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.
(If the Roman numerals don't appear neatly in your Web browser, then here's the total number of math tricks: 110)
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!
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:
(x is also equal to the number of rows & columns!)
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:
If the remainder in the quotient is 1, then Y = ten "X"-illion.
If the remainder in the quotient is 2, then Y = hundred "X"-illion.
If there's no remainder in the quotient, then Y = "X"-illion.
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 10^{100} - 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:
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:
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:
Another bonus fact!
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:
Example:
√(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:
log_{x}Y ÷ log_{x}Z = log_{z}Y; 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:
(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:
(log_{x}Y)^{-1} = log_{Y}X; neither X nor Y equals 1 or 0.
Yet another bonus fact about logarithms:
log_{10^X}(10^{Y}) = 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:
(3i)^{2} ≠ 3i^{2}; (3i)^{2} = -9 & 3i^{2} = -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 units^{2}!
Here's a bonus fact about the function y = x^{2}:
What those math symbols in T.V. Man's screen are saying is that for the function y = x^{2}, 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/(2^{m-1})!
n = the numerator of your choice, m = the power of 2 of your choice, x = n/2^{m}
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:
|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/(d^{x+1})) ÷ (n/(d^{x})) = 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^{-1}B = (AB^{-1})^{-1} & AB^{-1} = (A^{-1}B)^{-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 = (πD^{2})/4 = πR^{2}
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:
Here's a bonus fact about 2 specific functions: x^{2} & √(x):
x^{2} < x & √(x) when 1 > x > 0
Another bonus fact about logarithms:
log_{x}((x^{w})^{y}) = wy; x is equal to neither 0 nor 1 & must be positive
Here's a fact about prime numbers in fractions:
Here's a fact about differentiation:
And yes, the exclamation point is representing the factorial function!
Here's a fact about function inverses:
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:
Here's another bonus fact about the Law of the Cosines:
(I know that I'm using this image twice on the same Web page, but it's about the exact same formula!)
If Angle C = 360° then the formula simplifies to:
c = b - a or -a + b
...Because cos(360°) = 1. (However, triangles never have angles that large in Euclidean geometry!)
Note: This math fact refers to Math Trick #81: Cunning Cosine!
Speaking of numbers, here are some Web pages with interesting facts about specific numbers:
© Derek Cumberbatch