Order of operations and simplifying of brackets.
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Concept |
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Numbers |
Introducing numbers: from natural numbers to real numbers. |
Order of operations:
Brackets first (from inside to out)
Exponents
Division and Multiplication (from left to right)
Addition and Subtraction (from left to right)
There are some cases where brackets are implied by the notation:
The fraction line denotes an operation that requires brackets,
e.g.
An exponent itself is always in brackets, e.g.
\(\displaystyle{a^{x \pm y} = a^{(x \pm
y)}}\)
\(\displaystyle{a^{x \cdot y} = a^{(x
\cdot y)}}\)
\(\displaystyle{a^{\frac{x}{y}} =
a^{(\frac{x}{y})}}\)
Brackets
To override operator precedence, brackets must be used:
\[\begin{aligned}
1+3\cdot 5 &= 1+ 15 = 16 \\
(1+3)\cdot 5 &= 4 \cdot 5 = 20
\end{aligned}\]
For simplifying a minus in front of brackets, the identity \(- ( \ldots) = -1 \cdot (\ldots)\) is
useful.
Rules
Commutative law:
Summands or factors can be arbitrarily interchanged, i.e. \[a+b=b+a \quad \text{and} \quad a \cdot b = b
\cdot a.\]
Associative law:
For summands or factors, order does not matter, i.e. \[(a+b)+c = a+(b+c) \quad \text{and} \quad (a \cdot
b) \cdot c = a \cdot ( b \cdot c).\]
Distributive law:
Multiplication and addition combine as \[a
\cdot (b+c) = a \cdot b + a \cdot c.\]
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