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Learning intentions:
In this section we will examine:
- What is a rational number?
- How do we perform basic operations with rational numbers?
- How can fractions be simplified?
What is a rational number?
A rational number (ℚ) is any number that can be expressed as a fraction of two integers. Rational numbers include the following sets:
- Natural numbers, or counting numbers, ℕ = {1,2,3,4, ...}
- Whole numbers, 𝕎 = {0,1,2,3,4, ...}
- Integers, ℤ = {... ,-4,-3,-2,-1,0,1,2,3,4, ...}
- And any other number that can be expressed as a fraction of two integers (a and b):
In the fraction above, a is the numerator and b is the denominator. In fact, any fraction can be considered as a numerator divided by the denominator in the following general format:
Operations involving rational numbers
Addition and subtraction of fractions
When we perform addition and subtraction on fractions, they must have the same denominator!
When the denominators are the same we can directly add (or subtract) the numerator to find the solution.
When the denominators are not the same we cannot directly add (or subtract)the numerator to find the solution. Instead we must manipulate the equation to get a common denominator.
This cannot be calculated directly!
Please Note: We do not perform any operations on the denominator (unless we can simplify).
Multiplication and division of fraction
When we perform multiplication and division on fractions, they do not require the same denominators!
When the denominators are the same we can directly multiply (or divide) the numerator and denominator to find the solution.
When the denominators are not the same we can still directly multiply (or divide) the numerator and denominator to find the solution.
How to divide fractions:
What happens when we have the division of fractions? For example:
Instead of dealing with the division of fractions (which can get nasty) we can instead multiply the first fraction by the reciprocal (a "flipped fraction") of the second - or vice versa. That is, the expression below is equivalent to the one written above:
The reciprocal is found by raising the object to the power of negative 1:
Simplifying fractions
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For a fraction:
Success criteria:
You will be successful if you can:
- Recall the definition of a rational number and identify rational numbers.
- Perform basic operations (addition, subtraction, multiplication and division) with rational numbers.
- Simplify fractions by removing the highest common factor from the numerator and denominator.
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