Best intuitive metaphors for math concepts (of any level) Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)?
For example, I think the comparison of modular arithmetic to arithmetic with times, on a clock face, is fantastic, because it corresponds so perfectly and because clocks are so widely known.
 A: The number line. For whatever elementary school grade it is, it illustrates why you need zero, and why you need negative numbers.
The venerable cake, or probably pizza nowadays, to illustrate fractions.
The Riemann Sphere. 
The coin flip as a prototypical random event.
Markov chain as a frog hopping from lily pad to lily pad.
A: My brain only works well in Greekspace (that charming region of mathematics where everything has to correspond to a physical analog), so these are quite familiar to me.


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*If real numbers represent distance along a number line, imaginary numbers represent rotation. See the incredibly well-constructed explanation here: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

*If you want to push a crate, and you don't want it to spin while you're pushing it, the direction you have to push it in is an eigenvector. The amount it moves in proportion to how hard you push it is an eigenvalue.

*A matrix is a collection of vectors. If you have a matrix of some dimension less than four, you can visualize the vectors floating in 3D space. If not, you'll have to get more creative.

*Speaking of dimensions, a "dimension" is "anything that you can vary continuously," whether it's time, color, temperature, w/e. (I actually set up a 3D grid of different-colored vectors in my head for my linear algebra final. Very "A Beautiful Mind," in retrospect.)

*The Heaviside function is the curb that your response-function-car hits.

*Vector components/projections are shadows of the vector projected on x, and orthogonal vectors don't cast shadows on each other.

*Derivatives and integrals are getting distance/acceleration from the graph of your speed (which is, of course, drawn by a pen attached to your speedometer on a scrolling sheet of paper).

*Logarithms & their rules didn't really make sense to me until I saw & learned how to use a slide rule.

A: A graph (either the drawing of a function or the vertices-and-edges kind).
This might seem like a paltry or too general a metaphor but really, take any kind of algebraic expression (say in arithmetic); yes, one can manipulate it symbolically, but it often doesn't become meaningful until you draw something, something that captures a number of possibilities all at once in a picture (e.g. why is 1/x undefined at x = 0? the graph goes in two different directions there). 
The vertices/edges kind is so useful for capturing state/objects and transitions among them either for real world description or for mathematical concepts.
A: I find the concept of a digital sundial quite intriguing.  From wikipedia:

Roughly speaking, there exists a set
  with prescribed projections in almost
  all directions.

A: I can remember the following at school level -


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*Cartesian coordinates. Pythagoras theorem. (Metaphor - distance from the wall or from the edge of the floor.)

*Variables used in basic algebra. (Metaphor - common nouns, eg. "an animal is green and has two legs...")

*Irrationals and the completeness axiom. (Metaphor - can zoom in the real line without limit.)

*Complex number and complex plane. ("Add just solution to $x^2+1=0$ to the real line, suddenly every polynomial is solvable!")

*Sets, relations. (Venn diagrams are helpful for sets.)

A: Using parallel, vertical "gates" to visualize high-dimensional or even infinite cartesian products of intervals. It comes up in topological and measurable spaces (specifically in probability theory), and is nearly essential to intuiting convergence in these spaces.
