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Absolute Value

Definition

For a real number, the absolute value of \(x\), denoted \( \lvert x \rvert \), is defined as

\[ \left|x \right| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0. \end{cases} \]

Technique

Evaluate \( \left| 3\left( 17-55 \right) \right| \).

\[ \begin{align} \left| 3\left( 17-55 \right) \right| &= \left| 3\left( -38 \right) \right| \\ &= \left| -114 \right| \\ &= 114 _\square \end{align} \]

 

If \( x=\frac{5+\sqrt{11}i}{2} \) and \( y=\frac{5 -\sqrt{11}i}{2} \), what is the value of \( \left| x + 3y \right|^2 \)?

\[ \begin{align} \left| x + 3y \right|^2 &= \left| \frac{5+\sqrt{11}i}{2} + 3\left( \frac{5 -\sqrt{11}i}{2} \right) \right|^2 \\ &= \left| \frac{5+\sqrt{11}i + 15 -3\sqrt{11}i}{2} \right|^2 \\ &= \left| \frac{20-2\sqrt{11}i}{2} \right|^2 \\ &= \left| 10 - \sqrt{11}i \right|^2 \\ &= \left( \sqrt{10^2 + \sqrt{11}^2} \right)^2 \\ &= \sqrt{111}^2 \\ &= 111 _\square \end{align} \]

Application and Extensions

What is the absolute value of \( 2 - 10a \)?

The answer depends on the value of \( a \). From the definition above, we have

\[ \left|2-10a \right| = \begin{cases} 2-10a, & \text{if } 2-10a \geq 0 \\ -(2-10a), & \text{if } 2-10a < 0 \end{cases} \]

Solving the inequality on the right, we obtain

\[ \begin{align} 2-10a &\geq 0 \\ 2 & \geq 10a \\ \frac{1}{5} &\geq a. \end{align} \]

It follows that \[ 2-10a \geq 0 \Longleftrightarrow a \leq \frac{1}{5}. \]

Therefore,

\[ \left|2-10a \right| = \begin{cases} 2-10a, & \text{if } a \leq \frac{1}{5} \\ 10a - 2, & \text{if } a > \frac{1}{5} \end{cases} _\square \]

Note by Arron Kau
3 years, 2 months ago

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thank you very much :-) Shovik Nandy · 3 years, 1 month ago

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In ending step ..note that to take modulus of imag eniry number is different than simple modulas Puneet Mehra · 2 years, 7 months ago

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Excellent... Archiet Dev · 3 years, 2 months ago

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I didn't quite understand the problem having \( i \). At one stage the \( i \) disappears. Can you please explain? Shabarish Ch · 3 years, 2 months ago

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@Shabarish Ch It is because the absolute value of a complex number (\(a + bi\)) is defined as \(\sqrt{a^2 + b^2}\) , i.e., \(|a + bi|\) = \(\sqrt{a^2 + b^2}\). In the above question, \(|10 - \sqrt11|\) = \(\sqrt{10^2 + \sqrt{11}^2}\) Aman Bansal · 3 years, 1 month ago

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@Aman Bansal Didn't know that, thanks!! Shabarish Ch · 3 years, 1 month ago

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@Shabarish Ch \(i\) is defined as \(\sqrt{-1}\) i.e. imaginary unit. So, those were complex numbers (real number + imaginary numbers). Vishnuram Leonardodavinci · 3 years, 2 months ago

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@Shabarish Ch I am not sure but as far as my mind think I can say this is because when the sign of absolute value was removed.Then sign was changed to positive & i(iota) has nothing to do when there is is postive root.So,It would have been removed... Archiet Dev · 3 years, 2 months ago

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I like this format of note, I think I'll Start Share So !

Explanation More Exercises Solved ! Gabriel Merces · 3 years, 2 months ago

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