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TMSCA General Math Notes
Bases
To Base 10: First, separate the number into digits, multiply the digits by the
powers of the base, and add the products together.
From Base 10: Write powers of the base that are less than the main number
down, divide the main number by the powers of the base, in decreasing
order, and let the remainder of the larger power be divided by the smaller
power, and so on until the number is divided by 1. Then, put the quotients
into one number as place values in decreasing order, from greatest power
quotient.
Scientific Notation
Greater than 10: Move the decimal point based on the number of zeroes
there are in the number, then make the expression,”
a × 10
b
Less than 10: Same thing, but “ 10
− b
(Remember, for both situations, you must move the decimal point until a is
less than 10 but is still greater than or equal to 1.)
Sequences
Arithmetic Sequences: f ( n )= f ( 1 ) × D ( n − 1 )
Geometric Sequences: f ( n )= f ( 1 ) × R
n − 1
(f=first sequence number, n=the number place you are looking for in the
sequence, D=How much each number adds to itself in the sequence for the
next number, R=How many times each number multiplies itself in the
sequence for the next number)
Set Number Sequences: 1+2+3+4+…+20=
f + l
n
Exponential Functions
Formula:
y = m ( b )
x
m= y-intercept, b= growth factor, x= x-value
(Growth rate(c): b+1=c, Decay rate=1-b)
Measurements
Metric: kilo=1000, hecto=100, deca=10, deci=0.1, centi=0.01, milli=0.
Imperial: 1ft=2in, 1mi=1760yd & 640acre, 1yd=3ft, 1p=16floz, 1q=2p,
1g=4q/16 cups, 1lb=16oz, 1T=2000lb
Fahrenheit to Celsius:
° F → ° C =( ℉ − 32 ° ) ×
Celsius to Fahrenheit: ℃ → ℉ =
℃ ×
Square Roots
Simplifying: Put a divisible perfect square multiplied by the number of times
it goes into the root under the radical sign, find the root of the square put it
outside of the radical sign with the other number in it.
Adding & Subtracting: Simplify to same roots, if possible, operate on
coefficients of like roots, keep unlike roots the same, and make an
expression.
Multiplying & Dividing: Multiplying-
√a × √ b
Dividing-
a ÷
b
Rationalizing: First, multiply both the numerator and denominator by the
denominator, then simplify and divide if possible.
Geometry
of diagonals from 1 vertex: n-
of total diagonals formed in a shape:
n ( n − 3 )
Sum of angles measured: ( n − 2 ) × 180
Exterior angle measure: 360 ÷ n
Volume
Prisms: L × W × H
Triangular Prisms:
( L × W × H )
Cubes:
e
3
Sphere:
π r
3
Cylinder: ( π r
2
) × h
Pyramid:
( L × W × H )
Cone:
( π r
2
× h )
Area of Composite Figures
Turn the shape into multiple normal shapes, find the area of each shape, add
the areas together.
Surface Area
Total surface area: Surface area of a 3D figure
Lateral surface area: The surface area of a shape without the area of the
base.
Normal Surface Areas:
Cube-
Total:
6 a
2
Lateral:
4 a
2
Triangular prism-
Total: LSA + 2 ( area of one triangle )
Lateral: perimeter of base ∗ height
Cylinder-
Total:
2 πr ( r + h )
Lateral: 2 πrh
Pyramid-
Total: LSA + Area of the Base
Lateral:
1 / 2 ( Perimeter of base ∗ Slant height )
Special Surface areas:
Cone-
Normal= πr ( r + l )
Lateral= πrl
Sphere-
Total & Lateral=
4 π r
2
Hemisphere-
Total:
2
(lateral=half of the sphere formula)
Trigonometry
Transformations
Translation: The shape moves when reflected, but does not change
Rotation: The shape is moved based on the degree to another quadrant
when reflected
Reflection: The shape flips
Equations, Expressions, and Variables
Parts of an expression: 2x+4y-
Coefficients: 2x, 4y
Constant: -
Terms: 2x, 4y, -
∩ : Union (Cancels out common elements)
Number of subsets*: 2
n
Number of proper subsets: * 2
n
Miscellaneous
Formula for averages between people*:
a ( b )
a + b
( f + l ) n
(f=first number, n= number of numbers,
l=last number)
Mazes: Count every path and mark it with a number
Geometric mean: Multiply numbers instead of adding, find cube root of
product
Mi/hr = ft/sec:
m
h
Sum of 3 consecutive integers:
a − 3
Positive integral divisors*:
Total # of- Do prime factorization, then do equation
( a + 1 ) × ( a + 1 )
Sum-Raise power by 1, do equation
a
b
a − 1
c
d
c − 1
Combinations*:
nCr =
n!
( n − r )! r!
Permutations*:
nPr =
n!
( n − r )!
Largest unattainable sum of 2 numbers*:
mn −( m + n )
(*= need to do further study of topic or formula)
Magic Squares: Every row, every column, and the 2 diagonals of the square
add up to the same number.
Quadratic Factoring Formula: (
a + b
2
= a
2
− 2 ab + b
2
Quadratic Roots Formula: x =
− b ±
√
b
2
− 4 ac
2 a
Inverse:
-Additive inverse: -3>3, or 3>-
-Multiplicative inverse:
Distinct Prime Divisors: Prime numbers that occur during the factorization of
the number
