Absolute change (additive reasoning) & Relative change (multiplicative reasoning)

Measuring Errors

In this lesson we will learn how to quantify errors.

Learning objectives

identify true and relative true errors
identify approximate and relative approximate errors
explain the relationship between the absolute relative approximate error and the number of significant digits
identify significant digits

True and Relative True Errors

A true error ( $E_{t}$ ) is defined as the difference between the true (exact) value and an approximate value. This type of error is only measurable when the true value is available. You might wonder why we would use an approximate value instead of the true value. One example would be when the true value cannot be represented precisely due to the notational system or the limit of the physical storage we use.

true error ( $E_{t}$ ) = true value - approximate value

A true error doesn't signify how important an error is. For instance, a 0.1 pound error is a very small error when measuring a person's weight, but the same error can be disastrous when measuring the dosage of a medicine. Relative true error ( $\epsilon _{t}$ ) is defined as the ratio between the true error and the true value.

relative true error ( $\epsilon _{t}$ )   = true error / true value

Approximate and Relative Approximate Errors

Oftentimes the true value is unknown to us, especially in numerical computing. In this case we will have to quantify errors using approximate values only. When an iterative method is used, we get an approximate value at the end of each iteration. The approximate error ( $E_{a}$ ) is defined as the difference between the present approximate value and the previous approximation (i.e. the change between the iterations).

approximate error ( $E_{a}$ ) = present approximation – previous approximation

Similarly we can calculate the relative approximate error ( $\epsilon _{a}$ ) by dividing the approximate error by the present approximate value.

relative approximate error ( $\epsilon _{a}$ )  = approximate error / present approximation

Relative Approximate Error and Significant Digits

Assume our iterative method yield a better approximation as the iteration goes on. Oftentimes we can set an acceptable tolerance to stop the iteration at when the relative approximate error is small enough. We often set the tolerance in terms of the number of significant digits - the number of digits that carry meaning contributing to its precision. It corresponds to the number of digits in the scientific notation to represent a number's significand or mantissa.

An approximate rule for minimizing the error is as follows: if the absolute relative approximate error is less than or equal to a predefined tolerance (usually in terms of the number of significant digits), then the acceptable error has been reached and no more iterations would be required. Given the absolute relative approximate error, we can derive the least number of digits that are significant using the same equation.

 $|\epsilon _{a}|\leq 0.5\times 10^{2-m}\%$

Absolute Value

The absolute value of a number is its distance from zero (0) on a number line. This action ignores the "+" or "–" sign of a number because distance in mathematics is never negative. The symbol $|x|$ represents the absolute value of $x$ . It is also called modulus $x$ .

Lesson

The absolute value of a number is its distance from zero (0) on a number line. This action ignores the “+” or “–“ sign of a number because distance in mathematics is never negative.

You identify an absolute value of a number by writing the number between two vertical bars referred to as absolute value brackets: |number|.

A helpful way of thinking about absolute value is relating it to a railroad track. If you were to stand on a railroad track, more specifically on any one of the railroad ties and mark that spot as zero, railroad ties to the left would represent negative numbers and railroad ties to the right would represent positive numbers.

The number –7 is 7 units away from zero on the negative side of the railroad track. So, the following is true, |-7| = 7. The number 16 is 16 units away from zero on the positive side of the railroad track. So, |16| = 16. The number 0 is 0 units from zero on the railroad track. So |0| = 0 Therefore, the absolute value of any number is a positive number or zero.

In summary… THE ABSOLUTE VALUE OF A NUMBER

If x is a positive number, then $|x|=x$ . Example: $|5|=5$

If x is zero, then $|x|=0$ . Example: $|0|=0$

If x is a negative number, $|x|=-x$ . Example: $|-6|=-(-6)=6$

You can find the absolute value of expressions as well. When addressed with this you must treat the absolute value brackets as you would parentheses. You need to simplify everything inside the absolute value brackets by performing all the necessary operations by following the order of operations. Your last step once you have a single number inside the absolute value brackets is to take the absolute value. For example, $|-5+1\times 3|=|-5+3|=|-2|=2$ .

You may use the absolute value to find the distance between two numbers on the number line. Let a and b be variables. Then $|a-b|$ is the distance between a and b. For example, if $a=3$ and $b=7$ , then $|3-7|=|-4|=4$ . Because you used the absolute value, the distance is the same if you switch the order of the two numbers; if $a=7$ and $b=3$ , then $|7-3|=|4|=4$ .

Two things to watch out for are an opposite sign and/or an operation outside the absolute value brackets. As stated above, simplify everything inside the absolute value brackets by performing all the necessary operations by following the order of operations. Your last step once you have a single number inside the absolute value brackets is to take the absolute value. Once you have taken the absolute value then perform the other necessary operations by following the order of operations from left to right in the expression. For example, $-|5|=-5$ and $7+|-5+1\times 3|=7+|-5+3|=7+|-2|=7+2=9$ .