When LaTeX is processing your input text, it is always in one of three modes:
LaTeX changes mode only when it goes up or down a staircase to a different level, though not all level changes produce mode changes. Mode changes occur only when entering or leaving an environment, or when LaTeX is processing the argument of certain text-producing commands.
Paragraph mode is the most common; it's the one LaTeX is in when processing ordinary text. In that mode, LaTeX breaks your text into lines and breaks the lines into pages.
There are also several text-producing commands and environments for making a box that put LaTeX in paragraph mode. The box made by one of these commands or environments will be called a parbox. When LaTeX is in paragraph mode while making a box, it is said to be in inner paragraph mode. The normal paragraph mode, in which LaTeX starts out, is called outer paragraph mode.
LaTeX is in math mode when it's generating a mathematical formula. It is in math mode in the math, displaymath, equation, and eqnarray environments.
In math mode letters are assumed to be math symbols and spaced accordingly. Spaces in the input are ignored, except that spaces may be needed to delineate the end of commands.
In LR mode, as in paragraph mode, LaTeX considers the output that it produces to be a string of words with spaces between them. In LR mode, unlike paragraph and math modes, spaces are not ignored; an input space creates an output space. However, unlike paragraph mode, LaTeX keeps going from left to right; it never starts a new line in LR mode. Even if you put a hundred words into an \mbox, LaTeX would keep typesetting them from left to right inside a single box, and then complain because the resulting box was too wide to fit on the line.
LaTeX is in LR mode when it starts making a box with an \mbox command. You can get it to enter a different mode inside the box - for example, you can make it enter math mode to put a formula in the box.
Also see Math Formulas
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