Answers.Body: <p><span class="math-container">$\|x\| = |x|$</span> denotes the absolute value norm, which is a special case of the <span class="math-container">$L_1$</span> norm defined on the 1-D vector spaces formed by real or complex numbers.</p>
  
  <p><span class="math-container">$\|\textbf{x}\|_1 = \sum_{i=1}^n|x_i|$</span> denotes the Taxicab / Manhattan norm, relating to how a Taxi would drive along a rectangular grid of roads to reach a point <span class="math-container">$(x, y)$</span> from <span class="math-container">$(0,0)$</span>.</p>
  
  <p><span class="math-container">$\|\textbf{x}\|_2 = \sqrt{x_1^2 + \dots + x_n^2}$</span> denotes the Euclidean norm on an N-D Euclidean space, which is a result of the Pythagorean theorem (the shortest distance between two points).</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-01-27T05:18:41.517Z
Answers.Text: Could you please elucidate on absolute value norm computation.
Answers.UserDisplayName: Stephen Philip
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-01-27T10:54:27.220Z
Answers.Text: The Absolute value norm is a special case of the $L_1$ norm (defined in 1D). So, computing it would be equivalent to evaluating the absolute value of the single component of the vector.
Answers.UserDisplayName: s_bh
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-01-26T14:06:45.677Z
Answers.LastActivityDate: 2020-01-27T10:58:59.370Z
Answers.LastEditDate: 2020-01-27T10:58:59.370Z
Answers.UserDisplayName: s_bh
Body: <p>What is the difference between the notations <span class="math-container">$\|x\|_1, \|x\|_2$</span> and <span class="math-container">$|x|$</span>? I think <span class="math-container">$|x|$</span> is the magnitude of <span class="math-container">$x$</span>.</p>
  
Tags: <machine-learning><math><notation>
Title: What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?
UserDisplayName: Stephen Philip
ThreadText: # What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?
  
  (Asked by: username_0 on 2020-01-26)
  
  What is the difference between the notations $\|x\|\_1, \|x\|\_2$ and $|x|$? I think $|x|$ is the magnitude of $x$.
  
  **Tags:** machine-learning, math, notation
  
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  ## 1 answers
  
  ### Answer by username_1 on 2020-01-26. Score: 5 | Accepted answer
  
  $\|x\| = |x|$ denotes the absolute value norm, which is a special case of the $L\_1$ norm defined on the 1-D vector spaces formed by real or complex numbers.
  
  $\|\textbf{x}\|\_1 = \sum\_{i=1}^n|x\_i|$ denotes the Taxicab / Manhattan norm, relating to how a Taxi would drive along a rectangular grid of roads to reach a point $(x, y)$ from $(0,0)$.
  
  $\|\textbf{x}\|\_2 = \sqrt{x\_1^2 + \dots + x\_n^2}$ denotes the Euclidean norm on an N-D Euclidean space, which is a result of the Pythagorean theorem (the shortest distance between two points).
  
  > **Comments:**
  > **username_0**: Could you please elucidate on absolute value norm computation.  
  > **username_1**: The Absolute value norm is a special case of the $L_1$ norm (defined in 1D). So, computing it would be equivalent to evaluating the absolute value of the single component of the vector.  
  
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