Some Math Tricks I Discovered

Not only will you find tricks in this section, but also some tips to help you with math, & various information about numbers, logic, & geometry. Look out for my characters here, too; they'll make this section more entertaining & educating!

Here's a bonus cartoon for visiting this section:

If you can multiply big numbers with several digits quickly, then you're a super-genius! By the way, the correct answer to the math problem in the cartoon above is 5,332,114.

This section is divided into quarters. Which one do you want to view?

Please select a trick to view:

  1. The Super Special Function
  2. The Quasi-Sequence
  3. Rhymes in Math
  4. Trigonometry Tricks
  5. Diameters in Terms of Radii
  6. Matching Matrices
  7. Matrix Magic
  8. Average Antics
  9. Factorial Fractions
  10. Complex Numbers Meet the Pythagorean Theorem
  11. Consecutive Integer Addition
  12. Squaring Square Matrices
  13. Delightful Division
  14. Nifty Number Nine & Its Multiples
  15. The Formula for the Circumference of a Spiral
  16. Unit Fraction Uniqueness
  17. Super Summation
  18. The Formula for the Sum of Consecutive Integers From 1 to x
  19. Delightful Division 2: Number of Nines in a Divisor
  20. Trigonometry Tricks 2
  21. Trigonometry Tricks 3
  22. Trigonometry Tricks 4
  23. Dividing Numbers With Consecutive Digits
  24. Magic Cut
  25. Flipping Digits
  26. Roman Numeral Twist
  27. Pythagorean Power: Exponential Excellence
  28. Super Summation 2
  29. Unit Fraction Uniqueness 2
  30. Consecutive Integer Multiplication
  31. Trigonometry Tricks 5
  32. Trigonometry Tricks 6
  33. A Shortcut for Multiplying by 25
  34. Mega Matrix Multiplication
  35. Heavenly Half-Percentages
  36. Cuckoo For Conjugates!
  37. Mega Matrix Multiplication 2
  38. Cubing Square Matrices
  39. Consecutive Digit Slide
  40. Fraction Frantics
  41. Using All 4 Aritmetic Operations on the Same Number
  42. The Square Roots of Cubes
  43. When Square Roots & Absolute Value Match!
  44. Super Summation 3
  45. The Digits in Halves of Even Numbers
  46. Super Subtraction
  47. Logic Checkers
  48. Multiplication Vs. Division By Zero
  49. The Not-So-Naughty Not!
  50. Powerful Patterns: Square Roots
  51. Delightful Division 3: Reciprocals of Half-Integers
  52. Root Repetition
  53. Digit Cloning With Multiplication!
  54. Number Sandwiches
  55. Super Subtraction 2: The Formula for the Difference of Consecutive Integers From 1 to x
  56. Single Simple Statements Can Create Tautologies!
  57. Square Root Squabble
  58. The Staircase of Falsity
  59. Complex Number Kookiness!
  60. Powerful Patterns 2: The Negative Powers of 5
  61. Super Summation 4
  62. Super Summation 5
  63. Unit Fraction Uniqueness 3
  64. Super Summation 6
  65. Complex Number Kookiness 2: Adding Integers to the Imaginary Unit!
  66. Rad Radicals!
  67. Adding 2 Powers of 1 Half
  68. The Staircase of Truth
  69. Square Root Squabble 2
  70. Tricky Transposition!
  71. Super Summation 7
  72. Tricky Transposition 2!
  73. Multiplication Madness!
  74. Powerful Patterns 3: Summation with the Negative Powers of 10
  75. The Powers of Terrific Two!
  76. Super Summation 8
  77. Impossible Equations
  78. The Square Roots of the Powers of 2
  79. Trigonometry Tricks 7: Swapping A & B in a Complex Number
  80. Complex Number Kookiness 3: Dotted Distance
  81. Cunning Cosine!
  82. Cuckoo For Conjugates 2!
  83. Complex Number Kookiness 4
  84. Unit Fraction Uniqueness 4
  85. Rad Radicals 2!
  86. Matrix Magic 2: Squaring Square Matrices with Special Diagonals
  87. Lovely Logarithms!
  88. Super Summation 9
  89. Complex Number Kookiness 5
  90. Complex Number Kookiness 6
  91. Delightful Division 4
  92. Tangent Checkers!
  93. Powerful Patterns 4: Summation Fractions
  94. Square Root Surprise
  95. Perimeter Punch!
  96. Absolute Value Antics!
  97. The Halves of Odd Numbers: A Technique of Simplification
  98. Distance Dot Delight!
  99. Rad Radicals 3: Radical Multiplication!
  100. Logarithmically Lovely Line!
  101. Delicious Differences
  102. Determining Divisibility by Digits
  103. Super Subtraction 3: Changing the Minuend & Subtrahend
  104. Invincible Integration!
  105. Powerful Patterns 5: Fraction Freakiness!
  106. 2 Quick Multiplication Methods!
  107. Square Root Squabble 3 NEW!

Note: If I discover any more math tricks, then I'll add them to this list!

Here's a bonus fact: -40 degrees Farenheit is equal to -40 degrees Celsius!

Proof: The 2 conversion functions intersect at x = -40, y = -40

F = 9/5 × C + 32

C = 5/9(F - 32)

Another bonus fact! This is an interesting fact about the number 2, but you probably already knew it!

2+2 = 2*2 = 2^2 = 4

Another bonus fact!

The reciprocal of a number's square root is equal to the square root divided by the original number! (x is not equal to zero!)

Here's a fact about checkerboards! (Or you can also call them chessboards) You can calculate how many squares a checkerboard has with the 3 formulas below:

If n = the number of rows & columns of a checkerboard, then the total number of squares is equal to:

n2 + (n - 1)2 + (n - 2)2 + ... + 32 + 22 + 12

which is the same as:

(x is also equal to the number of rows & columns!)

which is the same as:

n3/3 + n2/2 + n/6

The 1st formula is in geometric series form, the 2nd one is in summation form and the 3rd is in cubic function form.

Here's a bonus fact about permutations & combinations:

If you pick 2 objects at a time from any set that has at least 2 objects, the permutation-to-combination ratio is always equal to 2. In other words, there will be half as many combinations as permutations.

Here's a bonus fact about the suffix -illion:

What does the suffix mean? Well, here's the -illion function!

Y = 10^(3X + 3)

X = the prefix's number, Y = the actual number

For example, 1 sexagintillion is a 1 followed by 183 zeroes! The prefix sexaginti- means "60"; 3 × 60 + 3 = 183.

Note: The number you pick for the variable x must be a counting number.

Here's the inverse of the -illion function!

X = (log(Y) - 3)/3

Click here to see the functions in Mathprint!

P.S.: About dividing by 3 in the inverse of the -illion function:

For example, the googol is a 1 followed by 100 zeroes, so according to the inverse of the -illion function, it's also 10 duotrigintillion since the prefix duotriginti- means "32" & (log 10100 - 3)/3 = (100 - 3)/3 = 97 ÷ 3 = 32 + 1/3; the remainder is 1 in this case.

Here's a bonus fact about imaginary/complex numbers:

The absolute value of a complex number (x + xi) is equal to x times the square root of 2, if x is greater than or equal to 0.

P.S.: If x is a negative number, then it's -x times the square root of 2.

Here's a bonus fact about triangles:

Let A, B & C be the 3 angles of a triangle. Then...

sin(A + B) = sin(C)

sin(B + C) = sin(A)

sin(C + A) = sin(B)

In other words, the sine of the sum of 2 angles in a triangle will always be equal to the sine of the 3rd remaining angle!

Here's a bonus fact about birth years & age, measured in full years of course:

If the number of your birth year is even, then your age will be even in even-numbered years & odd in odd-numbered years!

If the number of your birth year is odd, then your age will be even in odd-numbered years & odd in even-numbered years!

Another bonus fact!

The Mathematics of Pac-Man™:

After eating a power pellet...

If Pac-Man eats 1 ghost: If Pac-Man eats 2 ghosts: If Pac-Man eats 3 ghosts: If Pac-Man eats all 4 ghosts:
1st Ghost 200 200 200 200
2nd Ghost   400 400 400
3rd Ghost     800 800
4th Ghost       1600
Sums: 200 600 1400 3000

(This is referring to the original version of the video game)

The number of points earned for each ghost refers to this summation formula:

See? Even video games have math in them!

Here's a bonus fact about the Law of the Cosines:

If Angle C = 90°, then the formula simplifies to the original Pythagorean theorem (since the cosine of a right angle is zero); but if Angle C = 180°, then the formula simplifies to:

c = a + b

...Because cos(180°) = -1. (That's right! Side c simply becomes the sum of Sides a & b! However, the figure would be a straight line instead of a triangle!)

Here's a bonus fact that I call my Prime Number Test:

If an integer is not divisible by any prime numbers that are less than its square root, then it is also a prime number!


√(101) = 10.04987562112089...

7 is the last prime number we can pick since the next one, which is 11, is greater than the square root of 101. Therefore, 101 is a prime number!

Here's a bonus fact about fractions:

According to this cartoon, if the absolute value of the difference between the numerator & denominator is 1, then the fraction is already in its simplest terms! By the way, these 2 functions never intersect! Also, they're both discontinuous functions because of division by zero!

Here's a bonus fact about logarithms:

logxY ÷ logxZ = logzY; none of the 3 variables are equal to zero! Neither X nor Z equals 1.

Here's a bonus fact about trigonometry:

tan X = (sin X)/(cos X) = (sec X)/(csc X) = 1/(cot X)


cot X = (cos X)/(sin X) = (csc X)/(sec X) = 1/(tan X)

That's right! These are the classic Tangent & Cotangent Formulae! They both work with secants & cosecants, too! For more details, see the math tricks about trigonometry.

Here's another bonus fact about imaginary/complex numbers:

A complex number's square always has an angle that's twice the size of the 1st angle!

(If neither a nor b are negative, considering a + bi)


(1 + i)2 = 2i; angle(1 + i) = 45°, so angle(2i) = 90° and 45 × 2 = 90

Here's another bonus fact about logarithms:

(logxY)-1 = logYX; neither X nor Y equals 1 or 0.

Yet another bonus fact about logarithms:

log10^X(10Y) = Y/X; X ≠ 0

Note: You can substitute 10 with another number & get the same fraction! (But don't pick 0 or 1)

Here's another bonus fact:

Parentheses matter in math!


(3i)2 ≠ 3i2; (3i)2 = -9 & 3i2 = -3

Here's a bonus fact about right isosceles triangles:

P stands for the perimeter, A stands for the area & x = the length of 1 of the 2 equal sides(also called legs).

Note: The perimeter-to-area ratio will be 1:1 if x = 4 + 2 square roots of 2; the perimeter will be 12 + 8 square roots of 2 units & the area will be 12 + 8 square roots of 2 units2!

Here's a bonus fact about the function y = x2:

What those math symbols in T.V. Man's screen are saying is that for the function y = x2, if x is equal to a fraction that has a power of 2 as the denominator, then the numeric derivative is always equal to n/(2m-1)!

n = the numerator of your choice, m = the power of 2 of your choice, x = n/2m

Here's a bonus fact about integrals:

If B > A, then

The variable C is equal to any real number you want it to be!

Here's a bonus fact about dividing fractions:

A/B ÷ C/D = (C/D ÷ A/B)-1; none of the variables are equal to zero(0)


5/3 ÷ 6/7 = 35/18 and 6/7 ÷ 5/3 = 18/35

Notice how the quotients of the division problems are reciprocals of each other! Also, by the way...

5/3 = 1 + 2/3 and 35/18 = 1 + 17/18

(The numerator isn't always less than the denominator!)

You know that raising a number to the power of -1 gives you the reciprocal, don't you? Whenever negative numbers are exponents, numerators become denominators & vice versa!

By the way, fractions can be printed horizontally or vertically. The numerator is the number on top/to the left of the vinculum & the denominator is the number on the bottom/to the right of the vinculum.

In other words, remember:

X ÷ Y = X/Y

Another thing to remember about division:

Here's another bonus fact about imaginary/complex numbers:

When you multiply or divide complex numbers, the same is done to their absolute values! You also add up the sum of their angles if you're multiplying & if you divide, you'll also subtract their angles!


|1 + i| = √(2); angle(1 + i) = 45°

|2 + i| = √(5); angle(2 + i) ≈ 26.565°

(1 + i)(2 + i) = 1 + 3i; angle(1 + 3i) ≈ 71.565°; |1 + 3i| = √(10)

45° + 26.565° = 71.565° and √(2) • √(5) = √(10)

(1 + i) ÷ (2 + i) = 3/5 + 1/5i; angle(3/5 + 1/5i) ≈ 18.435°; |3/5 + 1/5i| = √(2/5)

45° - 26.565° = 18.435° and √(2) ÷ √(5) = √(2/5)

Here's another bonus fact about division & the powers of denominators of fractions:

Note: Watch the parentheses carefully!

(n/(dx+1)) ÷ (n/(dx)) = 1/d; neither d nor n are equal to zero(0)!

Here's a bonus fact about exponentiation with unit fractions:

Here's a bonus fact about multplying reciprocals:

A-1B = (AB-1)-1 & AB-1 = (A-1B)-1; neither A nor B are equal to zero(0)!

Here's a bonus fact about mixed numbers:

Note: Even if the variable x is not an integer, the 2 functions coincide!

If x = φ or -φ + 1, then so is y!

Here's a bonus fact to remember about negative numbers vs. positive numbers:

If X > Y, then -X < -Y

Here's a pair of bonus facts about trigonometry:

Here's a fact about the formula for the area of a circle:

D • (πR)/2 = (πD2)/4 = πR2

You can multiply the diameter by a quarter of the circumference to get the area! Also, Pi(π) times the diameter squared, divided by 4 equals Pi(π) times the radius squared!

Here's another fact about the formula for the area of a circle:

You can differentiate it to the formula for the circumference!

Here's a fact about the formula for the volume of a sphere:

You can differentiate it to the formula for the surface area!

(This is like the 3-dimensional version of the fact just above this one!)

Here's a fact about linear functions:

The derivative of a linear function is always equal to the slope!

Here's a bonus fact about 2 specific functions: x2 & √(x):

x2 < x & √(x) when 1 > x > 0

Another bonus fact about logarithms:

logx((xw)y) = wy; x is equal to neither 0 nor 1 & must be positive

Here's a fact about prime numbers in fractions:

If the denominator is a prime number & the numerator is less than the denominator, then the fraction is ALREADY in simplest terms!

Here's a fact about differentiation:

(y ≥ 1)

And yes, the exclamation point is representing the factorial function!

Here's a fact about function inverses:

The inverse of n - x is n - x; the value of n doesn't matter!


The inverse of n/x is n/x; the value of n doesn't matter!

Here's a fact about equilateral triangles:

The height of an equilateral triangle is equal to half of the square root of 3 times the length of 1 of the 3 equal sides!

Here's a fact about the square roots of pure imaginary numbers:

(x ≥ 0)

Speaking of numbers, here are some Web pages with interesting facts about specific numbers:

© Derek Cumberbatch