Math
Math, Maths or Mathematics, is the term used for the science of numbers, whether just for the purpose of discoveries and knowledge of Math (Pure Mathematics) or to be used in other fields (Applied Mathematics).
Counting and Numbers
Numbers are explained best to beginners as how many of an object there are: a farmer may be sad he only has one cow, if someone gave him another cow he would have two cows.
The use of numbers allows people to describe and predict the real world, for science, computers, trade, records and almost any field that needs description.
Whole numbers or naturals (ℕ) can be described by increasing them by one, starting at 0, known as counting. Mathematically, counting is a successor (S(a), where a is a number), and S(a) is a number is a is a number, if S(a) = S(b) then a = b (and other way around), and no natrual number, a, exists such that S(a) = 0.
In base 10, we count with the ones digit (0, 1, 3, 4, 5, 6, 7, 8, 9), but once we have to count from 9, we go to 0 and increase the digit to its left by 1 (if there is no digit, set it to 1). If that digit is at 9, we use another digit (ex. 99, 100). An example is shown below.
Counting Numbers Up To Fourteen
Zero 0
One 1
Two 2
Three 3
Four 4
Five 5
Six 6
Seven 7
Eight 8
Nine 9
Ten 10
Eleven 11
Twelve 12
Thirteen 13
Fourteen 14
There are more numbers, in fact infinitely. If negative numbers are allowed (such n + -n = 0, where n is a natural number), we can “negate” a number and get integers (ℤ). We can use rational numbers (ℚ) by allowing the division of any number by any other number, as long as the 2nd number isn’t 0. Real numbers (ℝ) exist to fill in the “gaps” (ex. sqrt(2)), and complex numbers (ℂ) are defined where i*i = -1. There are more number systems out there, but those are the most common ones.
Mathematical Symbols
There are some symbols that are utilized in Mathematics, such as:
- A=B : The equals sign. This symbol means that the values of the numbers on both sides of it are the same. It is symmetric (A=B implies B=A), reflexive (A=A), and transitive (A=B and A=C implies A=C).
- A≈B : Approximately equals. It is symmetric and reflexive, but not transitive. If A is not used, it is an approximate value.
- A+B : The plus sign. This means that the values of the numbers on either sides are going to be combined. In the natural numbers, A + 0 = B and A + S(B) = S(A+B).
- A-B : The minus sign. This means that the value of one number is going to be removed from the other, or, if A is not used, B is negated (ex. -(5) = -5 and -(-5) = 5).
- A%B : The percent sign. This dictates that the number preceding it is a percentage of something out of 100 or a remainder.
- * or × or ⋅ or AB: The multiplication or "times" sign. This means that the values of the numbers on either sides of it are going to be multiplied together, repeated addition. A and B represent numbers. In the natural numbers, A×0 = 0 and A×S(B) = A×B + A.
- A/B or ÷ : The division sign. This means that the value of the numbers are going to be reduced by how many times one number can be put into the other. Note that any two numbers can be divided (unless the 2nd number is 0), but you will not always get an integer.
- ^ or AB : The exponent sign. This is repeated multiplication. In the natural numbers, A0 = 1 and AS(B) = AB × A.
- √a or B√A : A square root (if b isn’t used), or a bth root if b is used, of a. This is one way to reverse exponents.
- logb(A) or log(A) or ln(A) : A logarithm, which is another way to reverse exponents. logb(a) is log base b, log(a) is log10(a), and ln(a) is loge(a), where e ≈ 2.7182818285…
- ↑ : A knuth up arrow. One up arrow (A↑B) is exponentiation, and adding more up arrows repeats more: A↑↑B is repeated exponents (tetration), a↑↑↑b is repeated tetration (pentation), and so on. It should be of note that a↑CB is a[c up arrows]b, where C is any natural number.
- A∧B or A&B : And. Only true if both A and B are true.
- A∨B or A|B : Or. Only true if at least one of A or B is true.
- ¬A or ~A : Not. Only true if A is false. If ~ is used, may also mean an approximate value.
- A⇒B or A→B : Implies. Only false if A is true and B is false.
- A⇔B or A⟷B : Iff or if and only if. Only true if A is true and B is true or A is false and B is false, like an equals sign.
- ∀A:B : Universal quantifier. For all A, B is true.
- ∃A:B : Existential quantifier. There exists an A such that B is true.
- ∃!A:B : Uniqueness quantifier. There exists exactly 1 A such that B is true.
- A∈B : In. A is in B.
- A∪B : Union. The set containing all elements from A and all elements from B.
- A∩B : Intersection. The set containing all elements in A that are also in B.
Basic Mathematical Functions
Succession
Increases the value of a number by one, also known as counting. After zero comes one.
Addition
Increases the value of one number by another number. Two and three is five, or 2 + 3 = 5.
Addition is equivalent to repeated succession: 2 + 3 = 2 + 1 + 1 + 1 = 5.
Subtraction
Decreases value from a number. Five subtracted from six is one, or 6 – 5 = 1.
Subtraction is equivalent to inverse addition: 2 + 3 = 5, and then 5 – 3 = 2.
Multiplication
Increases value by that many times. Five multiplied by three is fifteen, or 5 * 3 = 15. This is often said as "times", or "five times three", as in "if you take five three times, you have fifteen".
Multiplication is equivalent to repeated addition, 5 * 3 = 5 + 5 + 5 = 15.
Division
Decreases the value by how many times that value can go into it. Ten divided by two is five, or 10 ÷ 2 = 5.
Division is equivalent to inverse multiplication: 5 * 3 = 15, and then 15 ÷ 3 = 5.
Mod
The remainder of a division, loops numbers around.
Exponents
Repeated multiplication.
Advanced Mathematics
Functions
Functions are a part of Mathematics that describe the action(s) preformed on an "input". The result of a function is an "output". Any Mathematical operation done is a form of function, including counting (referred to formally as succession). There are many ways to represent functions, the most common being; Function Notation, Algebraic, Graphical, within a Table, or using Natural Language.
Axioms
Axioms are the most fundamental aspect of Mathematics, being the basic rules that are assumed to be always true. Most separate categories of Math are made distinct by additional or altered axioms being present, the most noticed example being in lower level Physics[1] where additional axioms are added to account for universal Laws and context based simplification.
Algorithms
Algorithms are sets of instructions or rules that allow mathematical problems to be solved, like the result of a hard to calculate function or the value of a constant.
Constants
Closely related to axioms are the less fundamental constants, simply put these are values that are extremely common to the point that they are assigned a Mathematical/Alphabetical symbol.
Variables
Also assigned letters or symbols. Can represent an constant unknown or a number that is not constant.
List of Mathematical Constants
| Symbol | Name | Origin | Example usage | Value (Rounded to 10 digits) |
|---|---|---|---|---|
| π | Pi | ratio of a circle's circumference divided by its diameter | Determining the area of circles using diameter | ~3.1415926535… |
| τ | Tau | ratio of a circles circumference divided by its radius, or 2π | Finding the time it takes radio waves to reach mars | ~6.2831853072 |
| ί | Imaginary Unit | the imaginary number or √-1 | The solution to x² + 4x + 5 = 0 is x = -2 ± i. | i |
| e | Euler's Number | The value that (1 + 1/n)n trends towards | Exponential growth or decay using ex | ~2.718281828 |
| φ | The Golden Ratio | The ratio between two numbers next to each other in the Fibonacci sequence | Finding a Fibonacci number without calculating every previous number | ~1.618033989 |
Useful Approximations
Some mathematical functions are difficult or impossible to calculate by hand, so it is useful to instead use an easier function that will give a value close to the real answer.
| Function | Approximation | Notes |
|---|---|---|
| sin(θ) | θ - θ3/3! + θ5/5! - θ7/7! + … | The more terms that you add, the more accurate the approximation will be for larger numbers. |
| cos(θ) | 1 - θ2/2! + θ4/4! - θ6/6! + … | Same note as above |
Mathematical tools
- An abacus is an analog calculator for basic arithmetic.