Savatingiz boʻsh
The reciprocals of numbers
Example 1
Perform the following multiplications, including two factors. What do you notice?
- \latex{2 \times\frac{1}{2}}
- \latex{\frac{2}{3}\times\frac{3}{2}}
- \latex{-\frac{8}{9}\times\biggr(-\frac{9}{8}\biggr)}
- \latex{0.7\times\frac{10}{7}}
- \latex{2\frac{1}{3}\times\frac{3}{7}}
Solution
- \latex{2 \times\frac{1}{2}=1}
- \latex{\frac{2}{3}\times\frac{3}{2}=1}
- \latex{-\frac{8}{9}\times\biggr(-\frac{9}{8}\biggr)=1}
- \latex{0.7\times\frac{10}{7}=\frac{7}{10}\times\frac{10}{7}=1}
- \latex{2\frac{1}{3}\times\frac{3}{7}=\frac{7}{3}\times\frac{3}{7}=1}
The product in each case is \latex{ 1 }.
If the product of two numbers is \latex{1}, they are called reciprocals.
The reciprocal of \latex{\frac{2}{3}} is \latex{\frac{3}{2}}, and vice versa; the reciprocal of \latex{\frac{3}{2}} is \latex{\frac{2}{3}}.
Similarly,
2 and \latex{\frac{1}{2}}; \latex{-\frac{8}{9}} and \latex{-\frac{9}{8}}; \latex{0.7} and \latex{\frac{10}{7}} are reciprocals.
Example 2
Write down the reciprocals of the following numbers.
- \latex{\frac{5}{6} }
- \latex{-\frac{4}{3}}
- \latex{-2\frac{4}{5}}
- \latex{-1}
- \latex{0.9}
- \latex{0}
Solution
In each case, you must find the number you have to multiply the given number by to get \latex{ 1 }.
- The reciprocal of \latex{\frac{5}{6}} is \latex{\frac{6}{5}}, because \latex{\frac{5}{6}\times\frac{6}{5}=1.}
- The reciprocal of \latex{-\frac{4}{3}} is \latex{-\frac{3}{4}}, because \latex{-\frac{4}{3}\times\biggr(-\frac{3}{4}\biggr)=1;}
- The reciprocal of \latex{2\frac{4}{5}} is \latex{\frac{5}{14}}, because \latex{2\frac{4}{5}\times\frac{5}{14}=\frac{14}{5}\times\frac{5}{14}=1}
- The reciprocal of \latex{-1} is \latex{-1}, because \latex{(-1)\times(-1)=1;}
- The reciprocal of \latex{0.9} is \latex{\frac{10}{9}}, because \latex{0.9\times\frac{10}{9}=\frac{9}{10}\times\frac{10}{9}=1;}
- \latex{ 0 } has no reciprocal, because there is no number which is \latex{ 1 } when multiplied by \latex{ 0 }.
\latex{\frac{\overset{1}{\cancel5}}{\underset{1}{\bcancel{6}} }\times\frac{\overset{1}{\bcancel6}}{\underset{1}{\cancel{5}} }=1}
\latex{-\frac{\overset{1}{\cancel4}}{\underset{1}{\bcancel3}}\times\biggr(-\frac{\overset{1}{\bcancel3}}{\underset{1}{\cancel4}}\biggr)=1}
\latex{\frac{\overset{1}{\cancel{14}}}{\underset{1}{\bcancel{5}}}\times\frac{\overset{1}{\bcancel{5}}}{\underset{1}{\cancel{14}}}=1}
\latex{\frac{\overset{1}{\cancel{9}}}{\underset{1}{\bcancel{10}}}\times\frac{\overset{1}{\bcancel{10}}}{\underset{1}{\cancel{9}}}=1}
Note that
\latex{\left|\frac{5}{6}\right|\lt1} and \latex{\left|\frac{6}{5}\right|\gt1};
\latex{\left|-\frac{4}{3}\right|\gt1} and \latex{\left|-\frac{3}{4}\right|\lt1};
\latex{\left|2\frac{4}{5}\right|\gt1} and \latex{\left|\frac{5}{14}\right|\lt1};
\latex{\left|-1\right|=1} and \latex{\left|-1\right|=1}.
The reciprocal of numbers with an absolute value smaller than \latex{ 1 } is a number with an absolute value greater than \latex{ 1. } This works the other way around as well.
\latex{0} has no reciprocal.
The reciprocal of \latex{1} is \latex{1}, and the reciprocal of \latex{-1} is \latex{-1.}
Exercises
{{exercise_number}}. Complete the multiplications. What is the relationship between the factors?
- \latex{\frac{1}{7}\times7}
- \latex{\frac{4}{9}\times\frac{9}{4}}
- \latex{(-3)\times(-\frac{1}{3})}
- \latex{\biggr(-3\frac{1}{3}\biggr)\times(-0.3)}
{{exercise_number}}. What are the reciprocals of the following numbers?
- \latex{1\frac{1}{2}}
- \latex{\frac{4}{8}}
- \latex{2}
- \latex{-\frac{2}{7}}
- \latex{\frac{2}{7}}
- \latex{-1}
- \latex{0}
- \latex{1.2}
{{exercise_number}}. What numbers should be written instead of the letters to make the equations true?
- \latex{\frac{5}{8}\times a=1}
- \latex{\frac{9}{7}\times b=1}
- \latex{3\times c =1}
- \latex{d\times\frac{1}{5}=1}
- \latex{e\times1\frac{1}{3}=1}
- \latex{f\times\biggr(-\frac{11}{13}\biggr)=1}
- \latex{\frac{501}{105} \times g=1}
- \latex{(-1)\times h=1}
{{exercise_number}}. A number was multiplied by \latex{\frac{2}{3}}. What number should the product be multiplied by to get the original number?
{{exercise_number}}. Decide whether the following statements are true or false.
- A number and its reciprocal have the same signs.
- There is a number that does not have a reciprocal.
- There is a number whose reciprocal is itself.
- There is no integer whose reciprocal is an integer.
- There is a number greater than \latex{ 1 } whose reciprocal is greater than \latex{ 1. }
{{exercise_number}}. Pair the numbers below with their reciprocals.
\latex{ 0 }
\latex{ 10 }
\latex{ 0.1 }
\latex{ -1.5 }
\latex{ 1.8 }
\latex{ 3.5 }
\latex{ \frac{7}{8} }
\latex{ \frac{9}{5} }
\latex{ -3\frac{1}{2} }
\latex{ \frac{0}{5} }
\latex{ -\frac{2}{3} }
\latex{ \frac{5}{9} }
\latex{- \frac{2}{7} }
\latex{1 \frac{4}{5} }
\latex{ \frac{8}{7} }
\latex{ \frac{3}{2} }
Q u i z
What is the reciprocal of the reciprocal of the reciprocal of the reciprocal of a number?
