Operations of Math
Overview
Operations of mathmatics are the main processes of math applied to a number or multiple numbers to produce new value(s). The fundamental operations of math are addition (the sum or total of 2+ numbers), subtraction (taking away a certain amount from another), multiplication (adding a number to itself multiple times, having multiple sets of a certain amount), and division (separating a quantity into a certain amount of equal sized groups).
There is an order of operations which states which operation to do first in a problem, known as PEMDAS (Partheses, Exponents, Multiplication & Division, then Addition & Subtraction) invented to create an objective way to solve equations (as doing operations in different orders will usually lead to different answers).
Exponents (^): Multiplying a number (the base, number on the bottom) by itself by a certain amount of times (the exponent, the small number on top)
Example: 5^4 = 5 * 5 * 5 * 5 = 625, 10^7 = 10 * 10 * 10 * 10 * 10 * 10 * 10 = 10,000,000
Logarithms (log()): One of two inverse of exponents (like how divison is to multiplication), finding out the number of times to multiply a number (the base) by itself to get another number (the argument)
Example: log_3(81) = 4, because you have to multiply 3 (the base) by itself 4 times in order to get the value of 81 (the argument) (3 * 3 * 3 * 3 = 81)
Factorials (!): Multiplying a number by all natural numbers (non fraction/decmials like 1, 2, 3, that are greater than zero) that come before it. (n! = n * (n-1) * (n - 2) * (n - 3)... * 2 * 1)
Example: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
Roots/Radicals (√): Basically fractional exponents and the other inverse of exponents, (e.g. 16 ^ (1/2)) with the opposite step of figuring out exponents, since we have to find out what number multiplied by itself a certain amount of times (the index/root/degree, number in the top left) will result in the radicand (number in the radical symbol)
Example: √(25) = 5, because 5 multplied by itself twice (5 * 5) is equal to 25 (radicals have a root of 2 by default as square roots), ∜(16) = 2, because 2 multplied by itself 4 times is 16 (2 * 2 * 2 * 2)
(Doing roots that aren't odd numbers of negative numbers will result in imaginary or complex numbers)
Modulo (mod): Finding the remainder of a division problem (finding out the value of when we subtract a certain amount (divisor/modulus) from a number (dividend) until we can't anymore without making the value negative)
Example: 26 (mod 7) = 5, because the remainder of 26 / 7 is 5 (26 - 7 = 19, 19 - 7 = 12, 12 - 7 = 5, and we can't subtract 7 from 5 without making it negative, so the answer is 5)
Absolute Value (| |) Basically how far a number is away from zero on the number line. Absolute values of positive numbers do not change, (e.g. the number 27 is 27 away from zero), but absolute values of negative numbers do change, bascially just removing the negative sign and making it its positive counterpart, since two things cannot be a negative distance away from each other (-39 is 39 away from zero). For imaginary numbers, it's basically the same (|4i| = 4), and complex numbers use the py
Example: |43| = 43, positive numbers do not change, |-49| = 49, negative numbers do change, |39 - 198| = |-159| = 150
Trigonometric Operations (sin(), cos(), tan()) The core functions of trigonometry based on the right triangle (triangle with a 90° angle) and its properties. As seen on the Concepts + Formulas page .
Sine: The sine of an angle (e.g. sin(47°)) is the length of the side opposite of the angle (side that doesn't touch it, "perp" on the C + F page) divided by length of the hypotenuse (longest side, (hyp)).
Cosine: The cosine of an angle is the length of the adjacent side of the angle (shorter side that touches the angle, "base") divided by the length of the hypotenuse.
Tangent: The tangent of an angle is just the length of the opposite side of the right triangle divided by the lenght of the adjacent, or the sine of the same angle divided by its cosine. (tan(x) = sin(x)/cos(x) or opp/adj)
There are also secant, cosecant, and cotangent. The secant of an angle is just 1 divided by the cosine of an angle, and vice versa (sec(x) = 1/cos(x), cos(x) = 1/sec(x)), cosecant is the same with sine (csc(x) = 1/sin(x)), and same with cotangent (cot(x) = 1/tan(x)).
Inverse functions for all of these exist as well, like arcsin(x) (AKA sin⁻¹(x)), which is basically the reverse of sin(x), figuring out the angle measure based what the opposite divided by hypotenuse is (sin⁻¹(sin(x)) = x, sin(sin⁻¹(x)) = x), same with arccos(x)/cos⁻¹(x) and arctan(x)/tan⁻¹(x).
Doing exponents of these operations involves writing the exponent on the function's word rather than the number or whole, (sin(x) * sin(x) = sin²(x), cos(x) * cos(x) * cos(x) * cos(x) = cos⁴(x)).
Note: This does not work with the negative one exponent (⁻¹) as it clashes with the inverse/reverse of the function (as seen above) and can just be subsituted with the co-functions (csc(x) for sin(x) to the -1st power, sec(x) for cos(x) to the -1st power, and cot(x) for tan(x) to the -1st power) since a number to the negative 1st power is basically just 1 divded by said number (n⁻¹ = 1/n).
Example: 5^4 = 5 * 5 * 5 * 5 = 625, 10^7 = 10 * 10 * 10 * 10 * 10 * 10 * 10 = 10,000,000
Logarithms (log()): One of two inverse of exponents (like how divison is to multiplication), finding out the number of times to multiply a number (the base) by itself to get another number (the argument)
Example: log_3(81) = 4, because you have to multiply 3 (the base) by itself 4 times in order to get the value of 81 (the argument) (3 * 3 * 3 * 3 = 81)
Factorials (!): Multiplying a number by all natural numbers (non fraction/decmials like 1, 2, 3, that are greater than zero) that come before it. (n! = n * (n-1) * (n - 2) * (n - 3)... * 2 * 1)
Example: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
Roots/Radicals (√): Basically fractional exponents and the other inverse of exponents, (e.g. 16 ^ (1/2)) with the opposite step of figuring out exponents, since we have to find out what number multiplied by itself a certain amount of times (the index/root/degree, number in the top left) will result in the radicand (number in the radical symbol)
Example: √(25) = 5, because 5 multplied by itself twice (5 * 5) is equal to 25 (radicals have a root of 2 by default as square roots), ∜(16) = 2, because 2 multplied by itself 4 times is 16 (2 * 2 * 2 * 2)
(Doing roots that aren't odd numbers of negative numbers will result in imaginary or complex numbers)
Modulo (mod): Finding the remainder of a division problem (finding out the value of when we subtract a certain amount (divisor/modulus) from a number (dividend) until we can't anymore without making the value negative)
Example: 26 (mod 7) = 5, because the remainder of 26 / 7 is 5 (26 - 7 = 19, 19 - 7 = 12, 12 - 7 = 5, and we can't subtract 7 from 5 without making it negative, so the answer is 5)
Absolute Value (| |) Basically how far a number is away from zero on the number line. Absolute values of positive numbers do not change, (e.g. the number 27 is 27 away from zero), but absolute values of negative numbers do change, bascially just removing the negative sign and making it its positive counterpart, since two things cannot be a negative distance away from each other (-39 is 39 away from zero). For imaginary numbers, it's basically the same (|4i| = 4), and complex numbers use the py
Example: |43| = 43, positive numbers do not change, |-49| = 49, negative numbers do change, |39 - 198| = |-159| = 150
Trigonometric Operations (sin(), cos(), tan()) The core functions of trigonometry based on the right triangle (triangle with a 90° angle) and its properties. As seen on the Concepts + Formulas page .
Sine: The sine of an angle (e.g. sin(47°)) is the length of the side opposite of the angle (side that doesn't touch it, "perp" on the C + F page) divided by length of the hypotenuse (longest side, (hyp)).
Cosine: The cosine of an angle is the length of the adjacent side of the angle (shorter side that touches the angle, "base") divided by the length of the hypotenuse.
Tangent: The tangent of an angle is just the length of the opposite side of the right triangle divided by the lenght of the adjacent, or the sine of the same angle divided by its cosine. (tan(x) = sin(x)/cos(x) or opp/adj)
There are also secant, cosecant, and cotangent. The secant of an angle is just 1 divided by the cosine of an angle, and vice versa (sec(x) = 1/cos(x), cos(x) = 1/sec(x)), cosecant is the same with sine (csc(x) = 1/sin(x)), and same with cotangent (cot(x) = 1/tan(x)).
Inverse functions for all of these exist as well, like arcsin(x) (AKA sin⁻¹(x)), which is basically the reverse of sin(x), figuring out the angle measure based what the opposite divided by hypotenuse is (sin⁻¹(sin(x)) = x, sin(sin⁻¹(x)) = x), same with arccos(x)/cos⁻¹(x) and arctan(x)/tan⁻¹(x).
Doing exponents of these operations involves writing the exponent on the function's word rather than the number or whole, (sin(x) * sin(x) = sin²(x), cos(x) * cos(x) * cos(x) * cos(x) = cos⁴(x)).
Note: This does not work with the negative one exponent (⁻¹) as it clashes with the inverse/reverse of the function (as seen above) and can just be subsituted with the co-functions (csc(x) for sin(x) to the -1st power, sec(x) for cos(x) to the -1st power, and cot(x) for tan(x) to the -1st power) since a number to the negative 1st power is basically just 1 divded by said number (n⁻¹ = 1/n).