# Numbers representation, tolerance and precision

On this page, you will find out how integers and decimal numbers are treated in CalcMe. You can also find a detailed explanation of Precision and Tolerance. You can choose between different options for the Notation used when displaying numbers. Finally, you can specify answers with repeated decimals or a concrete number of decimal places or significant figures.

## Integers and decimal numbers

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: $\frac{1}{2}$, $\pi$, $\sqrt{2}$, ...

### Tip

Decimal numbers are shown using the Precision setting and are compared using the Tolerance setting. What is understood as a decimal mark depends on defining the Separator symbols in the Input options section.

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 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

Rounded to significant figures

Rounded to decimal places

6

12.345

12.345000

5

12.345

12.34500

4

12.35

12.3450

3

12.3

12.345

2

12

12.35

1

12

12.3

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.

## Tolerance

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. ### Caution

The tolerance options only apply to decimal numbers! For instance, if the question's answer is 10, we may set 10.0 as the correct answer so that the validation system takes into account the defined tolerance.

The student answer must be exactly equal to the correct answer. It corresponds to what we could call zero-tolerance.

For instance, if the correct answer is $12.345$

Acceptance

$12.345$ $12.3450$ $12.3450000001$ $\frac{12345}{1000}$ ### Percent error

The student answer has an $n%$ error margin with respect to the correct answer. An answer will be marked as correct if it satisfies

$\frac{|ca-sa|}{|ca|}\le n·0.01$

For instance, if the correct answer is $12.345$

Percent error

Interval of acceptance

10

$\left[11.1105,13.5795\right]$

1

$\left[12.22155,12.46845\right]$

0.1

$\left[12.332655,12.357345\right]$

0.01

$\left[12.3437655,12.3462345\right]$

### Absolute error

The student answer has a margin of error of $n$. An answer will be marked as correct if it satisfies

$|ca-sa|\le n$

For instance, if the correct answer is $12.345$

Absolute error

Interval of acceptance

1

$\left[11.345,13.345\right]$

0.1

$\left[12.245,12.445\right]$

0.01

$\left[12.335,12.355\right]$

0.001

$\left[12.344,12.346\right]$

### 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

$\text{trunc}\left(ca·{10}^{m}\right)=\text{trunc}\left(sa·{10}^{m}\right)$ 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 $19.586$

Significant figures

20.01

19.6

19.59

19.58

1    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

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

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

Decimal places

19.6

19.59

19.587

19.586

1    2    3    ## Notation

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. ### Auto

Auto is the default notation if you use significant figures in the Precision setting. Auto notation chooses the best format for each value type: most numbers are shown without changes, but very big or small ones are shown in scientific notation. Decimal trailing zeros are removed too.

If you use $p$ significant digits as precision, the floating number $x$ will be shown in scientific notation when $\left|x\right|>{10}^{p}$ or $\left|x\right|<{10}^{-p}$. Note that this does not apply to integer numbers.

### Decimal

Decimal notation is like Auto but never shows scientific notation. Note that this notation can produce numbers with lots of digits. For example, you will see $1.·{10}^{99}$ in the Algorithm, but you will actually get a long number of hundred digits.

This is the only option when working with decimal places as precision.

### 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 $6.023·{10}^{23}$. 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

## Concrete number of decimals as assertion

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. ## Repeating decimals

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 $0.3\stackrel{⏜}{33}=0.\stackrel{⏜}{3}$, and of course, that $0.\stackrel{⏜}{9}=1$. 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. 