1.4.3 Decimals
e.g. \( 648.57807 \), \( 9.6375 \), \( 0.00576 \)
There are two types of decimals
Terminating
The decimal which has a finite number of digits is known as the terminating decimal.
e.g. \( 6.709 \), \( 0.25 \), \( 0.000752 \)
Non-terminating
The decimal which has an infinite number of digits is known as non-terminating decimal.
e.g. \( 3.141592654… \), \( 0.333333333333… \), \( 0.6785325786432896… \)
Non-terminating decimals are further divided into two categories.
Recurring decimals:
The decimal numbers which keep on repeating their values after decimal point are known as recurring decimals. Dot notation is used with recurring decimals.
The dot above the number shows which number recur.
e.g.
\( 0.3^\cdot =0.3333333333333. . . \)
\( 0.5^\cdot7^\cdot=0.575757575757. . . \)
Non-recurring decimals:
A non-recurring decimal is a decimal number that never ends and doesn’t repeat any group of digits.
e.g.
\( \pi = \frac{22}{7} = 3.141592653589793238462643383279 . . .\)
We can convert a recurring decimal into a fraction
Recurring decimal to fraction:
To convert a recurring decimal to fraction
Write the number equal to some variable
Use some repeats of recurring decimal here
Keep multiplying the equation by \( 10 \) until both equations have same recurring decimal
Subtract the lesser equation from bigger
Simplify
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Worked example:
Write the recurring decimal \( 0.2^ \cdot3^ \cdot \) as a fraction in its simplest form.
\( 0.2^ \cdot3^ \cdot \) means \( 0.232323232323… \)
Use a variable to represent this number
\( x=0.2323232323… \)
Multiply both sides by 100 to get another number with the same recurring decimal part.
\( 100x=0.23232323… \)
Subtract the two equations to cancel out the recurring decimal parts.
Divide both sides by \( 99 \)
\( x= \frac{23}{99} \)
\( x= \frac{23}{99} \)
Tip:
To verify your answer,
Type the fraction in your calculator and change it to a decimal.
