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I'm trying to help my daughter with the concept of lower and upper bounds of a number when given a specific accuracy, but I'm starting to realise how many holes there are in my own knowledge!

For a specific number of decimal places this is straightforward:

$x=1.23$ to 2 decimal places.

Therefore: $1.225 \le x \lt 1.235$

Equally, for numbers given to a number significant figures it is much the same. Eg:

$x=99$ to 2 significant figures

Therefore: $98.5 \le x \lt 99.5$

Or $x=110$ to 2 significant figures

Therefore: $105 \le x \lt 115$

However, this is what I can't figure out:

$x=100$ to 2 significant figures.

Since anything between 100 and 105 rounded to 2 significant figures would be 100, logically the upper bound is 105.

On the other hand, the lower bound cannot correspondingly be 95 because 95 to 2 significant figures is 95!

Therefore, should it be that in this case:

$99.5 \le x \lt 105$? If so, that rather lacks the symmetry I had expected.

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Your analysis is spot on. I suspect most people who talk of significant figures have not thought about this corner case at all. Another asymmetry is that $99$ to two significant figures is $\pm 0.5\%$ but $110$ to two figures is $\pm 5\%$

Most error analysis that I have done or seen has been rather informal. One doesn't worry too much about whether the error bounds are hard limits (like if you measure a length with a ruler marked in mm, the result should be $\pm 0.5$ mm with a uniform distribution) or a standard deviation. If it is a standard deviation, there is no hard limit. Now when you compute with the numbers, we just keep the same number of significant figures if we multiply, use the larger absolute error if we add, and so on. The result is a pretty good of the final error, but it could be outside that. To do a better job, you can use interval arithmetic, keeping track of the maximum and minimum possible. The bounds may well not be symmetric around your calculated value.

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When using significant figures and powers of $10$ ($10,100,1000,$ etc.) the upper bound is found as normal, However the gap to the lower bound is one tenth of the gap to the upper bound.

Hence $x=100$ to $2$ significant figures is $99.5\leq x <105$ and $x=100$ to $1$ significant figure is $95\leq x <150$ and $x=1000$ to $1$ significant figure is $950\leq x <1500$ and $x=1000$ to $2$ significant figures is $995\leq x <1050$

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In order to find the upper and lower bounds for a number which is to be rounded up to some number, the following procedure should be followed:

1) Let the given number be x and it is to be rounded up by a number y. Divide y by 2, i.e. y/2.

2) Now to find upper bound, add given number and y/2, i.e. upper bound = x + y/2.

3) To calculate lower bound, subtract y/2 from given number, i.e. lower bound = x - y/2.

Examples

Have a look at simple examples of upper and lower bounds:

Example 1:

The distance between two cities was approximated as 500 km which is rounded to nearest 100 km. Calculate the lower bound and upper bound.

Solution:

Roundup number = 100 km 100/2= 50

Upper bound = 500 + 50 = 550 km

Lower bound = 500 - 50 = 450 km

Example 2:

The length of a rope is 13 cm to the nearest cm. Calculate upper and lower bounds.

Solution:

Roundup number = 1 cm 1/2= 0.5

Upper bound = 13 + 0.5 = 13.5 cm Lower bound = 13 - 0.5 = 12.5 cm

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  • $\begingroup$ Hi Xaviour. I appreciate you adding your comment, but I think you've missed the point of the question. I know how to calculate lower and upper bounds, my question was about the asymmetry in doing so around certain numbers rounded to significant figures. Ross gave quite a good answer to that - 18 months ago :) $\endgroup$ – Richard Day May 10 '17 at 10:28

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