Numbers representation, tolerance and precision

CalcMe does calculations in exact mode and only falls to the approximate mode when any decimal number is involved. A decimal number is a number that has a decimal mark. A number without a decimal mark is an integer. Examples of exact expressions that are not integer numbers are: $\frac12$, $\pi$, $\sqrt2$, ...

You can convert an exact expression to approximate by making a simple operation with a decimal number, like multiplying by 1.0.

Beware that, despite the correct answer being an integer, the student can always answer a decimal number close enough, and it will be graded as correct. If you want to force the student to answer an integer, you better set the Symbolic answer validation option.

Also note some CalcMe commands sometimes return an exact number and sometimes an approximate one. For example: atan(), integrate(), solve(),... When programming the algorithm in Define random variables and functions, keep this in mind.

Precision is the number of significant figures or decimal places of variables shown after calculations. Precision affects the decimal numbers shown in wordings, feedback, and so. When rounding, the rule used for tie-breaking is half-up.

Consider the number 12.345

Precision is not the allowed deviation between students and correct answers; that is Tolerance. But precision can harm the comparison if set improperly. Precision affects the correct answer, how it's shown to the student, and also the actual value. The student answer will be compared with the correct answer that has been rounded using precision. If the precision number is smaller than the tolerance number, the correct answer will be rounded too much, probably outside the tolerance range. So remember to set the precision number always more significant than the tolerance number.

The Tolerance number is used to compare the student answer and the Correct answer. Denote by $sa$ the student's response, $ca$ the correct answer, and $n$ the tolerance.

Significant figures

Check that the first $n$ significant figures of student and correct answers are identical. An answer will be marked as correct if it satisfies

$\text{trunc}(ca·{10}^m)=\text{trunc}(sa·{10}^m)$ where $m$ is such that ${10}^{n-1}\le ca·{10}^m<{10}^n$, and $\text{trunc}$ is the truncation function.

For instance, if the correct answer is $19.586$

Student Answer
Decimal places	20.01	19.6	19.59	19.58
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2
3
4

Decimal places

Check that the first $n$ decimal places of student and correct answers are identical. An answer will be marked as correct if it satisfies

$\text{trunc}(ca·{10}^n)=\text{trunc}(sa·{10}^n)$

where $\text{trunc}$ is the truncation function. For instance, if the correct answer is $19.586$

Student Answer
Decimal places	19.6	19.59	19.587	19.586
1
2
3

These notations apply only to decimal numbers. Numbers and expressions without a decimal point are exact, so these notations don't apply to them. You can always convert an exact expression to decimal by multiplying by 1.0 , for instance.

Scientific notation

Scientific notation always shows numbers in normalized scientific notation, also called exponential notation. The precision number here means significant digits of the mantissa to show.

Consider $6.023·{10}^{23}$ as the correct answer. By default, the only accepted notation is $6.023 \cdot 10^{23}$. If you want to allow students to answer in other formats, you must check the Computer scientific notation option in the Input options section. This way $6.023E23$ and $6.023e23$ is also accepted.

Note that the following answers are not accepted:

• 12.3e+2: because there are too many digits before the decimal point

• 1.0 e+2: because there is an extra space in the middle

As with any other assertion, we can choose if the student answer has a concrete number of significant figures or decimal places.

Imagine that we want the student to ask with 1 number after the decimal point. Suppose that the correct answer is 21.5, then.

As you may have noticed, both significant figures and decimal places comparison methods use the truncation function. Conversely, you may be interested in replicating the same behaviour with the rounding function. Below you can find two alternative procedures to do so.

Using absolute error tolerance and a concrete number of decimals assertion

The first option will require combining two assertions we have previously analyzed: absolute error and a concrete number of decimals. If we want to check that the student's answer exactly has $n$ decimal places, and these are identical to the correct answer first $n$ decimal places after rounding, we need to set the error margin to $0.5·{10}^{-n}$ and define the precision to $n$ decimal places.

For instance, if the correct answer is $19.586$

Student Answer
Decimal places	19.6	19.59	19.58	19.587	19.586
1
2
3
4

Using a grading function

The second option will require some programming in CalcMe through a grading function. If we want to check that the first $n$ decimal places of the student and correct answers are identical after rounding, we need to define the following user-defined routine (for $n=2$).

Tip

You can see more about grading functions here.

For instance, if the correct answer is $19.586$

Student Answer
Decimal places	19.6	19.59	19.58	19.587	19.586
1
2
3
4

WirisQuizzes understands decimal numbers with repeating decimals. Repeating decimals is a notation used to represent decimal numbers with periodic expansion, rational numbers. There are three symbols available to mark repeating decimals; you can use any of them. The system is smart enough to know that $0.3\overset\frown{33}=0.\overset\frown3$, and of course, that $0.\overset\frown9=1$.

Thus, students can answer numbers with repeating decimals at any time. The system will recognize it like any other decimal number. If the answer fits the Validation criteria (Tolerance, Mathematically equal, ...), it will be graded as Correct.

Conversely, teachers can also set for a Correct answer an endless number with repeating decimals. The student can answer any number, with or without repeating decimals, even a fraction, and it can be graded as correct, depending on the Validation criteria.

Finally, remark there is no Additional property to validate if a student answer has or has not repeated decimals nor the possibility to generate variables with repeating decimals. But you can use Literally equal to check if the student answer and the correct answer have the same form or if they both have or have not repeated decimals.