mozaMedical by Mozaik Education

The forms of rational numbers

Numbers are similar to humans. Some belong in the same family, like the 'family' of integers or rational numbers. There are famous numbers as well as common ones. Numbers can even disguise themselves.

The introduction of the decimal system was suggested during the French Revolution. At this time, the supporters of the innovations used decimals, while people against the reforms calculated with fractions.

Natural numbers \latex{(\N)}

\latex{\mid}subtraction

\latex{\downarrow}\latex{2-5=-3}

Integers \latex{(\Z)}
\latex{\mid}division
\latex{\downarrow}\latex{2\div 5=\frac{2}{5}}

Rational numbers \latex{(\Q)}

(numbers that can be written as the quotient of two integers)

Example 1

Convert the following fractions to decimals. What type of decimals do you get?

\latex{-\frac{12}{3}}

\latex{3\frac{3}{4}}

\latex{\frac{1}{9}}

Solution

\latex{-\frac{12}{3}=(-12)\div 3=} \latex{\color{blue}-4} → integer;

\latex{3\frac{3}{4}=\frac{15}{4}=15\div 4=} \latex{\color{blue}3.75} → terminating decimal;

\latex{\frac{1}{9}=1\div 9=} \latex{\color{blue}0.\dot{1}} → repeating decimal.

Based on the results in Example 1, you can see that rational numbers can have three decimal forms:

integer;
terminating decimal;
repeating decimal.

Example 2

Convert the following decimals to fraction form.

\latex{0.2}

\latex{-3.25}

\latex{1.\dot{1}}

Solution

\latex{0.2=\frac{2}{10}=\frac{1}{5}}.

\latex{-3.25=-\frac{325}{100} =-\frac{13}{4}}.

Based on the result of the previous exercise, \latex{\frac{1}{9}=0.\dot{1}.} So:

\latex{1.\dot{1}=1+0.\dot{1}=1+\frac{1}{9}=\frac{10}{9}.}

Based on the results, you can see that rational numbers can be expressed in various forms.

For example, \latex{1=\frac{9}{9}=0.\dot{9}} stand for the same rational number.

Example 3

Arrange the following numbers in ascending order and show them on a number line.

\latex{0.3;\;-0.6;\;\frac{3}{8};\;-\frac{35}{50};\;0.\dot{6}.}

Solution

Convert the numbers to decimal form.

\latex{0.3 ;\;-0.6 ;\;\frac{3}{8}=3\div 8=0.375 ;\;-\frac{35}{50}=-\frac{7}{10}=-0.7;\;\;0.\dot{6}=0.666...}

Arranged in ascending order:

\latex{-0.7\lt -0.6\lt 0.3\lt 0.375\lt 0.\dot{6}.}

The original numbers arranged in ascending order:

\latex{-\frac{35}{50}\lt -0.6\lt 0.3\lt \frac{3}{8}\lt 0.\dot{6}.}

Shown on a number line:

\latex{-1}

\latex{-\frac{35}{50} }

\latex{0.3 }

\latex{0}

\latex{-0.6 }

\latex{\frac{3}{8} }

\latex{0.\dot{6} }

\latex{1}

Example 4

Write down two rational numbers that are between \latex{\frac{2}{3}} and \latex{\frac{3}{4}.}

Solution 1

You can find a rational number that is between two other rational numbers in several ways. For example, you can divide the distance between them into three equal parts. The difference between the original fractions (distance between them on a number line):

\latex{\frac{3}{4}-\frac{2}{3}=\frac{9}{12}-\frac{8}{12}=\frac{1}{12} .}

Divide this difference into three equal parts.

\latex{\frac{1}{12}\div 3=\frac{1}{36} .}

\latex{\frac{8}{12}=\frac{24}{36}}

\latex{\frac{9}{12}=\frac{27}{36} }

\latex{\frac{1}{12}}

\latex{+\frac{1}{36}}

\latex{+\frac{2}{36}}

The two numbers marked on the number line:

\latex{\frac{8}{12}+\frac{1}{36}=\frac{24}{36}+\frac{1}{36}=\frac{25}{36}} and \latex{\frac{8}{12}+\frac{2}{36}=\frac{24}{36}+\frac{2}{36}=\frac{26}{36}.}

So:

Solution 2

The average of any two numbers falls between them on a number line.

The average of \latex{\frac{2}{3}} and \latex{\frac{3}{4}}: \latex{ \left(\frac{2}{3}+\frac{3}{4} \right)\div 2=\left(\frac{8}{12}+\frac{9}{12} \right)\div 2=\frac{17}{12}\div 2=\frac{17}{24}.}

Now calculate the average of \latex{\frac{2}{3}} and \latex{\frac{17}{24}.}

\latex{\left(\frac{2}{3}+\frac{17}{24} \right)\div 2=\left(\frac{16}{24}+\frac{17}{24} \right)\div 2=\frac{33}{24}\div 2=\frac{33}{48}=\frac{11}{16}.}

So:

\latex{\frac{2}{3}}

\latex{\frac{11}{16}}

\latex{\frac{17}{24}}

\latex{\frac{3}{4}}

Exercises

{{exercise_number}}. Convert the following numbers to decimal form.

\latex{\frac{1}{4}}

\latex{\frac{2}{5}}

\latex{-\frac{5}{16}}

\latex{2\frac{3}{7}}

\latex{-\frac{20}{8}}

\latex{-5\frac{1}{8}}

\latex{\frac{4}{20}}

\latex{-\frac{5}{8}}

{{exercise_number}}. Convert the following decimals to fractions.

\latex{0.2}

\latex{0.125}

\latex{1.15}

\latex{1.\dot{6}}

\latex{-2.5}

\latex{-0.16}

\latex{2.\dot{9}}

\latex{-3.875}

{{exercise_number}}. Which of the following numbers is the odd one out?

\latex{0.25}

\latex{\frac{1}{4} }

\latex{\frac{2}{9} }

\latex{\frac{10}{40}}

\latex{1.2}

\latex{\frac{12}{6}}

\latex{\frac{120}{10}}

\latex{\frac{240}{20}}

\latex{-\frac{3}{5} }

\latex{- 0.6 }

\latex{\frac{-6}{12} }

\latex{-\frac{18}{30} }

\latex{\frac{1}{6} }

\latex{1.6}

\latex{0.1\dot{6} }

\latex{\frac{8}{42} }

a)

b)

c)

d)

{{exercise_number}}. Decide whether the following statements are true or false.

If the signs of the numerator and denominator are different, the fraction is negative.

There is an integer whose reciprocal is larger than the number itself.

If the numerators and denominators of two fractions are equal, the fractions are also equal.

If the decimal forms of two fractions are equal, their denominators are also equal.

There is only one rational number that is equal to its reciprocal.

{{exercise_number}}. Write a number in the square so that the fraction \latex{-\frac{3}{\Box} } is

at least \latex{ 1 };

at most \latex{-1};

positive and less than \latex{ 1 };

negative and greater than \latex{-1.}

{{exercise_number}}. Which number is greater?

\latex{0.17} or \latex{\frac{1}{6}}

\latex{0.92} or \latex{\frac{12}{13}}

\latex{-\frac{225}{11}} or \latex{- 20.5}

{{exercise_number}}. Arrange the following numbers in ascending order and show them on a number line.

\latex{1.25;\;1\frac{1}{2};\;\frac{15}{6};\;2.5}

\latex{0.2;\;\frac{2}{5};\;\frac{1}{3};\;0.\dot{6}}

\latex{- 0.2;\;-\frac{2}{5};\;-\frac{1}{3};\;-0.\dot{6}}

\latex{- 1.25;\;-1\frac{1}{2};\;-\frac{15}{6};\;-2.5}

\latex{-0.2 ;\;\frac{2}{5};\;-\frac{1}{3};\;-0.\dot{6}}

\latex{1.25;\;-1\frac{1}{2};\;\frac{15}{6};\;-2.5}

{{exercise_number}}. At what places after the decimal point can you find the digit \latex{ 2 } in the decimal form of fraction \latex{\frac{1}{7}?}

{{exercise_number}}. In the decimal form of fraction \latex{\frac{1}{7}} , what number is after the decimal point in the

\latex{ 2 }nd place;

\latex{ 20 }th place;

\latex{ 2,008 }th place?

{{exercise_number}}. What is the \latex{ 100 }th digit after the decimal point in the decimal forms of the following fractions?

\latex{\frac{1}{12}}

\latex{\frac{2}{7}}

\latex{\frac{2}{11}}

\latex{\frac{33}{13}}

{{exercise_number}}. Anna completed the sailing race in an \latex{ hour } and a quarter, Bora in \latex{\frac{5}{6}} of an \latex{ hour }, Kate in \latex{ 85 } \latex{ minutes }, Dora in \latex{ 1.6 } \latex{ hours }, and Ester in \latex{\frac{8}{6}} of an \latex{ hour. }

What was their order at the finish line?

{{exercise_number}}. Write down two rational numbers that are between

\latex{\frac{2}{3}} and \latex{1};

\latex{\frac{2}{5}} and \latex{\frac{3}{5};}

\latex{\frac{1}{10}} and \latex{\frac{1}{5}.}

Mark the position of the numbers on a number line as well.

{{exercise_number}}. Plot the following points in a coordinate system and connect them in the given order. Continue the list with \latex{ 4 } additional points.

\latex{A\left(0;-\frac{1}{4} \right)}

\latex{B\left(\frac{1}{2};0 \right) }

\latex{C(0;1)}

\latex{D(-2;0)}

\latex{E(0;-4)}

\latex{F(8;0)}

{{exercise_number}}. Play a game. Two students take turns taking one of the following cards: 1 2 3 4 5 6 7 8 . When they have picked up all the cards, both of them make a fraction, substituting the squares with their cards \latex{ \frac{\fcolorbox{black}{white}{\textcolor{white}{O}} \, \fcolorbox{black}{white}{\textcolor{white}{O}}} {\fcolorbox{black}{white}{\textcolor{white}{O}} \, \fcolorbox{black}{white}{\textcolor{white}{O}}} }. The player who can make the larger fraction wins the game. When is the value of the fraction the largest? When is it the smallest?

Quiz

What is the smallest and largest possible result of the expression \latex{1\div 2\div 3\div 4\div 5\div 6\div 7\div 8\div 9} if you can put brackets anywhere you want?

Mathematics 7.

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