1A - Organising numbers
1A - Content video: Organising numbers
This video covers the theory and several examples relating to how we organise numbers.
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Summary of notation:
- ∈ is an element of...
- ∉ is not an element of...
- ⊂ is a subset of...
- ⊆ is not a subset of...
- ∪ the union between...
- ∩ the intersection between...
- \(A'\) the complementary set of \(A\); that is, all element in the universal set that are not in \(A\).
- \ except or excluding...
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The real number system:
In mathematics, a set is a collection of objects which are referred to as element of the set. In many cases, these objects are numbers. The real number system, \(\mathbb{R}\), is the set of numbers that you would be most familiar with. The real number system contains both rational and irrational numbers. The real number system, \(\mathbb{R}\), contains the following sets of numbers
In mathematics, a set is a collection of objects which are referred to as element of the set. In many cases, these objects are numbers. The real number system, \(\mathbb{R}\), is the set of numbers that you would be most familiar with. The real number system contains both rational and irrational numbers. The real number system, \(\mathbb{R}\), contains the following sets of numbers
- Rational numbers (\(\mathbb{Q}\)): any number that can be expressed as a fraction of two integers (\(\mathbb{Z}\)).
- Irrational numbers (\(\mathbb{Q}'\)): any number that cannot be expressed as a fraction of two integers (\(\mathbb{Z}\)).
- Integers: \(\mathbb{Z}=\left \{ ...,-3,-2,-1,0,1,2,3,... \right \}\)
- Whole numbers: \(\mathbb{W}=\left \{ 0,1,2,3,4,5,... \right \}\)
- Natural numbers: \(\mathbb{N}=\left \{ 1,2,3,4,5,... \right \}\)
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■ Recall that zero is neither positive or negative; therefore, \(\mathbb{R}^+\) and \(\mathbb{R}^-\) do not include zero. To include: \(\mathbb{R}^+ \cup \left \{ 0 \right \}\) |
Review of inequalities:
There are four inequality signs (\(<,\leq,>,\geq\)):
There are four inequality signs (\(<,\leq,>,\geq\)):
On the number lines above:
- An open circle (○) is used for \(<\) or \(>\) where the point is not included.
- An closed circle (●) is used for \(\leq\) or \(\geq\) where the point is included.
Set notation:
We often need to create new sets of numbers; we can do this using set notation.
We often need to create new sets of numbers; we can do this using set notation.
- \(\left \{ x:a<x<b \right \}\) "\(x\) is any number greater than \(a\) and less than \(b\)"
- \(\left \{ x:a\leq x\leq b \right \}\) "\(x\) is any number greater than or equal to \(a\) and less than or equal to \(b\)"
- Using “\(+\)” and “\(-\)” will restrict the set of positive or negative numbers (not including zero).
- Using “\(\cup\)” (union) will include another element, or set of elements in the set.
- Using “\” will exclude another element, or set of elements from the set.
Interval notation:
Interval notation is very useful to create continuous sets of numbers.
Interval notation is very useful to create continuous sets of numbers.
- \(x\in (a,b)\) "\(x\) is any number greater than \(a\) and less than \(b\)"
- \(x\in [a,b]\) "\(x\) is any number greater than or equal to \(a\) and less than or equal to \(b\)"
- (round brackets) are used to exclude a number, similar to \(<\) and \(>\) symbols.
- [square brackets] are used to include a number, similar to \(\leq\) and \(\geq\) symbols.
1A - Example 1: Using set and interval notation to organise numbers
Consider the interval \(A\), where \(x\in (-\infty,6]\), and the interval \(B\), where \(x\in (-4,8]\).
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1A - Example 1: Video solution
1A - Example 1: Practice
Question 1: ABC Question 2: ABC 1A - Example 1: Solutions
Question 1: ABC Question 2: ABC |