OCME · Bilingual mathematics & derivations
Seven-Round Algebra and Probability Foundations: Powers, Identities, Functions, and Finite Models
On this page
- Arithmetic operations
- Real number
- Variables and algebraic expressions
- Rational number
- Equation
- Function mapping
- Linear equation in one variable
- Coordinate plane
- Event
- Ratio and proportion
- Natural-number powers and principal square roots
- Univariate polynomials over the rationals
- Polynomial factorization over the rationals
- Quadratic equations over the reals
- Affine functions and degree-one functions
- Graphs of real-valued functions
- Probability on finite sample spaces
Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Variables and algebraic expressions
Existing canonical prerequisite for the new objects in this pulse.
Rational number
Existing canonical prerequisite for the new objects in this pulse.
Equation
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Linear equation in one variable
Existing canonical prerequisite for the new objects in this pulse.
Coordinate plane
Existing canonical prerequisite for the new objects in this pulse.
Event
Existing canonical prerequisite for the new objects in this pulse.
Ratio and proportion
Existing canonical prerequisite for the new objects in this pulse.
Natural-number powers and principal square roots
This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct.
Univariate polynomials over the rationals
A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree.
Polynomial factorization over the rationals
Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values.
Quadratic equations over the reals
For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct.
Affine functions and degree-one functions
A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant.
Graphs of real-valued functions
In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output.
Probability on finite sample spaces
Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes.
Path source & status
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