Topic outline
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In this course you will learn about:
- The real number system.
- Rational and irrational numbers.
- How to convert terminating decimals to rational numbers.
- How to convert recurring decimals to rational numbers.
- Rounding off decimal numbers.
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The number system is made up of real numbers and imaginary numbers. Real numbers \(\mathbb R\) are all the numbers on a number line. The real number line is infinite, which means it continues forever in both directions. Real numbers are infinite.
Real numbers are made up of all rational \(\mathbb Q\) and all irrational \(\mathbb {Q}^|\) numbers.
Beginning with the natural numbers, we expand each set to form a larger set of numbers.
We use the following definitions.
The set of natural numbers \(\mathbb N\) includes the numbers used for counting: {\( {1, 2, 3, ...} \)} .
The set of whole numbers \(\mathbb N_o\) is the set of natural numbers plus zero: {\( {0, 1, 2, 3, ...} \)} .
The set of integers \(\mathbb Z\) adds the negative of the natural numbers to the set of whole numbers
{\( {..., -3, -2, -1 ,0, 1, 2, 3, ...} \)}. It is important to remember that integral values are not fractions.The set of rational numbers \(\mathbb Q\) includes fractions that can be written as \( \frac{a}{b} \) where \( a, b \in \mathbb{Z}, b \neq0 \).
The set of irrational numbers \(\mathbb {Q}^|\) are numbers that cannot be written as a fraction with the numerator and denominator as integers. These fractions give us non-repeating and non-terminating (they do not end) decimals.
Test your understanding of real numbers by trying the next activity.
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Exercise: Classify real numbers
Test your knowledge of real numbers by trying this activity.
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More on real numbers
To learn more about real numbers watch the video below.
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Rational numbers
Rational numbers can be written as fractions. All integers and natural numbers are rational since they can be written with a denominator of 1.
Example
\( \frac{2}{1}, \frac{3}{4}, \frac{5}{2}, 6\frac{1}{3} \) and \( \frac{28}{5} \) are all examples of rational numbers.
Irrational numbers
Irrational numbers cannot be written as fractions with integer numerators and integer denominators.
Example
\( \sqrt{2}, \pi \) and \( - \sqrt{12} \) are examples of irrational numbers.
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How to tell the difference between rational and irrational numbers
The video explains the difference between rational and irrational numbers in more detail.
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Exercise: Rational or irrational?
Check your understanding of rational and irrational numbers by trying the exercise.
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Since all rational numbers can be written as fractions, this means that any rational number can be written in decimal form.
A rational number can be converted to either:
A terminating decimal. For example, \( \frac{3}{4} = 0.75 \).
OR
A recurring decimal. For example, \(\frac{4}{11}\)=\(0.\overline{36}\)
We use a line drawn over the repeating block of numbers to show that those numbers are repeated.
NOTE: You can write any rational number as a decimal number but not all decimal numbers are rational numbers.
To check if you understand the difference between terminating and recurring decimals try the next exercise.-
Exercise: What type of decimal?
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Steps to convert a terminating decimal to a rational number
These are the steps you can use to convert from a terminating decimal to a rational number.
- Rewrite the terminating decimal as a decimal divided by one.
- Multiply both numerator and denominator by \( 10 \) for every number after the decimal point in the numerator. (For example, if there are three numbers after the decimal point, then multiply by \( 1 000 \), if there are four then multiply by \( 10 000 \), and so on.)
- Simplify the fraction if possible.
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To convert a recurring decimal to a fraction, we need to do a bit more work on the fractional part of the decimal number.
The next video uses examples to explain the steps to convert from a recurring decimal to a fraction. When you are sure that you understand the concepts taught, try the quiz after the video.
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Rounding off a decimal number to a given number of decimal places is the quickest way to get an estimate of the number.
Watch the video for some examples on rounding off decimals and then try the quiz on estimating answers.