TMSCA High School Number Sense, Calculator, General Math

Largest value of k such that x(C)y = z

(x!)/(y!(x-y)!) = z(Choose highest value)

# of relatively prime integers less than n, where n can be factorized a s (x^a)(y^b)...

(x-1)(y-1)(x^(a-1))(y^(b-1))

sum of integral divisors less than n, if n = (x^a)*(y^b)*...

(x^(a+1) - 1)/(x-1) * (y^(a+1) - 1)/(y-1) * ...

Number in form x0y(205,307,408,etc.) squared

(x^2)(2xy)(y^2)(Adjust # places if needed)

0.xyzyzyzyzyzyz(yz repeating) as fraction

(xyz as number - first digit)/(990)

Maclaurin Series cos(x)

1-x^2/2! + x^4/4! - x^6/6! + ...

How to find all integer sides of a right triangle given one integer leg length n

n^2/2 is between the integer lengths of the other leg and the hypotenuse

MacLaurin Series: e^x

x + x^2/2! + x^3/3! + x^4/4! + ...

If x and y are positive integers and x^2-y^2 = k(where k is positive and odd), then find x and y.

x = k/2 + 0.5; y = k/2 - 0.5(if it doesn't work space out the numbers further)

MacLaurin Series sin(x)

x-x^3/3!+x^5/5! - x^7/7! + ...

a * a/b(mixed number form)

|a-(b-a)|((a-b)^2/b)

Each nth triangular number squared is equal to

∑(k = 1,n+1) k^3

sum of relatively prime numbers less than an integer

# of relatively prime numbers less than the integer * the integer/2

x/(a*b) + x/(b*c) + x/(c*d) + ... + x/(y*z) =

(# of terms * x)/(a*z)

x+(x+d)+(x+2d)+...+L

(F+L) * ((L-F)/d + 1)/2

Mean value of a functiokn

(Integral evaluated from b to a)/(b-a)

Positive Integral Divisors of an integer, x^a * y^b * ..., where x and y and z... are the prime factors of the integer

(a+1) * (b+1)

a/b + b/a =

(a^2+b^2)/ab OR 2 + (a-b)^2/ab

1*1! + 2*2!+3*3!+... + n*n! =

(n+1)! - 1

0.xyyyyyyyyyyyyyy...(repeating y) to decimal form

(number value in numerical form - first digit)/90

894^2(show process)

1. Last digit is last digit of 4^2 = 16 = 6 2. Next digit is last digit of 2(9)(4) + 1 = 73 = 3 3. Next digit is last digit of 2(8)(4)+9^2+7 = 152 = 2 4. Next digit is last digit of 2(8)(9) + 15 = 159 = 9 5. First digit is 8^2 + 15 = 79(or first two) 894^2 = 799236(result)

Square any 3 digit number(xyz)

1. Last digit is last digit of z^2 2. Next digit is last digit of 2yz + carryover 3. Next digit is last digit of 2xz + y^2 + carryover 4. Next digit is last digit of 2xy + carryover 5. First digit is x^2 + carryover

.444444444444444444(base 9) to fraction(base 10)

1/2

Number in form 10x(101,102,103,...) squared

10(2x)0(x^2)

23^3

12167

77th Heptagonal Number

14707

feet per second to miles per hour conversion

15/22

128^2

16384

784/884 + 884/784(mixed number)

2 2500/173889

Shell method(around vertical line)

2*pi*integral((x-line of integration)(f(x)-g(x))^2) from limit a to limit b where f(x) is greater than g(x) in all points between a and b and the x-coordinates of the functions are greater than the x-coordinate of the line of integration. dx

0.48888888888888888 as fraction(8 repeating)

22/45

cubic inches in 1 gallon

231

22^4(*hint: multi-step problem)

234256

5/8 + 10/48 + 1/16 + 5/120 + 10/96 =

25/24

3rd Decagonal Number

27

0.54545454545454545454545454545454 as fraction

27272727272727272727272727272727/50000000000000000000000000000000

The nth pentagonal plus the nth triangular number is

2n^2

8/3 + 3/8( mixed number)

3 1/24

7 * 7/11

3 16/11 = 4 5/11(Simplified)

8 * 8/13

3 25/13 = 4 12/13(Simplified)

What is the difference between the range and the average of the relatively prime numbers less than 75?

34.5

Challenge: Find the sum of integral divisors less than 3645(show process)

3645 = 3 * 1215 = 3 * 3 * 405 = 3 * 3 * 9 * 45 = 3 * 3 * 9 * 9 * 5 = 3 ^2 * 9^2 * 5 = 3^6 * 5(prime factorization) (3^7-1)/(3-1) * (5^2-1)/(5-1) = 2186/2 * 24/6= 1093 * 6= 6558

more practice: 626^2(just answer)

391876

9 * 9/14

4 25/14 or 5 11/14(simplified)

26 + 10 + 50/13 + 250/169 + ... (find the sum of all the terms starting from term 3 of the sequence described above)

6.25

782^2(just answer)

611524

0.764646464646464...(64 repeating) as fraction

757/990

2+5+8+11+14+17+20

77

0.167345673456734567345...(67345 repeating) as fraction

83672/499995

more practice: 992^2(use method outlined in this set)

984064

more practice: 993^2(use alternate method) - similar to 93^2

993^2 = (1000-7)(1000-7) = 1000000 - 14000 + 49 = 986049

Area of circumscribed triangle

A = (abc)/(4r), a, b, and c are 3 sides and r is radius of circle

Evaluate the antiderivative of sin(3+5x)-(-cos(3))+9/x, where the constant of integration is equal to the largest prime number less than the least common multiple of 2 and (3+4)/(7/3).

F(x) = -1/5cos(5x + 3) + cos(3) x + 9 ln(x) + 5

how to find maximum value of function

First derivative test; use second derivative test to check which ones are truly maximums(not just critical points or minimums)

Integral in number sense

Integrate using integration rules and plug in the limits of integration

Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

Roman Numerals

M= 1000 D= 500 C= 100 L= 50 X=10 V= 5 I= 1

Maclaurin Series a(x) + b(x)

Maclaurin a(x) + Maclaurin b(x)

.nnnnnnnnnnnnnnnnnn(base x) to base 10 fraction = n/x + n/x^2 + n/x^3 + ...(sum of infinite geometric series)

Miscellaneous changing bases

What type of function is the triangular number one?

Quadratic(all such number functions are quadratics)

y! = x(mod z), 0 <= x <= n; x = ?

Use Wilson's Theorem to find x: For prime p, (p-1)! = (p-1)mod(p)

Washer method

V = pi(integral) R^2-r^2 dx

Disc method

V= pi (integral) r^2 dx

Infinite Geometric Series sum(a+an+an^2+an^3+...)

a/(1-n)

Triangular, Square, Pentagonal, Hexagonal, ... Numbers

n((second such number-2) * n - (second such number - 4))/2

Finite Arithmetic sequence sum(1+2+3+...+n)

n(n+1)/2

Diagonals of a polygon

n(n-3)/2

If you sum the nth triangular number and the (n-1)th one, what do you get?

n^2