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Algebra

Exercises in C++

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Algebra: 0% done

Module 1 · Numbers and what they count

Numbers began as a way to count

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1 / 260 · Numbers and what they count

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Numbers began as a way to count
, 2 to practiseTen digits write every natural number
, 1 to practiseCode in this course: your first C++ program
, 1 to practiseFunctions, decisions, loops and lists in C++
, 1 to practiseTry it: split a number into its places
, 1 to practiseNegative numbers fill the gap below zero
, 1 to practiseFurther right is bigger
, 2 to practiseAbsolute value is distance from zero
, 1 to practiseFractions fill the gaps between integers
, 1 to practiseSome lengths are not fractions
Fractions get close to the square root of 2, never there
, 1 to practiseThe real numbers fill the whole line
, 4 to practisePractice problems
Recap

, 2 to practiseAn agreed order makes a sum mean one thing
, 2 to practiseProgramming languages use the same order
, 1 to practiseOrder and grouping do not matter for adding and multiplying
, 1 to practiseMultiplying spreads over a sum
, 1 to practiseZero and one change nothing
, 1 to practisePrimes are the numbers with no smaller factors
, 1 to practiseEvery number is a product of primes in one way
, 2 to practiseThe gcd and the lcm
, 1 to practiseEuclid’s algorithm finds the gcd fast
, 1 to practiseEstimate before you calculate
, 4 to practisePractice problems
Recap

A fraction is a division waiting to happen
, 1 to practiseMany fractions name the same number
, 2 to practiseAdd fractions over a common denominator
, 1 to practiseMultiplying is easier than adding
, 1 to practiseTry it: add two fractions exactly
, 1 to practiseDecimals are fractions over powers of ten
, 1 to practiseRecurring decimals turn back into fractions
, 1 to practiseA percentage is a fraction out of one hundred
, 2 to practiseTry it: percentage change
, 1 to practiseRatios compare amounts; proportion keeps them fixed
, 1 to practiseRates and unit conversion are fractions too
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA power is repeated multiplication
, 2 to practiseThe laws of exponents follow from counting factors
, 1 to practiseZero and negative exponents keep the pattern going
, 1 to practiseTry it: powers by repeated multiplication
, 1 to practiseA root undoes a power
, 2 to practiseA fractional exponent is a root
, 1 to practiseSimplify a root by taking out squares
, 1 to practiseTry it: a square root by halving the gap
, 1 to practiseScientific notation writes huge and tiny numbers
, 1 to practiseOrders of magnitude compare by powers of ten
, 1 to practiseSignificant figures say how precise a number is
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

A letter stands for a number
, 1 to practiseAn expression names a value; an equation makes a claim
, 2 to practiseSubstitute to find an expression’s value
An expression is a small function
, 2 to practiseOnly like terms can be added
, 2 to practiseExpanding brackets multiplies every term
, 2 to practiseFactoring puts the brackets back
, 2 to practiseTurning words into expressions
, 1 to practiseAn identity is true for every value
, 4 to practisePractice problems
Recap

An equation is a balance
, 1 to practiseUndo what was done to the unknown, in reverse order
, 2 to practiseGather the unknowns on one side
, 1 to practiseAlways check by substituting back
, 1 to practiseExpand brackets first
, 2 to practiseClear fractions by multiplying through
, 1 to practiseSome equations have no solution, some have every number
, 1 to practiseTry it: a solver for every linear equation
, 2 to practiseRearranging a formula is solving for a letter
, 1 to practiseWord problems: name, write, solve, check
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseAn inequality has a range of solutions
Solve an inequality like an equation, with one exception
, 2 to practiseMultiplying by a negative flips the inequality
, 1 to practiseFind the missing flip
, 1 to practiseInterval notation writes a range in brackets
, 1 to practiseCompound inequalities: and, or
, 1 to practiseTry it: solve any linear inequality
, 3 to practiseAn absolute value equation usually has two answers
, 1 to practiseAbsolute value inequalities: inside or outside
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

A function gives exactly one output for each input
, 2 to practisef(x) names the output
A graph draws every input–output pair
, 1 to practiseDomain is what goes in, range is what comes out
, 1 to practiseThe vertical line test
, 1 to practisePiecewise functions use different rules on different inputs
, 2 to practiseComposition feeds one function into another
, 3 to practiseAn inverse function runs the machine backwards
, 1 to practiseShift, stretch and reflect a graph
, 4 to practisePractice problems
Recap

, 1 to practiseTwo number lines make a plane
, 1 to practiseSlope is rise over run
, 1 to practiseSlope-intercept form: y = mx + c
, 2 to practiseTry it: the line through two points
, 1 to practisePoint-slope form starts from any point
, 1 to practiseStandard form handles every line, vertical ones too
, 2 to practiseParallel lines share a slope; perpendicular slopes multiply to −1
, 1 to practiseDistance and midpoint between two points
, 1 to practiseA line is a model of steady change
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseTwo unknowns need two facts
The solution is where the lines cross
, 2 to practiseSubstitution: solve one, put it into the other
, 2 to practiseElimination: add the equations to cancel a variable
, 1 to practiseOne solution, none, or infinitely many
, 1 to practiseTry it: solve any two-by-two system
, 1 to practiseWord problems with two unknowns
, 1 to practiseThree unknowns need three equations
, 1 to practiseSystems of inequalities mark out a region
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA polynomial is a sum of powers of one letter
, 1 to practiseAdding and subtracting means collecting like terms
, 1 to practiseMultiplying means every term times every term
, 2 to practiseThree products worth knowing by heart
, 1 to practiseEvaluate a polynomial the fast way
, 1 to practiseFactoring undoes multiplying
, 2 to practiseDifferences of squares and trinomials
, 1 to practiseDividing one polynomial by another
, 3 to practiseThe remainder and factor theorems
, 1 to practiseRoots, touching and the ends of the graph
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA quadratic draws a parabola
, 1 to practiseVertex form shows the turning point
, 1 to practiseSolving by factoring
, 1 to practiseA square has two square roots
, 2 to practiseCompleting the square
, 1 to practiseThe quadratic formula, derived
, 2 to practiseUsing the formula, step by step
, 1 to practiseThe discriminant counts the roots
, 1 to practiseWhen the textbook formula fails a computer
, 1 to practiseThe vertex answers biggest and smallest
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA rational expression is a fraction of polynomials
, 2 to practiseSimplify by cancelling factors, never terms
, 1 to practiseExcluded values are holes or walls
, 1 to practiseMultiply and divide as you would fractions
, 1 to practiseAdd and subtract over a common denominator
, 2 to practiseSolve rational equations by clearing the denominators
, 1 to practiseSimplify radicals and rationalise the denominator
, 2 to practiseRadical equations: isolate, square, check
, 1 to practiseFar from zero, the leading terms decide
, 4 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseExponential growth multiplies by the same factor
, 1 to practiseThe base decides growth or decay
, 2 to practiseCompound interest is exponential growth
, 1 to practiseThe number e is compounding without end
, 1 to practiseA logarithm asks which power
, 1 to practiseThe laws of logarithms come from the laws of exponents
, 2 to practiseChange of base: any logarithm from one you have
, 2 to practiseSolve exponential equations by taking logarithms
, 2 to practiseHalf-life: decay measured in halvings
, 1 to practiseLog scales put huge ranges on one ruler
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA sequence is a list that follows a rule
, 1 to practiseArithmetic sequences add the same amount each time
, 1 to practiseGeometric sequences multiply by the same amount
, 2 to practiseRecursive rules build each term from the last
, 1 to practiseSigma notation writes a long sum in a line
, 3 to practiseAn arithmetic series is pairs that add to the same total
, 1 to practiseA geometric series collapses when you multiply it by r
, 2 to practiseSome infinite sums have a finite total
, 1 to practiseInduction proves a rule for every whole number
, 1 to practisePascal’s triangle expands any power of a sum
, 1 to practiseCompute the triangle, row by row
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseAn angle measures a turn
, 2 to practiseRadians measure an angle by the arc it cuts
, 2 to practiseIn a right triangle, the angle fixes the ratios of the sides
, 2 to practiseSolve a right triangle step by step
, 1 to practiseThe unit circle defines sine and cosine for any angle
, 1 to practiseSine and cosine are waves; tangent repeats with gaps
, 1 to practiseSine squared plus cosine squared is always 1
Angle-sum identities split an angle in two
, 3 to practiseThe laws of sines and cosines solve any triangle
, 2 to practiseInverse functions turn a ratio back into an angle
, 3 to practisePractice problems
, 3 to practiseMore practice problems
Recap

A number whose square is minus one
, 1 to practiseAdding and multiplying work as you expect
, 2 to practiseThe conjugate makes division possible
Every complex number is a point in a plane
, 3 to practiseSize and direction: modulus and argument
Polar form and Euler’s formula
, 1 to practiseMultiplying multiplies sizes and adds angles
, 1 to practiseTry it: complex arithmetic in code
, 2 to practisePowers, roots and the roots of unity
, 2 to practiseEvery polynomial has all its roots in the complex numbers
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

A vector is a size and a direction
, 1 to practiseAdding vectors goes tip to tail
, 1 to practiseThe length of a vector comes from Pythagoras
, 2 to practiseThe dot product measures how much two vectors agree
, 1 to practiseTry it: the angle between two vectors
, 1 to practiseA matrix is a grid of numbers
, 4 to practiseMatrix multiplication is rows times columns
, 1 to practiseThe identity matrix and the transpose
, 1 to practiseA matrix moves every point of the plane
, 1 to practiseA system of equations is one matrix equation
, 1 to practiseGaussian elimination solves any system step by step
, 1 to practiseTry it: solve a system by elimination
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseMixing vectors: combinations and span
, 1 to practiseIndependent vectors add something new
, 1 to practiseA basis is a coordinate system
, 2 to practiseA linear map keeps lines straight and the origin still
, 1 to practiseThe determinant is how much area scales
, 1 to practiseTry it: a three by three determinant
, 1 to practiseAn inverse undoes the map
, 1 to practiseEigenvectors keep their direction
, 2 to practiseFinding eigenvalues with a determinant
, 1 to practiseDiagonalising: the matrix in its own basis
Markov chains settle on an eigenvector
, 1 to practiseTry it: run a Markov chain
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA proof covers every case at once
, 2 to practiseProving the other way round, and by contradiction
, 1 to practiseSets and maps: the words every structure uses
, 2 to practiseModular arithmetic: numbers on a clock
, 1 to practiseTry it: fast powers modulo n
, 1 to practiseA group: one operation, four rules
, 1 to practiseSymmetries form groups, and order matters
, 2 to practiseRings and fields: two operations that get along
, 1 to practiseIsomorphic structures are the same in disguise
, 1 to practiseWhere algebra goes next
, 4 to practiseMixed problems from the whole course
, 3 to practisePractice problems
, 3 to practiseMore practice problems
Recap