So I have been thinking about which ideas in mathematics I use most often and I have listed them below. Please feel free to comment because I would like to use this list as a starting point for a future list of “The most useful ideas in Mathematics for Scientists and Engineers”. This is only a draft, so I expect to make a number of revisions over the next few days.
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I showed this list to Carl and he said his list would be completely different. (He choked a bit when he saw primes at #70.) Hopefully we can post his list as well as others.
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 counting
 zero
 integer decimal positional notation 100, 1000, …
 the four arithmetic operations + – * /
 fractions
 decimal notation 0.1, 0.01, …
 basic propositional logic (Modus ponens, contrapositive, Ifthen, and, or, nand, …)
 negative numbers
 equivalence classes
 equality & substitution
 basic algebra – idea of variables, equations, …
 the idea of probability
 commutative and associative properties
 distributive property
 powers (squared, cubed,…), – compound interest (miracle of)
 scientific notation 1.3e6 = 1,300,000
 polynomials
 first order predicate logic
 infinity
 irrational numbers
 Demorgan’s laws
 statistical independence
 the notion of a function
 square root (cube root, …)
 inequalities (list of inequalities)
 power laws (i.e. $a^b a^c = a^{b+c}$ )
 Cartesian coordinate plane
 basic set theory
 random variable
 probability distribution
 histogram
 the mean, expected value & strong law of large numbers
 the graph of a function
 standard deviation
 Pythagorean theorem
 vectors and vector spaces
 limits
 real numbers as limits of fractions, the least upper bound
 continuity
 $R^n$, Euclidean Space, and Hilbert spaces (inner or dot product)
 derivative
 correlation
 central limit theorem, Gaussian Distribution, Properties of Guassains.
 integrals
 chain rule
 modular arithmetic
 sine cosine tangent
 $\pi$, circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,…
 linear regression
 Taylor’s theorem
 the number e and the exponential function
 Rolle’s theorem, Karush–Kuhn–Tucker conditions, derivative is zero at the maximum
 the notion of linearity
 injective (onetoone) / surjective (onto) functions
 imaginary numbers
 symmetry
 Euler’s Formula $e^{i \pi} + 1 = 0$
 Fourier transform, convolution in time domain is the product in the frequency domain (& vice versa), the FFT
 fundamental theorem of calculus
 logarithms
 matrices
 conic sections
 Boolean algebra
 Cauchy–Schwarz inequality
 binomial theorem – Pascal’s triangle
 the determinant
 ordinary differential equation (ODE)
 mode (maximum likelihood estimator)
 cosine law
 prime numbers
 linear independence
 Jacobian
 fundamental theorem of arithmetic
 duality – (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, …)
 intermediate value theorem
 eigenvalues
 median
 entropy
 KL distance
 binomial distribution
 Bayes’ theorem
 $2^{3.32} \approx 10$
 compactness, Heine – Borel theorem
 metric space, Triangle Inequality
 Projections, Best Approximation
 $1/(1X) = 1 + X + X^2 + \ldots$
 partial differential equations
 quadratic formula
 Reisz representation theorem
 Fubini’s theorem
 the ideas of groups, semigroups, monoids, rings, …
 Singular Value Decomposition
 numeric integration – trapezoidal rule, Simpson’s rule, …
 mutual information
 Plancherel’s theorem
 matrix condition number
 integration by parts
 Euler’s method for numerical integration of ODEs (and improved Euler & Runge–Kutta)
 countable vs uncountable infinity
 pigeon hole principle
There is a long list of mathematical ideas that I use less often. Here’s a sampling: Baire category theorem, Banach Spaces, Brouwer Fixed Point Theorem, Carathéodory’s Theorem, Category Theory, Cauchy integral formula, calculus of variations, closed graph theorem, Chinese remainder theorem, Clifford algebra (quaternions), Context Free Grammars, Cramer’s Rule, cohomology, Euclidean algorithm, fundamental group, Gauss’ Law, Grassmannian algebra , Graph Theory, HahnBanach Theorem, homology, Hairy Ball Theorem, Hölder’s inequality, inclusionexclusion, Jordan Decomposition, Kalman Filters, Markov Chains (Hidden Markov Models), modules, nonassociative algebras, Picard’s Great Theorem, Platonic/Euclidean solids, Probabilistic Graphical Models (Bayesian Networks, Markov Random Fields), Pontryagain duality, Spectral Theorem, Sylow p subgroup, repeating decimals equal a fraction, ring ideals, sine law, tensors, tessellation, transcendental numbers, Uniform Boundedness Theorem, Weierstrass approximation theorem, …
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Interesting list, especially since geometry plays a distinctly secondary role. I’d add:
proof
fractal
graph
optimization
model
partially ordered set
algorithm
dynamical system 
You might find this interesting.
http://onehappybird.com/2012/12/03/whatsthemostimportanttheorem/

Might be buried in there, as there is a lot in there, but a basic math bit is factors. Pi should probably be clearly stated. And maybe I missed it, but the ordinal rules for doing a calculation (as in, things in parentheses get done first, etc.) seem pretty important.
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