# Mathematics Fundamentals

Below is an overview of some of the fundamentals of mathematics that are useful for understanding Machine Learning.

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Mathematics Fundamentals Exponents Overview Raising a base number x to the power of n => multiplying the base number x by itself n times Shorthand for multiplying the base number x by itself n times: x Superscript n question-mark Baseline equals y Exponent (n) notation: x Superscript n Operations Procedure to MULTIPLY numbers with exponents, given they have the same base => ADD the exponents Multiplication example: x squared times x Superscript 5 Baseline equals x Superscript 2 plus 5 Baseline equals x Superscript 7 Procedure to DIVIDE numbers with exponents, given they have the same base => SUBTRACT the exponents Division example: StartFraction x squared Over x cubed EndFraction equals x Superscript 2 minus 3 Baseline equals x Superscript negative 1 Examples Exponent example: 3 squared equals 3 times 3 equals 9 Radicals / Roots Overview Opposite of exponent - finding the base number x, given y (the result) and the exponent n of the base number - e.g. square root, cube root To find the base number x: x question-mark Superscript n Baseline equals y Notations nth Root notation: RootIndex n StartRoot y EndRoot Square Root notation: StartRoot y EndRoot Cube Root notation: RootIndex 3 StartRoot y EndRoot Examples Square Root example: x squared equals 9 x equals StartRoot 9 EndRoot equals 3 Logarithms Overview The exponent n for a given base number x and result y To find the exponent n to determine how many times to multiply a base number by itself to get the given result: x Superscript n question-mark Baseline equals y Log notation: n equals log Subscript x Baseline left-parenthesis y right-parenthesis Examples Log example: 5 Superscript n question-mark Baseline equals 25 n equals log Subscript 5 Baseline left-parenthesis 25 right-parenthesis n equals 2 Common Logarithm The "common logarithm" of a number is an exponential for the base 10 "Common Log notation - default of base 10: log left-parenthesis y right-parenthesis equals log Subscript 10 Baseline left-parenthesis y right-parenthesis Log example: 10 Superscript n Baseline equals 1000 n equals log Subscript 10 Baseline left-parenthesis 1000 right-parenthesis equals log left-parenthesis 1000 right-parenthesis n equals 3 Natural Logarithm The "natural logarithm (ln)" of a number is an exponential for the base e, where e is a constant ~2.718 Natural Log notation: ln left-parenthesis y right-parenthesis equals log left-parenthesis e right-parenthesis Natural Log example: log Subscript e Baseline left-parenthesis 64 right-parenthesis equals ln left-parenthesis 64 right-parenthesis equals 4.1589 Inverse of Natural Log: e Superscript x Derivative of ln x: StartFraction 1 Over x EndFraction Properties / Laws Distributive Property Allows algebraic expressions in the form of a(b + c) to be reformulated to an equivalent expression Distributive law: a left-parenthesis b plus c right-parenthesis equals a b plus a c Khan Academy: Distributive Property Monomials Monomial - an algebric expression that contains only one term Trinomials Trinomial - an algebric expression that contains only three terms Factoring Rewriting a number or term as a product of several smaller factors or values in common Difference of Squares Pattern Every polynomial that is a difference of square terms can be factored with this formula: a squared minus b squared equals left-parenthesis a plus b right-parenthesis left-parenthesis a minus b right-parenthesis Khan Academy: Factoring Binomials Overview Binomial - an algebric expression that contains only two terms Operations Difference of Squares Pattern - special products of binomials (applying the Distributive Property twice): left-parenthesis a plus b right-parenthesis left-parenthesis a minus b right-parenthesis equals a left-parenthesis a plus b right-parenthesis minus b left-parenthesis a plus b right-parenthesis equals a squared plus a b minus a b minus b squared equals a squared minus b squared Squaring binonmials of the form (a+b)^2: left-parenthesis a plus b right-parenthesis squared equals left-parenthesis a plus b right-parenthesis left-parenthesis a plus b right-parenthesis equals a squared plus 2 a b plus b squared Squaring binonmials of the form (a-b)^2: left-parenthesis a minus b right-parenthesis squared equals left-parenthesis a minus b right-parenthesis left-parenthesis a minus b right-parenthesis equals a squared minus 2 a b plus b squared Squaring binonmials of the form (ax+b)^2: left-parenthesis x plus a right-parenthesis squared equals left-parenthesis x plus a right-parenthesis left-parenthesis x plus a right-parenthesis equals x squared plus 2 a x plus a squared Polynomials Overview Polynomial - an algebric expression that contains many terms - constants, variables with coefficients, and exponentials with a positive exponent Notations Standard Format - sorted by highest exponent variables first, then non-exponential variables (x to the power of 1), then constants: a Subscript n Baseline x Superscript n Baseline plus a Subscript n minus 1 Baseline x Superscript n minus 1 Baseline plus period period period plus a 2 x squared plus a 1 x plus a 0 Summation notation: sigma-summation Underscript k equals 0 Overscript n Endscripts a Subscript k Baseline x Superscript k Examples Rearranging into standard format (note the negative!): 6 plus 3 x minus 2 x squared equals minus 2 x squared plus 3 x plus 6 Operations Addition/Subtraction of polynomials - add or subtract the coefficients of the like terms Addition example: left-parenthesis 2 x squared plus 3 x plus 4 right-parenthesis plus left-parenthesis x squared minus 2 x plus 3 right-parenthesis equals 3 x squared plus x plus 7 Multiplication - multiply each term in the first expression with each term in the second expression Multiplication Example: left-parenthesis 2 x squared plus 3 x plus 4 right-parenthesis times left-parenthesis x squared minus 2 x plus 3 right-parenthesis equals 2 x Superscript 4 Baseline minus x cubed plus 4 x squared plus x plus 12 Monomial Division (by a single term expression) - convert each term to a fraction and simplify Divison Example: left-parenthesis 4 x squared plus 8 x right-parenthesis division-sign 2 x equals StartFraction 4 x squared Over 2 x EndFraction plus StartFraction 8 x Over 2 x EndFraction equals 2 x plus 4 Division by more than one term - use long division Wikipedia: Polynomial Long Division Khan Academy: Intro to long division of polynomials Quadratic Equations Overview Graphs as a parabola. Vertex on top (A shaped) if the a in ax^2 is negative. Vertex on bottom (U shaped) if the a in ax^2 is positive For every y value, there are two x values, equidistant from the line of symmetry Line of symmetry x coordinate of vertex: x equals StartFraction negative b Over 2 a EndFraction Notation for a quadratic equation: y equals a x squared plus b x plus c