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Subsection 1.1.3 Finding reciprocals of real numbers using division
Activity 1.1.4 .
In groups of
\(5\text{,}\) write down numbers from
\(2\) to
\(15\) in ascendinding order
\((2,...,15)\text{.}\) Label this as the first list.
Rearrange the numbers from
\(15\) to
\(2\) in descending order
\((15, 14, 13, ..., 2)\text{.}\) Label this as the Second List.
In turns, each group member to create a fraction by:
Using the first number from the First List as the numerator.
Using the first number from the Second List as the denominator.
Find the reciprocals of the fractions you have formed using division.
For example: Fractions formed are
\(\frac{2}{15},\frac{3}{14} , \frac{5}{12}, \frac{10}{7}, \frac{7}{10}\) then the reciprocal of
\(\frac{2}{15}\) using division is
\(15 \div 2\)
From the whole numbers
\(2\) to
\(15\) you wrote down, pick any
\(3\) numbers and find its reciprocal using division.
Discuss your work with fellow learners.
\(\textbf{Key Takeaway}\)
The reciprocal of a number is the number that, when multiplied by the original number, gives a product of
\(1\text{.}\)
In other words the reciprocal of a real number
\(x\) is
\(\frac{1}{x}\text{,}\) except when
\(x=0\) since division by zero is undefined.
1. The reciprocal of
\(2\) is
\(\frac{1}{2}\text{.}\) Using division, the reciprocal of
\(2\) can be found as
\(1 \div 2\)
2. To find the reciprocal of
\(0.25\) using division, you divide
\(1\) by
\(0.25\) i.e
\(1 \div 0.25\text{.}\)
3. The reciprocal of
\(\frac{3}{5}\text{,}\) using division, is
\(1 \div \frac{3}{5}\text{.}\)
\(\textbf{To find the reciprocal of a real number using division, follow these steps:}\)
Understand the Reciprocal: The reciprocal of a real number
\(x\) is
\(\frac{1}{x}\text{.}\) This means that when you multiply a number by its reciprocal, the result is always
\(1\text{:}\)
\begin{align*}
x \times \frac{1}{x}=\amp 1
\end{align*}
Next, use division to find the reciprocal of a number.
Example 1: The reciprocal of
\(5 = \frac{1}{5}\)
Using division to find reciprocal of
\(5\) we have:
\(1 \div 5\text{.}\)
\(0.2\)
\(5\) \(10\)
-
\(10\)
\(0\)
Example 2: The reciprocal of
\(-3\) is
\(1 \div -3 \text{.}\)
Using division to find reciprocal we have:
\(0.333\)
\(3\) \(10\)
-
\(9\)
\(10\)
-
\(9\)
\(10\)
-
\(9\)
\(1\)
Hence the reciprocal of
\(-3\) using division is
\(= -0.333...\) also written as
\(-0.\dot{3}\) to mean
\(3\) is recurring.
Example 1.1.11 .
Find the reciprocal of the following numbers using division.
Solution .
The reciprocal of
\(256\) according to definition is
\(\frac{1}{256}\) .
\(0.003906\)
\(256\) \(1000\)
\(-768\)
\(2320\)
-
\(2304\)
\(1600\)
-
\(1536\)
\(64\)
Hence, the reciprocal of
\(256\) using division is
\(1\div 256 = 0.003906\)
The reciprocal of
\(4.2\) according to definition is
\(\frac{1}{4.2}\text{.}\)
We can rerwrite \(\frac{1}{4.2}\) as \(\frac{1}{4.2}\times \frac{10}{10} \)
\begin{align*}
=\amp \frac{10}{42} \\
=\amp \frac{5}{21}
\end{align*}
Dividing
\(5\) by
\(21\) we have:
\(0.238\)
\(21\) \(50\)
-
\(42\)
\(80\)
-
\(63\)
\(170\)
-
\(168\)
\(2\)
Therefore, using division, the reciprocal of
\(4.2\) is
\(15\div 21 = 0.238\)
Checkpoint 1.1.12 . Using Division to Find Reciprocals.
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Checkpoint 1.1.13 . Finding Value of a Number Given its Reciprocal.
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Checkpoint 1.1.14 . Finding Unknown Number Given Sum with Reciprocal.
Load the question by clicking the button below.
Using division, find the reciprocal of following numbers:
\(\displaystyle \frac{2}{7}\)
\(\displaystyle -\frac{5}{8}\)
If the reciprocal of
\(x\) is
\(\frac{1}{6}\text{,}\) find the value of
\(x\text{.}\)
Movin, a Grade
\(10\) learner covers
\(\frac{1}{x}\) km from home to school. He realizes that the distance he covers each day from home to school is
\(0.2\) km. Find the value of
\(x\text{.}\)
A cyclist covers a distance of
\(12 \) km in
\(1\) hour. Using division to find reciprocals, determine the time taken to cover
\(1\) km at the same speed.
