02. Absolute Values
Dated: 30-10-2024
This plays an important role in determining distance between 2 points on a coordinate line.1
Definition
The absolute value of a real number \(a\) is denoted as \(\lvert a \rvert\).
Example
\[|a+b| \le |a| + |b|\]
Reason behind this is that one of the numbers can be negative.
Hence, decreasing the overall sum.
Example
\[|x - 5| = 4\]
Look at this equation, it can have 2 possibilities.
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\[x -5 = 4\]
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\[x - 5 = -4\]
Example
\[|x + 4| = |x - 2|\]
This equation can have 4 possibilities.
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\[(x+4) = (x - 2)\]
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\[- (x+4) = (x - 2)\]
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\[(x+4) = -(x - 2)\]
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\[-(x+4) = -(x - 2)\]
Notice how 1 is related to 4 and 2 is related to 3.
Relationship between Square Root and Absolute Value
\[\sqrt{a^2} \ne (\sqrt{a})^2\]
Example
Let's take \(a = -4\) and take its square root.
- Case \(\sqrt{a^2}\)
\[\sqrt{(-4)^2}= \sqrt{16} = 4\]
- Case \((\sqrt{a})^2\)
\[(\sqrt{-4})^2 = (\sqrt{-1 \times 4})^2\]
\[= (\sqrt{-1} \times \sqrt{4})^2\]
\[\because \iota = \sqrt{-1}\]
\[= (\iota \times \sqrt{4})^2\]
\[= \iota^2 \times (\sqrt{4})^2\]
\[\because \iota^2 = -1\]
\[= -1 \times 4 = -4\]
Notice how the first case resembles the absolute value.
\[|a| = \sqrt{a^2}\]
Properties of Absolute Values
\[\lvert a\rvert = \lvert -a \rvert\]
\[\lvert a \cdot b\rvert = \lvert a\rvert \cdot \lvert b\rvert\]
\[\left| \frac{a}{b} \right| = \frac{\lvert a \rvert} {\lvert b \rvert}\]
\[\lvert a^n\rvert = \lvert a\rvert^n\]
Geometric Interpretation of Absolute Value
We can define distance as an absolute value
References
Read more about notations and symbols.
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Read more about coordinate line. ↩