# MATHEMATICAL OBJECTS #
- Real Number -
A real number is an assigned value, with a sign, digits, & decimal places. A real number with no decimal places is an integer.
Examples: -3990, -5, -1.1, 0, 0.5, 1, 14, 41, 237.332, 8343902
- Variable -
A variable is a symbol or letter, used to express an unknown real or integer value.
Examples: n, x, y, z, a, b, c, d
- Expression -
An expression is a string of operations, numbers, or variables, showing an algebraic form.
_
Examples: 2 + 5, 67x, x² + x - 1, 5 * 34, 1/√2
- Equation -
An equation is a statement to show that 2 expressions have the same value.
Examples: 5 + 4 = 9, 2x = 5, 3.4a² - 2a + 0.66 = 0, 4x + 2 = -2x - 7
- Set -
A set is an unordered collection of mathematical objects, with unique values.
Examples: {}, {3, 7, 8}, {x, y, z}, {1, 2, 3, 4, 5, 6, 7}
- Function -
A function takes an input value & returns an output value. Many input values can share an output, but not the other way around.
Common function names are f(x) =, a(n) =, y =, but we can use any variables.
Example: f(x) = 2x + 3
f(0) = 3, f(1) = 5, f(2) = 7, etc.
- Data Set -
A data set is a collection of numbers, like a Set, but with key differences:
* A data set describes only numbers.
* The order of the numbers matter.
* Repeat terms are allowed.
* Data sets have to be finite.
Examples: [4, 5, 6, 1, 4], [14.2, 16.4, 11.9, 15.2, 18.5, 22.1], [1, 2, 3]
# BASIC MATH #
- Plus (+) -
The plus sign is commonly used to represent the addition of 2 numbers or expressions.
Examples: 1 + 1 = 2, -73 + 0.5 = -72.5
It can also be used to express a minimum number.
Examples: 50+ means "greater than or equal to 50"
- Minus (-) -
The minus sign is used to represent lots of things, like:
* Subtraction of 2 numbers or expressions.
* Negative numbers.
* A range of 2 values (no space character).
* The difference between 2 sets.
Examples: 4 - 2 = 2, 49 - 0.1 = 48.9
Examples: -193, -35, -3.24, -0.7
Example: 6-7 is any number between 6 & 7 (inclusive)
Example: {3, 5, 7, 12, 13} - {5, 7, 8, 9} = {3, 12, 13}
- Times (*) -
To represent multiplication, you use the asterisk for 2 numbers.
Examples: 7 * 5 = 35, 2 * -0.01 = -0.02
When dealing with variables or expressions, we put the terms next to each other, or we use parentheses.
Examples: 2x, xy, 3(x + 1), (a + b)(a - b)
- Divide (/) -
The division line --- can be used to represent fractions, or dividing 2 numbers or expressions.
5 x
Examples: --- = 1.66666666…, ---
3 y
Alternatively, we can use the slash symbol for inline typing.
Example: 6/3.2 = 1.875
- Parentheses (, ) -
Parentheses are normally used for changing the order of operations (BODMAS), but is also used for multiplication. See above.
Example: 41 + 2 * 8 = 57, but (41 + 2) * 8 = 344
- Root (√) -
A root is used to obtain the nonnegative result for xⁿ = y, where y is the target value, & n is the exponent.
The default root is the square root, but it can be changed using a left superscript.
_ __
Examples: √5 = 2.23606797…, ³√64 = 4
- Exponents -
An exponent is a right superscript of a number or expression, indicating the number of multiplications required to get the result.
Examples: 0⁰ = 1, (-4.5)² = 20.25, 10⁻⁶ = 0.000001
- Percents (%) -
A percent is a common way to write "out of 100". Has many uses in statistics or increase & decrease.
Example: 23% is 0.23, which means 0.23x out of x, where x is any value.
Example: 6 + 2% = 6 * 1.02 = 6.12
- Ratios -
A ratio is a way of representing a proportion. The right number should usually be 1.
Example: 5.2 : 1 is 5.2x of something out of every x.
- Money -
While money technically doesn't appear in pure math, it appears very often in adult life with its own vast subject.
It is like a real number, except it only has 2 decimal places for cents. Negative numbers represent debt.
Common currencies include: $ (dollar), € (euro), £ (pound), etc.
Examples: -$500, $0.50, $3, $94.99, $1000000
# ALGEBRA #
- Absolute Value -
The absolute value of a number is the distance of a number from 0, where negative numbers become positive.
Example: |5| = 5, |-3| = 3, |0| = 0
- Inequalities -
These 4 symbols (<, >, ≤, ≥) are used in inequalities.
Less than (<) is strictly smaller, while greater than (>) is strictly bigger.
Less than or equal to (≤) is "no more than", while greater than or equal to (≥) is "no less than".
Examples: 5 > 3, 1.41 ≤ x ≤ 1.43, 1/3 < 0.5
- Infinite Decimals -
3 dots (…) is used to represent real numbers with infinite decimal places.
Examples: 3.1415926…, -9999.8888888…, 7.12345679012…
The expansion 0.4444… is normally assumed to be 4/9.
But it can even be 0.44448975849…, or 0.4444333344443333…, or expand to anything.
It is a good idea to give context, so people know exactly what constant you're talking about.
- Coordinates -
In graphing, we have a 2D coordinate system: (x, y) is the point x units to the right, & y units up.
A negative x is left, while a negative y is down. The origin point is (0, 0).
Examples: (3, 5), (5, 3), (103, 11), (-2, -2), (1.2, 4.303)
- Euler's Number (e) -
Euler's number (e) is a constant with a value of 2.718281828459045…
It is related to exponential growth & many areas in math, such as calculus.
- Logarithms -
A logarithm is the solution to aˣ = b, where x is our desired exponent value.
The default logarithm base is Euler's number (2.718281828…) but it can be changed by a right subscript.
Examples: log(2) = 0.6931471805…, log₁₀(1000) = 3, log₇(10) = 1.1832946624…
- Pi (π) -
The constant pi (π) is equal to 3.141592653589793…
It is defined, geometrically, as a circle's circumference divided by its diameter.
It shows up in many areas of math too, by surprise, almost like the "easter egg" of math.
- Degrees -
The degree symbol ° is used in trigonometry.
Real numbers are radians by default, while 1° is π/180, or 0.017453292519…
360° is one full revolution.
- Trigonometry -
Introducing 3 functions: sin(x), cos(x), tan(x)
A sine is the horizontal position (x degrees clockwise about the coordinate (0, 1)) of the unit circle.
Cosine & tangent are defined as: cos(x) = sin(x + 90°), tan(x) = sin(x)/cos(x)
Examples: sin(30°) = 0.5, tan(1) = 1.55740772…
- Inverse Trigonometry -
Arcsine, arccosine, & arctangent return the inverses of sine, cosine & tangent respectively.
The functions asin(x), acos(x), atan(x) return the original value a where -π ≤ a ≤ π, or -180° ≤ a ≤ 180°.
Example: atan(1) = 45°
# NUMBER THEORY #
- Factorial (!) -
The factorial (!) function is defined where n! = n * (n - 1) * (n - 2) … 4 * 3 * 2 * 1. Nonnegative integers only.
Examples: 0! = 1, 5! = 120, 9! = 362880
- n choose k -
n
The function ( ) is the number of ways to choose k items from a set of n total items. Nonnegative integers only.
k
5
Example: ( ) = 10
3
- Greatest Common Factor -
The greatest common factor between 2 numbers gcf(n, k) is the biggest number that both inputs are divisible by. Positive integers only.
Examples: gcf(7, 4) = 1, gcf(36, 75) = 3
- Least Common Multiple -
The least common multiple between 2 numbers lcm(n, k) is the smallest number that is a multiple of the 2 inputs. Positive integers only.
Example: 6 * 8 = 48, but it can be factored out while still being a multiple: lcm(6, 8) = 24
# SET THEORY #
- Union -
The union between 2 sets (A ∪ B), merges all terms into one set, while leaving no duplicates.
Example: {2, 3, 5, 7, 11} ∪ {1, 4, 7, 10} = {1, 2, 3, 4, 5, 7, 10, 11}
- Intersection -
The intersection between 2 sets (A ∩ B), gives the terms that both sets have in common.
Example: {1, 4, 9, 16, 25, 36} ∩ {2, 4, 6, 8, 10, 12, 14, 16} = {4, 16}
- Difference -
The difference between 2 sets removes all terms from the 2nd set from the 1st. See above in the "Basic Math" section.
- Cardinality -
The cardinality of a set n(A), is just the number of terms inside the set. 'Nuff said.
Examples: n({}) = 0, n({33}) = 1, n({a, b, c, d}) = 4
- Natural Numbers -
The symbol ℕ refers to the set of all nonnegative integers. It is an infinite set.
ℕ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25…}
- Real Numbers -
The symbol ℝ refers to the set of all real numbers. It is an uncountably infinite set, meaning we can not list them all in a "normal" infinite way.
# CALCULUS #
- Sigma Sum -
The sigma symbol Σ is used to write a long, potentially infinite addition expression with finite symbols.
The bottom equation is the starting value, the top number is the value to end at, & the right expression is what we're adding.
10
__
Example: \ n² = 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 380
/_
n = 3
- Capital Pi Product -
The capital Pi symbol Π is used to write a long, potentially infinite multiplication expression with finite symbols.
It has a similar foundation to the Sigma Sum. Not to be confused with the constant π (3.1415926…).
10
__
Example: || 0.5n = 2.5 * 3 * 3.5 * 4 * 4.5 * 5 = 2362.5
||
n = 5
- Infinity (∞) -
Infinity refers to the idea of a value being greater than any real number. It is not a number, but it's a useful tool for math.
∞ > n
-∞ < n
for all values of n
- Limits -
Sometimes you can't reach a defined value (1/0 is undefined), so we may use limits to approach a desired value.
The number we're heading to narrows in from the lesser & greater side of the value.
1
Example: lim --- = -∞ OR ∞
x -> 0 x
x² - 1
Example: lim -------- = 2
x -> 1 x - 1
1
Example: lim 1 - --- = 1
x -> ∞ x
# DATA #
- Count -
The "count" of a data set refers to the length of the data set. Similar idea to cardinalities in Sets.
Example: count([1, 5, 3, 5]) = 4
- Total -
The sum of all numbers in a data set is the "total".
Example: total([6, 3, 5, 7, 8, 11]) = 40
- Mean -
The mean of a data set is the average - it is the same as total(n)/count(n).
Use the mean of a data set when distribution is important.
Example: mean([123, 131, 100, 104, 193]) = 130.2
- Median -
To take the median of a data set, sort the list from lowest to highest & find the middle value.
If the count is even, take the mean (average) of the 2 middles.
It is the more accurate idea of finding a central value. Use if distribution doesn't matter.
Examples: median([4, 5, 9, 13, 20]) = 9, median([11.2, 11.46, 12.01, 12.56]) = 11.735
# CONCLUSION #
x _
The 6 basic operations are +, -, *, ---, ʸ√x, xʸ. Numbers & expressions are the basis of math.
y
All numbers use 14 symbols: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -, ., ---, …}
There are currently 45 definitions in the cook book.
We have many tools in algebra, number theory, set theory, calculus, & more.
Thank you for reading my math cook book, & have fun exploring! - @horizontal_shading