Systems {arithmetic} can idealize amount and individuation logic. Arithmetic operations add, subtract, multiply, and divide discrete magnitudes.
An arithmetic branch {mensuration}| calculates geometric-shape length, area, and volume.
Maximum numbers {modulus, arithmetic} can define arithmetics {modular arithmetic} {modulo arithmetic}. If arithmetic-operation result is more than modulus minus one, divide modulus into result until you get remainder less than modulus.
set
In modular arithmetic, modulus is highest number, and zero is lowest number. Modular arithmetic has no negative numbers or fractions. Therefore, many equations have no solutions.
periodicity
Modular arithmetic makes circular set and makes mathematical operations periodic.
congruency
An infinite number of positive integers {congruent integer} have same remainder when divided by modulus. For modulus, all congruent integers form set {residue class, congruency} [x]n or Zn.
Operations {mu operation}| {µ operation} can find the smallest natural number that satisfies an arithmetic equation or inequality. Turing machines, and formal systems and algorithms that simulate Turing-machine computations, must have the mu operation to reach the stop command.
For digits to right of decimal point, dots over digits indicate a repeating digit sequence {repeating decimal}.
Horizontal line {vinculum, arithmetic} above terms indicates terms to collect.
An arithmetic operation {addition, arithmetic} involves number {addend}, second-number addend, and total number {sum}.
To sum numbers {casting out nines}, first add the digits, for each number. Add sums. Use modulo nine to find result. Casting out nines can check additions.
Uppercase Greek letter {sigma, uppercase} can indicate continual addition of indexed variable or function.
An arithmetic operation {subtraction} involves minuend, subtrahend, and remainder number {difference, subtraction} left after subtracting subtrahend from minuend. Subtraction inverts addition.
Subtraction has a number {minuend}, from which to remove subtrahend.
Subtraction removes a number {subtrahend} from minuend.
An arithmetic operation {multiplication} involves number {multiplicand}, second number {multiplier, multiplication}, and total number {product, multiplication}. Multiplication adds multiplier number of multiplicands. For example, 3*4 = 3 + 3 + 3 + 3 = 12 adds four threes.
Products {continued product} can have more than two factors, for example, 3*4*5.
Minus times minus is plus, plus times plus is plus, and plus times minus is minus, and minus times plus is minus {law of signs}|.
Uppercase Greek letter {pi, uppercase} can indicate continued multiplication of indexed variable or function. Inverted uppercase Greek letter pi indicates continued division of indexed variable or function.
Process {rationalization, arithmetic} can eliminate irrational quantities or numbers, by removing non-repeating decimals.
An arithmetic operation {division, arithmetic} involves number {dividend, division}, second number {divisor}, and result number {quotient}. Division subtracts divisor number from dividend until remainder {remainder, division} is greater than or equal to zero and less than divisor. For example, 12/4 = 12 - 4 - 4 - 4 = 3 subtracts three fours from twelve. Quotient is number of repeated differences required. Division inverts multiplication. Quotient and divisor can be equal {square root}.
Quotients {fraction} {simple fraction} {common fraction} {ordinary fraction} {ratio of terms} have top expression {numerator} {antecedent, ratio} divided by bottom expression {denominator, fraction} {consequent, ratio}. Term can divide another term: a/b, x/y, and a*x^b / c*y^d.
simplification
Simplify fraction by factoring same number, variable, coefficient, or polynomial from both numerator and denominator. Then cancel factor. After all common factors cancel, smallest possible denominator {least common denominator} remains, and numerator is simplest.
addition
To add or subtract two simple fractions, first make denominators equal by multiplying each-fraction top and bottom by other-fraction denominator. Then add resulting numerators. Simplify fraction by factoring out terms, to make smallest possible denominator.
multiplication
To multiply two fractions, multiply numerators and multiply denominators. Simplify resulting fraction.
division
To divide two fractions, multiply top-fraction numerator by bottom-fraction denominator to make numerator. Multiply top-fraction denominator by bottom-fraction numerator to make denominator. Simplify fraction by factoring.
rational numbers
Integers divided by integers make rational numbers.
Two fractions can have same denominator {common denominator}.
An oblique line {solidus, arithmetic} / denotes fraction.
Fractions {aliquot part} {unit fraction} can have the number one in numerators: 1/n.
Polynomials can divide into other polynomials {complex fraction}: (a*x + b) / (b*y + d). Alternatively, fractions can have numerator and/or denominator fractions: (3/4) / (7/8).
A fraction {continued fraction}| can have an integer plus a numerator-1 fraction, and that fraction can have a denominator with an integer plus a numerator-1 fraction, and so on: a + 1/(b + 1/(c + 1/(...))), where a, b, and c are integers. a + (b + (c + (...)^-1)^-1)^-1. Rational numbers can be terminating continued fractions. Quadratic irrational numbers can be non-terminating periodic continued fractions. Real numbers can be non-terminating non-periodic continued fractions.
Two fractions, a1/b1 and a2/b2, can make a fraction {mediant fraction} whose value is between the original fractions: (a1 + a2) / (b1 + b2).
Fractions {partial fraction} can have form A / (a*x + b)^n or (A*x + B) / (a*x^2 + b*x + c)^2, and so on. Proper fractions can be sums of partial fractions whose denominators are proper-fraction denominator factors: 16/12 = 1/2 + 1/3 + 1/2.
Numerator can be less than denominator {proper fraction}. Numerator can be greater than denominator {improper fraction}.
Fractions {similar fraction} can have same denominator.
Variable or constant raised to power {exponent, arithmetic} indicates to multiply power number of times. For example, 2^4 = 2*2*2*2. Fractional exponents are roots. For example, square root has exponent 0.5: 4^0.5 = 2.
Constant or variable can have power {involution, mathematics}.
Constant or variable can have nth root {evolution, mathematics}.
Bases can have exponent zero {zeroth power}: x^0. Bases, except zero, to zeroth power equal one: x^0 = 1 and 2^0 = 1. Zero to zeroth power has no definition: 0^0 = null.
Exponents {fractional exponent} can be fractions. If exponent is 1/2, find square root. If exponent is 1/3, find cube root. If exponent is 1/n, where n is integer, find nth root. Exponentials with fractional exponents, of form 1/n, have main roots {principal nth root}.
Fractional exponent can have special root symbol {radical sign}, with fraction denominator placed in crook of radical sign.
Fractional exponent can have the base {radicand}, raised to fraction numerator, inside radical sign.
Powers {logarithm, exponential} of bases make numbers. For number e^2, ln(e^2) = 2. For number 100, log(100) = log(10^2) = 2. Taking logarithms and finding exponentials are inverses.
Logarithms {Naperian logarithm} can use base 10. For example, log(100) = 2, because 10^2 = 100.
Logarithms {natural logarithm, exponential}| {hyperbolic logarithm} can use base e. For example, ln(e^2) = 2.
Numbers can multiply logarithms with one base to give logarithms with another base {modulus, logarithm}.
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Date Modified: 2022.0225