter numbers with an unlimited number of significant digits, embrace the enigma, have a good laugh, and remember that in the world of limitless precision, things can get a bit loopy. But hey, who doesn't enjoy a little bit of mathematical madness from time to time?
Some numerical values have an unlimited number of significant digits because they are considered exact or defined values.
In general, significant digits are a way of expressing the accuracy or precision of a measured or calculated value. The number of significant digits in a number indicates how many digits are known with certainty, plus one estimated digit.
For example, consider the number 2.5000. This number has six significant digits because all the zeros between the non-zero digits are known with certainty. However, if we wrote the number as 2.5, it would have only two significant digits, indicating that the zeros are not known with certainty.
In the case of some numerical values, such as mathematical constants or conversion factors, they are considered exact or defined values. These values are not obtained through measurements or calculations and thus do not have any inherent uncertainties. As a result, they can be represented with an unlimited number of significant digits.
For instance, the mathematical constant π (pi) is an irrational number that represents the ratio of a circle's circumference to its diameter. Its decimal representation begins with 3.14159 and continues infinitely without repeating. Since π is an exact value, it can be written with as many significant digits as desired.
Similarly, conversion factors like the speed of light in a vacuum (299,792,458 meters per second) or the Avogadro constant (6.02214076 x 10^23 particles per mole) are defined values. They are established through scientific agreement and are considered to be exact without uncertainties, allowing them to have an unlimited number of significant digits.
In summary, numerical values with an unlimited number of significant digits are typically exact or defined values that do not involve uncertainties arising from measurements or calculations.