# Norm (mathematics)

In mathematics, the norm of a vector is its length. For the real numbers the only norm is the absolute value. For spaces with more dimensions the norm can be any function $p$ with

1. Scales for real numbers $a$, that is $p(ax) = |a|p(x)$
2. Function of sum is less than sum of functions, that is $p(x + y) \leq p(x) + p(y)$ or the triangle inequality
3. $p(x) = 0$ if and only if $x = 0$.

## Examples

1. The one-norm is the sum of absolute values: $\|x\|_1 = |x_1| + |x_2| + ... + |x_N|.$ This is like finding the distance from one place on a grid to another by summing together the distances in all directions the grid goes; see Manhattan Distance
2. Euclidean norm is the sum of the squares of the values: $\|x\|_2 = \sqrt{x_1^2 + x_2^2 + ... + x_N^2}$
3. Maximum norm is the maximum absolute value: $\|x\|_{\infty} = \max(|x_1|,|x_2|,...,|x_N|)$

This short article about mathematics can be made longer. You can help Wikipedia by .
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8819395899772644, "perplexity_flag": "head"}