PROOF LAB
Prove it, one line at a time.
Every line cites an axiom or an earlier result. Nothing outside the list may be used — and at the end of each proof, you can remove one axiom and watch exactly which lines lose their footing.
Numbers
- (−1) × (−1) = 101
- a ÷ 0 cannot be defined02
- a/b + c/d = (ad+bc)/bd03
- a ÷ (b/c) = a × (c/b)04
- Every repeating decimal can be written as a fraction05
- 0.999… = 106
- a − b = a + (−b)07
- The Euclidean algorithm always terminates08
- √2 is not rational09
- What an extension preserves determines the extension10
- There are infinitely many primes11
- Prime factorization is unique12
- 1 is not prime13
- i² = −1 is a definition14
- Reducing a fraction leaves its value unchanged70
- Fraction comparison is settled by cross-multiplication71
- A decimal is a fraction whose denominator is a power of 1072
- Shifting the decimal point multiplies by 10 or by 1/1073
- In decimal multiplication the decimal places add74
- Rounding error never exceeds half a unit75
- Adding rounded values accumulates error76
- A fraction terminates exactly when its denominator's prime factors are 2 and 577
- The repetend is shorter than the denominator78
- a × 0 = 079
- The additive inverse is unique80
- (−a)b = −(ab)81
- gcd(a,b) × lcm(a,b) = ab82
- The last remainder of the Euclidean algorithm is the gcd83
- If n is not a perfect square, √n is irrational84
- The sum of a rational and an irrational is irrational85
- What each extension gains86
- A composite number has a prime factor at most √n87
- The complex numbers cannot be ordered by size88
- The product of a complex number and its conjugate is real89
- Significant figures do not increase under multiplication186
- There exist integers with gcd(a,b) = ax + by187
- The rationals are dense yet have gaps188
- What each extension loses189
- Gaps between primes can be arbitrarily wide190
- Every degree-n equation has a root in ℂ191
Language
- a⁰ = 115
- a⁻ⁿ = 1/aⁿ16
- log(ab) = log a + log b17
- Transposition is justified18
- The quadratic formula19
- If a < b and c < 0, then ac > bc20
- The distance between two points21
- Perpendicular lines have slopes whose product is −122
- The product formulas come from the distributive law23
- If f(a) = 0, then (x−a) is a factor24
- Only the contrapositive is equivalent to the original statement25
- Proof by contradiction is justified26
- One counterexample brings a statement down27
- Only like terms can be added90
- Substitution is valid91
- Identities and equations are different92
- a^(1/2) is √a93
- (aᵐ)ⁿ = aᵐⁿ94
- √a is fixed as the nonnegative one95
- The condition under which √(ab) = √a √b holds96
- √(a²) is the absolute value of a97
- log(aⁿ) = n log a98
- The change-of-base formula99
- The argument of a logarithm must be positive100
- The discriminant determines the number of roots101
- The relation between roots and coefficients102
- Squaring creates extraneous roots103
- An inequality cannot be squared without a condition104
- Elimination does not change the solution set105
- When there is no solution or infinitely many106
- Substitution and elimination give the same answer107
- The midpoint formula108
- The section formula109
- Factoring is reading the product formulas backwards110
- If AB = 0, then A = 0 or B = 0111
- De Morgan's laws112
- Necessary and sufficient conditions are containment113
- The arithmetic mean is at least the geometric mean192
Relations
- 1 + 2 + … + n = n(n+1)/228
- The graph of the inverse function is symmetric about y = x29
- The graph of a linear function is a line30
- The vertex of a quadratic function31
- Trigonometric ratios are determined by the angle alone32
- sin²θ + cos²θ = 133
- The law of sines and the law of cosines34
- The angle addition theorems35
- The general term of an arithmetic sequence is linear114
- In a proportion, the product of the extremes equals the product of the means115
- The graph of inverse proportion never touches the axes116
- The grounds of percentage computation117
- Why a function must have exactly one output118
- An inverse function requires injectivity119
- Composition is associative but not commutative120
- Parallel lines have equal slopes121
- The distance between a point and a line122
- The graph of a quadratic function is symmetric about its axis123
- The discriminant and the number of x-intercepts124
- The maximum and minimum come from the vertex125
- The exponential and logarithmic functions are symmetric about y = x126
- The exponential function is always positive127
- The period of the trigonometric functions comes from one full turn128
- e arises where the growth rate equals the function itself193
Change
- The sum of an arithmetic sequence36
- The sum of a geometric sequence37
- If the common ratio is less than 1, the infinite geometric series converges38
- The ε-δ definition of the limit39
- Differentiable implies continuous40
- The product rule41
- The quotient rule42
- The chain rule43
- The mean value theorem44
- The intermediate value theorem45
- The definite integral is the limit of Riemann sums46
- The fundamental theorem of calculus47
- Verifying a general term by induction129
- The limit is unique130
- The limit of sin x divided by x is 1131
- A convergent sequence is bounded132
- The definition of continuity comes from the definition of a limit133
- The left and right derivatives must agree134
- The derivative of xⁿ135
- The sign of the derivative determines increase and decrease136
- At an extremum the derivative is 0137
- The derivative can be 0 without an extremum138
- The second derivative determines convexity139
- The intermediate value theorem fails in the rationals140
- If the upper and lower sums meet, the function is integrable141
- Substitution is the chain rule read backwards142
- Integration by parts is the product rule read backwards143
- Properties of the definite integral144
- 0.999… = 1 is a question of limits194
Space
- Vertical angles are equal48
- The angles of a triangle sum to 180°49
- The area of a circle is πr²50
- The inscribed angle is half the central angle51
- The angle in a semicircle is a right angle52
- Triangle similarity conditions53
- Similarity ratio a:b gives area ratio a²:b²54
- The centroid divides each median in the ratio 2:155
- a² + b² = c²56
- The converse of the Pythagorean theorem is also true57
- The volume of a cone is 1/3 of the cylinder58
- Removing the parallel postulate59
- Alternate and corresponding angles at parallel lines are equal145
- The interior angles of an n-gon sum to 180(n−2)146
- The exterior angles always sum to 360°147
- Translations and reflections preserve lengths and angles148
- Two reflections make a rotation149
- The triangle area formula comes from the rectangle150
- Derivation of the trapezoid area formula151
- A tangent to a circle is perpendicular to the radius152
- The power of a point153
- Parallel lines divide segments in equal ratios154
- Vector addition is commutative155
- The dot product is the product of magnitudes times cos156
- If the dot product is zero, the vectors are perpendicular157
- The volume of a sphere158
- The surface area of a sphere159
- There are only five regular polyhedra195
- Without the parallel postulate there is no Pythagoras196
- The ratio of circumference to diameter is a single number201
Uncertainty
- nCr = nPr / r!60
- The binomial theorem61
- The pigeonhole principle62
- Bayes' theorem63
- The product rule and the sum rule of counting160
- Probability lies between 0 and 1161
- The probability of the complement is one minus the original162
- The addition rule for mutually exclusive events163
- The definition of conditional probability shrinks the sample space164
- Expectations add even without independence165
- The expected value may never actually occur166
- The birthday problem is counted by the complement181
- The minimum number of moves for the Tower of Hanoi is 2ⁿ−1182
- The rice on the chessboard is a geometric series183
- An Eulerian trail requires 0 or 2 odd vertices184
- If the number of divisors is odd, the number is a square185
- With many repetitions the average approaches the expected value197
- Even an accurate test is often wrong before the base rate198
Information
- The mean makes the deviations sum to zero64
- Why variance squares the deviations65
- Why squares rather than absolute values66
- Correlation is not causation67
- Matrix multiplication is function composition68
- AB ≠ BA69
- The median is not swayed by extreme values167
- When the mean, median, and mode diverge168
- Changing the bin width changes the shape169
- The ambiguity in defining quartiles170
- The standard deviation returns to the original unit171
- The correlation coefficient lies between −1 and 1172
- Standardization makes comparison possible173
- What a confidence interval says and does not say174
- Moving against the gradient decreases the value175
- A large learning rate diverges176
- Backpropagation is the chain rule177
- Without activation functions, stacked layers are still linear178
- A matrix is invertible only if its determinant is nonzero179
- The determinant is the scaling factor of area180
- The sample variance divides by n−1199
- The more you add, the closer to the normal distribution200