The Glade 4.0

Off the bat, I am bad at math, so if these seems dumb feel free to smack the idea down.

So the number 1 due to rounding and being verifiable is actually between 0.5 and 1.49(repeating)?

So 1+1 can be 0.5 +0.5 to 1.49 + 1.49? so the actual answer to 1+1 is any where from 1.0 to 2.98?

So 1+1+1 can now be 1.5 to 4.47? and 1+1+1+1 now has a range of 2.0 to 5.96?

Am I figuring this out correctly? and yes I know if I go any further I run into the bullet skit from Clue.

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"Life isn't divided into genres. It's a horrifying, romantic, tragic, comical, science-fiction cowboy detective novel. You know, with a bit of pornography if you're lucky."
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1.49999... Would actually be equivalent to 1.5.
But in any case, for scientific/engineering purposes, you track error/precision separately. So:

1 +/-0.5 * 4 = 4 +/-2.0

Also, bear in mind that there are differing standards for rounding, depending on application.

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Rounding is not addition. It is basically half of one away from a whole number. If you are rounding from a whole number, you use that basis. If you are adding the possible value of a group of rounded numbers, then you can add the possibilities, but the likelihood in averaging would suggest that the end result is going to be close to the standard 0.5 up or down.

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"A mind needs books like a sword needs a whetstone." -- Tyrion Lannister, A Game of Thrones

You have just stumbled upon the reason why we always carry measurements out further than a single digit. Yes, a measured value of 100 is anywhere between 50 and 150. If you use several such measurements, then you arrive at a number between 2 and 6 in your last example. This is the reason why scientific notation was developed, and the ease of communicating one's degree of uncertainty is why it sees continued use, despite being less useful than engineering notation. (The difference between the two being that engineers only use powers of 10 that correspond to divisions of one thousand).

Propagation of error and uncertainty is a topic of great interest in the scientists, and there are a number of ways to communicate your degree of uncertainty to others. We typically use introductory chemistry courses as a platform for beating the notion of significant figures into people's heads. Parts manufacturers can use another method, the one that springs most readily to mind is resistor color codes. A first semester chemistry student would interpret red-purple-red-gold as 27x10^2, or 2700 ohms. Having two significant figures means the actual value is anywhere between 2650 and 2750. In reality, the gold band at the end means the error is 5%, so you're as low as 2565 and as high as 2835.

You may notice that I have not used repeating 9s at any point. Due to the way decimal notation works, 0.9 repeating is not a distinct number from 1. This can be shown with the fraction 1/9 and its conversion to decimal form. If I divide 1 by 9, I get 0.1 repeating. If I multiply 1/9 by 9, I get 1. If I multiply 0.1 repeating by 9, I must also get 1. Since 0.1 repeating * 9 yields 0.9 repeating, 0.9 repeating has the same relation to the number 1 as colour has to color. This exercise points out the shortcomings of our decimal number system.

Error and uncertainty is also introduced inherently by certain numbers. Take, for example, the number π, which people commonly know to be 3.14. It is actually more accurate to use 22/7 as opposed to the commonly memorized decimal. The fraction 355/113 is more accurate than 3.1416 (the "good" decimal approximation of pi) by over a full order of magnitude. Arithmetic operations on numbers are problematic as well. If I have to take the square root of a number like 2, I introduce error depending on where I round off the decimal. Much of the error accumulated in calculations can be eliminated by working with numbers in their exact forms, but since fractions and radical signs scare the **** out of algebra students, you're going to see fewer people working with exact forms the further left you go on this scale.

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Buckle your pants or they might fall down.

This is why math sucks.

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The number 1 as a scientific quantity is nearly useless unless you know more about it. Such as the instrument that acquired it, what you're measuring, the accuracy of past measurements, etc.

Using 1 to count the number of apples in a basket means it is discrete and can't be 1.3. It's different than measuring the temperature.

Saying that 1 can range from 0.5 to 1.5 depends on the scenario. If it does, chances are there's a probability distribution centered around 1.0, or something. It all depends on what you're looking at. Maybe you really messed something up and 1 means 759.

Perhaps the possibilities could be 0.01, 0.1, 0, 1, 100, 1000, or 10000. If you measure 1, maybe you meant either 0 or 10 when you were twisting the knob. An example of this could be telescopic magnification.



Short answer: Depends on the problem.

I'd never though of that before. quite intriguing in and of itsself.

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I prefer to think of them as "Fighting evil in another dimension"

Math is always true.

Rounding is for engineers. This is why bridges fall down.