Conjugates & Dividing by Radicals | Purplemath

There’s a lot of math out there. Some things show up all the time on the ACT. Other things don’t. I need to know this information in order to make the questions and question-selection algorithms for Mathchops. So I went through every question from the last 10 ACTs and figured out which skills showed up. Then one of my partners helped me make a Python script and we did a bunch of data analysis. What follows is a list of the 75 most common skills, along with an estimate of how likely they are to appear on your actual test.
Guaranteed To Show Up: These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.
Fractions – All four operations. Mixed numbers.
Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
Percents – Know all basic variations. More advanced ones are common also.
Exponents – All operations. Fractional and negative exponents are very common too (see below).
Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
Picking Numbers – You never have to use this but it will be a useful option on every test.
Ratio – Part:part, part:whole.
Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
Area/Perimeter of basic shapes – Triangles, rectangles, circles.
Negatives – Be comfortable with all operations.
SOHCAHTOA – Every variation of right triangle trig, including word problems.
Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
Extremely Likely (> 80% chance):
Function shifts – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was 6th test score?
MPH – The concept of speed in miles per hour shows up every time.
Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
Radicals – Basic operations. Translate to fractional exponents.
System of Equations – Elimination. Substitution. Word problems.
Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
Time – Hours to minutes, minutes to seconds
Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
Composite function – As in g(f(x)).
Factoring – Mostly the basics. Almost never involves a leading coefficient.
Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
Very Likely (> 50% chance):
Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
Negative exponents – Know what they do and how to combine them with other exponents.
Shaded area – The classic one has a square with a circle inside.
Counting principle – License plate questions.
Logarithms – Rewrite in exponential form. Basic operations.
Imaginary numbers – Powers of i. What is i^2? The complex plane.
LCM – Straight up. In word problems. In algebraic fractions.
FOIL – This has to be automatic.
Worth Knowing (>25% chance):
Ellipses – Know how to graph basic versions.
Scientific notation – Go back and forth between standard and scientific notation. All four operations.
Vectors – Add, subtract, multiply (scalar), i and j notation.
Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
c = product of roots, -b = sum of roots – Use when in x^2 + bx + c form. Usually not required but often helpful.
Difference of two squares – (x + y)(x - y) = x^2 - y^2
Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
Triangle opposite side rule – There is a relationship between an angle and the side across from that angle?
Change the base – If 9^x = 27^5, what is x?
Similar triangles – Relate the sides with a proportion.
Probability with “or” – 3 reds, 5 blue, 6 green. Probability of picking a red or blue?
Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
Factors – The basic concept and greatest common factor, with numbers and variables.
30:60:90 – Know the basic relationships. Sometimes required for advanced trig questions.
Volume of a cylinder – They’ll usually give it to you but not always.
Trapezoid – Usually basic area questions.
Domain – Usually you can think of it as “possible x values”.
Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
Exponential Growth/Decay – Be comfortable with this: Final = Initial(1+/- rate)^time.
Weighted average – Class A has 8 kids and an average of 70. Class B has 12 kids and an average of 94. What is the combined average of the two classes?
Inverse trig – Use right triangle ratios to find angles.
Parallelogram – Know that adjacent angles add to 180. Area formula.
Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
Value/frequency charts – They’ll tell you the value and frequency and then ask about mean or median.
3:4:5 – Recognize 3:4:5 right triangle relationships.
Algebra LCD – Find the lowest common denominator, then combine the numerators.
5:12:13 – Recognize 5:12:13 right triangle relationships.
System of equations with three equations – Usually a word problem. Involves substitution.
Compare numbers – Radicals, fractions, decimals, absolute value.
Translate points – Images, reflections.

There’s a lot of math out there. Some things show up all the time on the ACT. Other things don’t. I need to know this information in order to make the questions and question-selection algorithms for Mathchops. So I went through every question from the last 10 ACTs and figured out which skills showed up. Then one of my partners helped me make a Python script and we did a bunch of data analysis. What follows is a list of the 75 most common skills, along with an estimate of how likely they are to appear on your actual test.
Guaranteed To Show Up: These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.
Fractions – All four operations. Mixed numbers.
Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
Percents – Know all basic variations. More advanced ones are common also.
Exponents – All operations. Fractional and negative exponents are very common too (see below).
Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
Picking Numbers – You never have to use this but it will be a useful option on every test.
Ratio – Part:part, part:whole.
Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
Area/Perimeter of basic shapes – Triangles, rectangles, circles.
Negatives – Be comfortable with all operations.
SOHCAHTOA – Every variation of right triangle trig, including word problems.
Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
Extremely Likely (> 80% chance):
Function shifts – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was 6th test score?
MPH – The concept of speed in miles per hour shows up every time.
Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
Radicals – Basic operations. Translate to fractional exponents.
System of Equations – Elimination. Substitution. Word problems.
Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
Time – Hours to minutes, minutes to seconds
Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
Composite function – As in g(f(x)).
Factoring – Mostly the basics. Almost never involves a leading coefficient.
Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
Very Likely (> 50% chance):
Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
Negative exponents – Know what they do and how to combine them with other exponents.
Shaded area – The classic one has a square with a circle inside.
Counting principle – License plate questions.
Logarithms – Rewrite in exponential form. Basic operations.
Imaginary numbers – Powers of i. What is i^2? The complex plane.
LCM – Straight up. In word problems. In algebraic fractions.
FOIL – This has to be automatic.
Worth Knowing (>25% chance):
Ellipses – Know how to graph basic versions.
Scientific notation – Go back and forth between standard and scientific notation. All four operations.
Vectors – Add, subtract, multiply (scalar), i and j notation.
Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
c = product of roots, -b = sum of roots – Use when in x^2 + bx + c form. Usually not required but often helpful.
Difference of two squares – (x + y)(x - y) = x^2 - y^2
Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
Triangle opposite side rule – There is a relationship between an angle and the side across from that angle?
Change the base – If 9^x = 27^5, what is x?
Similar triangles – Relate the sides with a proportion.
Probability with “or” – 3 reds, 5 blue, 6 green. Probability of picking a red or blue?
Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
Factors – The basic concept and greatest common factor, with numbers and variables.
30:60:90 – Know the basic relationships. Sometimes required for advanced trig questions.
Volume of a cylinder – They’ll usually give it to you but not always.
Trapezoid – Usually basic area questions.
Domain – Usually you can think of it as “possible x values”.
Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
Exponential Growth/Decay – Be comfortable with this: Final = Initial(1+/- rate)^time.
Weighted average – Class A has 8 kids and an average of 70. Class B has 12 kids and an average of 94. What is the combined average of the two classes?
Inverse trig – Use right triangle ratios to find angles.
Parallelogram – Know that adjacent angles add to 180. Area formula.
Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
Value/frequency charts – They’ll tell you the value and frequency and then ask about mean or median.
3:4:5 – Recognize 3:4:5 right triangle relationships.
Algebra LCD – Find the lowest common denominator, then combine the numerators.
5:12:13 – Recognize 5:12:13 right triangle relationships.
System of equations with three equations – Usually a word problem. Involves substitution.
Compare numbers – Radicals, fractions, decimals, absolute value.
Translate points – Images, reflections.

I'm quite frustrated. This thread is a call for help and a bit of a rant. I'm sorry, it's been a really long day with precalc.
​
I need to self-study mathematics up to differential equations. I started from pre-alg (Khan Academy, Prof. Leonard), went through alg1 (Khan Academy), then intermediate alg (Prof. Leonard + the book he was using). I learned things quite well and enjoyed the process overall. I studied a lot and made sure that I got good at everything from linear and quadratic equations, to exponentials, logs, radicals, complex numbers, conjugates. Prof. Leonard's course ended with conic sections.
That's when things got terrible.
Worst thing in self-studying is trying to find a proper study path. I'm in a country that doesn't at all attune to the route of alg1, 2, precalc, calc 1, 2, 3, which, because I don't speak the language, I need to take, then learn everything in time so that I can ace the classes here. The syllabus of the 3 next math classes I need to take is REALLY messed up. They do differential equations and intro to stats and geometry at the same time. They do limits before they do trig. Some things they don't even teach. It's a mess.
But that's okay, as long as I can study effectively, in good tempo, I can cover everything in time to take classes and basically just ace them. I did that until now, with the system I've had.
But when it got to precalculus it got really bad. Every single precalc book I've seen teaches things I've already learned in intermediate algebra. Sometimes they expand the concepts. Sometimes they don't. Halfway through the 2nd chapter there's damn slope-intercept, are you kidding me?
Every precalc book is over a thousand pages too. I didn't know that course was so huge. But oh well. Quite pissed off that I could just start from precalculus and learn everything I know rather than now having to skim, skip, and pick apart each topic asking myself if I know this or not. It just doesn't make sense that people take a class to then take another class and just half of it is revision?
I'm constantly questioning my resources, more than ever now. It's really bad. I'm badly in need of advice and help, perhaps even mentorship or a tutor who would guide me through the process of setting up the right path, choosing a book. I'm not sure what topics in precalc are the most important, and which ones should I just skim through without spacing repetition of exercises, in order to have a shot at acing integrals, derivatives, trigonometry and differential equations. Given it's almost 1500 pages, my flashcards are well into hundreds now and I haven't even reached page 300 yet. I get stuck on things I'm certain I'll never use, and perhaps pay too little attention to things I will use all the time.

Basic Math Symbols

SymbolSymbol NameMeaning / definitionExample=equals signequality5 = 2+35 is equal to 2+3≠not equal signinequality5 ≠ 45 is not equal to 4≈approximately equalapproximationsin(0.01) ≈ 0.01,x ≈ y means x is approximately equal to y>strict inequalitygreater than5 > 45 is greater than 4Download the printable chart here- Basic Math Symbols

2. Algebra Symbols

SymbolSymbol NameMeaning / definitionExamplexx variableunknown value to findwhen 2x = 4, then x = 2≡equivalenceidentical ton/a≜equal by definitionequal by definitionn/a:=equal by definitionequal by definitionn/a~approximately equalweak approximation11 ~ 10≈approximately equalapproximationsin(0.01) ≈ 0.01∝proportional toproportional toy ∝ x when y = kx, k constant∞lemniscateinfinity symboln/a≪much less thanmuch less than1 ≪ 1000000≫much greater thanmuch greater than1000000 ≫ 1( )parenthesescalculate expression inside first2 * (3+5) = 16[ ]bracketscalculate expression inside first[(1+2)*(1+5)] = 18{ }bracessetn/a⌊x⌋floor bracketsrounds number to lower integer⌊4.3⌋ = 4⌈x⌉ceiling bracketsrounds number to upper integer⌈4.3⌉ = 5x!exclamation markfactorial4! = 1*2*3*4 = 24| x |single vertical barabsolute value| -5 | = 5f (x)function of xmaps values of x to f(x)f (x) = 3x+5(f ∘ g)function composition(f ∘ g) (x) = f (g(x))f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)(a,b)open interval(a,b) = {x | a < x < b}x∈ (2,6)[a,b]closed interval[a,b] = {x | a ≤ x ≤ b}x ∈ [2,6]∆deltachange / difference∆t = t1 - t0∆discriminantΔ = b2 - 4acn/a∑sigmasummation - sum of all values in range of series∑ xi= x1+x2+...+xn∑∑sigmadouble summation📷∏capital piproduct - product of all values in range of series∏ xi=x1∙x2∙...∙xnee constant / Euler's numbere = 2.718281828...e = lim (1+1/x)x , x→∞γEuler-Mascheroni constantγ = 0.5772156649...n/aφgolden ratiogolden ratio constantn/aπpi constantπ = 3.141592654...is the ratio between the circumference and diameter of a circlec = π⋅d = 2⋅π⋅rDownload the printable chart here- Algebra Symbols

3. Geometry Symbols

SymbolSymbol NameMeaning / definitionExample∠angleformed by two rays∠ABC = 30°📷measured angle n/a📷ABC = 30°📷spherical angle n/a📷AOB = 30°∟right angle= 90°α = 90°°degree1 turn = 360°α = 60°degdegree1 turn = 360degα = 60deg′primearcminute, 1° = 60′α = 60°59′″double primearcsecond, 1′ = 60″α = 60°59′59″📷lineinfinite line n/aABline segmentline from point A to point B n/a📷rayline that start from point A n/a📷arcarc from point A to point B📷= 60°⊥perpendicularperpendicular lines (90° angle)AC ⊥ BC∥parallelparallel linesAB ∥ CD≅congruent toequivalence of geometric shapes and size∆ABC ≅ ∆XYZ~similaritysame shapes, not same size∆ABC ~ ∆XYZΔtriangletriangle shapeΔABC ≅ ΔBCD|x-y|distancedistance between points x and y| x-y | = 5πpi constantπ = 3.141592654...is the ratio between the circumference and diameter of a circlec = π⋅d = 2⋅π⋅rradradiansradians angle unit360° = 2π radcradiansradians angle unit360° = 2π cgradgradians / gonsgrads angle unit360° = 400 gradggradians / gonsgrads angle unit360° = 400 gDownload the printable chart here- Geometric Symbol

4. Set Theory Symbols

SymbolSymbol NameMeaning / definitionExample{ }seta collection of elementsA = {3,7,9,14}, B = {9,14,28}|such thatso thatA = {x | x∈📷, x<0}A⋂Bintersectionobjects that belong to set A and set BA ⋂ B = {9,14}A⋃Bunionobjects that belong to set A or set BA ⋃ B = {3,7,9,14,28}A⊆BsubsetA is a subset of B. set A is included in set B.{9,14,28} ⊆ {9,14,28}A⊂Bproper subset / strict subsetA is a subset of B, but A is not equal to B.{9,14} ⊂ {9,14,28}A⊄Bnot subsetset A is not a subset of set B{9,66} ⊄ {9,14,28}A⊇BsupersetA is a superset of B. set A includes set B{9,14,28} ⊇ {9,14,28}A⊃Bproper superset / strict supersetA is a superset of B, but B is not equal to A.{9,14,28} ⊃ {9,14}A⊅Bnot supersetset A is not a superset of set B{9,14,28} ⊅ {9,66}2Apower setall subsets of A n/a📷power setall subsets of A n/aA=Bequalityboth sets have the same membersA={3,9,14}, B={3,9,14}, A=BAccomplementall the objects that do not belong to set A n/aA'complementall the objects that do not belong to set A n/aA\Brelative complementobjects that belong to A and not to BA = {3,9,14}, B = {1,2,3}, A \ B = {9,14}A-Brelative complementobjects that belong to A and not to BA = {3,9,14}, B = {1,2,3}, A - B = {9,14}A∆Bsymmetric differenceobjects that belong to A or B but not to their intersectionA = {3,9,14}, B = {1,2,3}, A ∆ B = {1,2,9,14}A⊖Bsymmetric differenceobjects that belong to A or B but not to their intersectionA = {3,9,14}, B = {1,2,3}, A ⊖ B = {1,2,9,14}a∈Aelement of, belongs toset membershipA={3,9,14}, 3 ∈ Ax∉Anot element ofno set membershipA={3,9,14}, 1 ∉ A(a,b)ordered paircollection of 2 elements n/aA×Bcartesian productset of all ordered pairs from A and B n/a|A|cardinalitythe number of elements of set AA={3,9,14}, |A|=3#Acardinalitythe number of elements of set AA={3,9,14}, #A=3📷aleph-nullinfinite cardinality of natural numbers set n/a📷aleph-onecardinality of countable ordinal numbers set n/aØempty setØ = {}A = Ø📷universal setset of all possible values n/a📷0natural numbers / whole numbers set (with zero)📷0 = {0,1,2,3,4,...}0 ∈📷0📷1natural numbers / whole numbers set (without zero)📷1 = {1,2,3,4,5,...}6 ∈📷1📷integer numbers set📷= {...-3,-2,-1,0,1,2,3,...}-6 ∈📷📷rational numbers set📷= {x | x=a/b, a,b∈📷and b≠0}2/6 ∈📷📷real numbers set📷= {x | -∞ < x <∞}6.343434 ∈📷📷complex numbers set📷= {z | z=a+bi, -∞<a<∞, -∞<b<∞}6+2i ∈📷Download the printable chart here- Set Theory Symbols

5. Calculus & Analysis Symbols

SymbolSymbol NameMeaning / definitionExample📷limitlimit value of a function n/aεepsilonrepresents a very small number, near zeroε → 0ee constant / Euler's numbere = 2.718281828...e = lim (1+1/x)x , x→∞y 'derivativederivative - Lagrange's notation(3x3)' = 9x2y ''second derivativederivative of derivative(3x3)'' = 18xy(n)nth derivativen times derivation(3x3)(3) = 18📷derivativederivative - Leibniz's notationd(3x3)/dx = 9x2📷second derivativederivative of derivatived2(3x3)/dx2 = 18x📷nth derivativen times derivation n/a📷time derivativederivative by time - Newton's notation n/a📷time second derivativederivative of derivative n/aDx yderivativederivative - Euler's notation n/aDx2ysecond derivativederivative of derivative n/a📷partial derivative n/a∂(x2+y2)/∂x = 2x∫integralopposite to derivation ∫ f(x)dx∬double integralintegration of function of 2 variables ∫∫ f(x,y)dxdy∭triple integralintegration of function of 3 variables ∫∫∫ f(x,y,z)dxdydz∮closed contour / line integral n/a n/a∯closed surface integral n/a n/a∰closed volume integral n/a n/a[a,b]closed interval[a,b] = {x | a ≤ x ≤ b} n/a(a,b)open interval(a,b) = {x | a < x < b} n/aiimaginary uniti ≡ √-1z = 3 + 2iz*complex conjugatez = a+bi → z*=a-biz\* = 3 + 2izcomplex conjugatez = a+bi → z = a-biz = 3 + 2i∇nabla / delgradient / divergence operator∇f (x,y,z)📷vector n/a n/a📷unit vector n/a n/ax * yconvolutiony(t) = x(t) * h(t) n/a📷Laplace transformF(s) =📷{f (t)} n/a📷Fourier transformX(ω) =📷{f (t)} n/aδdelta function n/a n/a∞lemniscateinfinity symbol n/a

Hi folks.
7 months ago, I started learning French again. I had 10 years of French lessons in English public school as a child in Canada. I'm in my 50's now, and it is finally time for me to become conversational in French. I posted my review of using Assimil: New French with Ease over on learnfrench https://www.reddit.com/learnfrench/comments/fzltsz/my_experience_using_assimil_new_french_with_ease/
Taking this long to go from a false beginner at an A2 level to a B1 level got me thinking of when I was in my mid 20's and had taken Spanish in Guatemala in 1993.
When I joined a friend of my mine on her trip to Central America, she had said she was going for 2 months, but that she was going to start her trip with 1 to 2 weeks in a Spanish school. She was doing this so that she could have a deeper cultural experience while there. I didn't speak a word of Spanish.
When we arrived, in Guatemala, the cost of the school was $100US per week and that gave you 4 hours per day of one on one tutoring, and then room and board with a family. There were cultural excursions that we could go on. These were in a mix of English and Spanish.
I remember my Spanish teacher getting mad at me because I wasn't doing any homework in the evenings. I also remember feeling very frustrated at how slow I was learning. Speaking with the host family was always frustrating but they would make a lot of effort to understand and encourage me.
At the 1.5 week mark, I went out on a walk around the town with my friend plus a new friend we had made at the school. We had lunch and did some shopping, all in Spanish. After a while, the new friend ask me how long I had been speaking Spanish. I said for 10 days. She was astonished that I could speak so well in such as short time. It was at this point that I realized that with a lot of effort, I was getting by in Spanish.
By the end of the second week, my friend and I left the school and started travelling around Guatemala and Honduras. We could make ourselves understood to do shopping and to travel around the country. I could understand when I was spoken to. Two weeks after leaving the school I was having simple conversations with locals. I was only speaking in present tense. Would refer to a time or date in the past or present and then conjugate with the present tense to make the past and future tense. Example (in English). "Next week, we eat dinner". "Yesterday I see a monkey". When other travellers would mention how good my Spanish was, I would say, I only know 200 words, but I know them well. I think this wasn't far off. To locals, I am sure I sounded like a toddler but they understood me. I developed good skills at describing things using the words I knew to make up for my lack of vocabulary.
I think I was able to learn so quickly for the following reasons:
- being immersed in a language for 24 hours per day is very effective for learning
- having a personal tutor for 4 hours per day, gave me lots of practice and lots of feedback on what I was doing incorrectly
- in Central America people spoke clearly and slower than they do in Spain, this made it easy for me to understand them
- I didn't know enough Spanish, to know that I was making errors, so I was never embarrassed
- Since I knew so little Spanish, I never had to pause to search for a word to use, since the list of what I knew was so small
- at that time, not a lot of Guatemalans spoke English, so I was forced to use Spanish to accomplish things in daily life
- for an English speaker, written Spanish is phonetic, so if I can see it, I can say it.
I think this experience learning to quickly speak (very rustic) Spanish sort of cursed me when it came to French. I thought I could go to school for 2 weeks and be conversational in French as well. In fact after spending 3 months in Central America with my friend, we flew to her home province of Quebec where she returned to her college degree in French as a francophone and I signed up for French classes at the community centre.
I was shocked that my French didn't magically become conversational after 3 months there. The problems were
- I was only in a group class for 3 hours per week. half of that time, the professor spoke English
- What French I had was getting mixed up with Spanish words
It has been a long time, but I returned to studying French 7 months ago. I have been shocked how long this takes studying 1 hour per day, but I have stuck with it and it is bearing fruit. I have gone from an A2 to a B1 over that time. I am able to have conversations with my Italki tutors but it takes a lot of work on my part. My goals is to get to a high B1 or a low B2 by July for a 3 week full day immersion group class in French.
After travel opens up again, I would like to go to Paris and take 2 weeks of immersion to give me that final push to speaking more naturally.
Anyhow, I just thought I would share these 2 radically different language experiences that I had. Let me know if you have any question, thoughts or advice.
EDIT: After making this post...and doing some quick math, speaking Spanish for 16 hours per day for 2 weeks is16 x 14 = 224 hours. Study 1 hour of French for 7.5 months is 1 x 7.5 x 30 = 225 hours. The same!

This is a long post on study materials I have used over the past 2 years, and my learning journey. May be of help to someone.
​
My study material: A comprehensive overview
Outline

1. Introduction/objective
   
2. Journey Overview
   
3. Tae Kim Guide
   
4. Ankideck & Kanji
   
5. Other material
   
6. Where to from here?
   

(TLDR's are at the end of each material overview section)
​
1. Introduction/Objective
I first studied Japanese for half a year about six years ago, and learnt Hiragana during this time. However, after that I stopped and did nothing related to Japanese until two years ago. One day I thought I should learn a language, and since then I've been studying everyday so far. I am typing this up to perhaps help other people with their studies, and to also serve as a reminder of what I have gone through. Keep in mind that my goal is probably different from most language learners. If one's goal is to be able to speak and converse in Japanese, my study methodology is most likely not the best course of action. My own goal is to be able to do this, but also be a fluent writer and reader, being able to thoroughly understand the grammar and such as well. Feel free to skip to the relevant sections to hear my experience only on particular material.
​
2. Journey Overview
Once I had decided to dedicate time to learn Japanese, it took some months before I discovered Tae Kim's Guide. During those months, I experimented with different material. This included study of radicals, and (obviously) watching anime. These things seemed futile at the time so once I got to work on the study guide, it was exciting to see the progress. I studies Tae Kim's guide for one year. Of course other material was used, but largely it was this. After completion, I may've mucked around with some other sources, but at this stage I thought it best to expand my vocabulary. I found a great resource with Ankideck, which I just completed. The next sections look exclusively at my experience with these sources, and I will give some suggestions and advice. These should be useful to those learning Japanese.
Note: For all my kanji stroke order and word meanings, the Takoboto Japanese Dictionary has been essential
​
3. Tae Kim Guide
So I began looking through this guide at the start of 2018. Initially, I was looking through the grammar guide, however switching to the complete guide was a good decision.
Tae Kim's Complete Guide is split into different sections that focus on a topic, such as verbs. Within these sections are chapters. My method goes as follows
• Read the chapter
• Write down the chapter (kanji and all)
• Spend time learning most of the kanji and words from that chapter
I made sure to do this most days, and would say that the guide is very useful. It is a great introduction to Japanese, especially for complete beginners. This is because, from an English speaker's perspective, the sentence structure was very confusing. Yet, the guide explains things in an easy to understand manner, all in logical order. The outcome after one year of doing this goes as follows
• A (fairly) solid of grammatical functions such as…
Tenses, conjugations, particles, clauses, te-forms and so-on
• A basic set of vocabulary
• A foundation to branch off into other areas of study
• Good study habits
• A generally better idea of Japan and Japanese culture
Expanding on this last point, I have found that for me, learning Japanese isn't just about increasing vocabulary and understanding grammar. There are important implications, suggestions, and nuances within the language. I feel Tae Kim's Guide explains these well.
I recommend going through Tae Kim's Complete Guide. Depending on your purpose for study, even just reading through and/or taking brief notes is really helpful. The explanations are easy to understand, and I still go back and reread chapters often, even several months after. In addition to the complete guide, the grammar guide and his blog posts are useful reads that I'd also recommend.
TLDR (for this part)
Read the complete guide to gain a solid basis for your language learning. It is useful as a quick read or a more intensive study.
4. Ankideck & Kanji
Learning kanji is, from what I have read online, perhaps an intimidating task. I suggest Ankideck for this, and to expand vocabulary. After completing Tae Kim's Guide, I decided I wanted to increase my vocabulary. This was because I want to be able to read articles, novels, and manga in Japanese as soon as possible.
The deck I used was the core 2000 one. My method goes as follows
• X amount of new cards per day, plus old cards
• For new words, search up in dictionary and write down a few times with correct stroke order
• For old words, only write kanji for difficult or hard to remember words
I wrote the kanji in a maths textbook in the squares. After completing the deck in this way, I know a slew of kanji, and 2007 words with a 97% recollection rate for mature cards.
I started off using the deck at 20 new words per day. This eventually made it too much, so I decreased to no new words a day for a week or so. After, I left it at 10 new words per day. On average, I did about 100 reviews daily.
I recommend using the Ankideck core 2000 (maybe 6000?) as it does an incredible job of increasing vocabulary (and kanji if you choose to write down) with a surprising recollection rate. Each card has a sentence, audio recording, and translation (once flipped). It took 158 days, each study on average took 38 minutes and the average space between words is 2 months.
TLDR
Use the ankideck core cards for a rapid improvement in kanji (written and recognition), reading improvement, and vocabulary enhancement. However, it does require dedication
5. Other material
While I didn't do much other study when going through Tae Kim's Guide, I used a range of other material infrequently as I was using the ankideck.
Benjiro:
Benjiro is a youtuber who has conversations with native Japanese speakers. I listened to one or two videos around the time of starting my learning journey, and understood nothing really. But after finishing the guide, and doing some vocabulary work, I understand about 60 - 70% of the conversation, being able to follow along with topics and so forth. His speaking is slow and easy to understand, plus he does a helpful vocabulary list in conjunction with his video.
I recommend listening to at least a video or two each month, or more if you have the time. They are usually entertaining which helps
Anime
I didn't really use anime a lot. Much less so to improve my Japanese skills. But, after finishing my aforementioned study ventures, I am able to understand a lot more. I recommend watching as much as you can
Raw Manga
Recently, I have been reading a fair bit of raw manga. It requires a bit of effort, makes the story a bit less interesting, and I don't understand as much as I'd like. My method is having two tabs open, one with the raw version and one translated. However, there has been the occasional chapter where I have been able to read completely without any translation. If you have the time, and want to improve reading skills, I recommend this.
(side note on raw manga, my most memorable moments in learning Japanese have come from reading. First example is seeing a kanji I had never studied, but somehow knew the meaning and reading of. The other example is learning the word for foreign trade (貿易) and thinking it was useless. I then read that word later in the day in a Jojo manga)
NHK Easy News
I read these every now and again. These are helpful and would recommend as a supplement to studies.
Podcasts
A few hours have been spent listening to podcasts, where I understood essentially nothing. At my current stage, I would not recommend (unless the speaking is low)
Satori Reader
Highly recommended. Especially for improving reading. I do not do as much of this as I'd like. Satori Reader has short stories in Japanese. These stories are available in audio, translated, and also explanations for many sentences. Great resource, and many articles are free. Download for offline use on your phone, or read on a PC.
6. Where to from here?
I am unsure. For the next month I think I will just read and listen to a lot of Japanese content. Ideally I would find a Japanese acquaintance to practice with, but things don't really work like that. However, after delving into some quality Japanese content, I think revising and exploring other material is useful, perhaps trying out placement tests. Either way, it is a fun journey that, if you have read this far, some may connect with.
Although my goal may be different to the apparent slew of Japanese learners coming in, I think at least trying out these sources is very useful, and in some cases enjoyable.
Since the end of the 2k ankideck was in sight, I wanted to write this. It seems dumb now, and I don't think anyone will actually read the entire thing, but getting my thoughts down could help someone else wanting to learn or progress their studies further.

Please don't hate me for posting this Mods lol.
So, basically I have 5 months until I start college. Recently, I have made it a challenge to myself and my abilities to see how much math I can really handle.
To be honest, I've always been mediocre at math, but math has never ceased to amaze me with all its equations and levels of complexity. And I've also realized that it was mainly by lack of focus, dedication, diligence, patience, and commitment which has deterred my progress in math, not so much my intelligence (so I believe haha).
However, now that I've been reinvigorated to confront the new possibilities that await through the process of learning, I'm ready for a disciplined approach to not just understanding maths, but absorbing and utilizing the concepts of math fundamentally. In doing so, I've laid out a path to follow: from basic Algebra to college level stuff like Calc 3 and Differential Equations. This is the exact order and listing from calcworkshop.com, the resource I'll be making tremendous use of on this adventure. For anyone interested in an intuitive and simple to comprehend way of teaching, I highly suggest you check out the website (albeit it is a paid subscriber platform).
I obviously do not intend to finish this entire monstrosity of a course/program/ultimate math killer bootcamp or even half of it, but the crucial thing is that we try our best everyday in pursuit of some arbitrary (or specific) ,personal (or public) goal. And I don't think I'm naive either in thinking I'll just be able to breeze past difficult topics in just 5 months. On the contrary, I am more so inclined to master the basic fundamentals like algebra, trig, precalc than to half-heartedly tackle obscure problems in Linear Algebra just for the sake of doing "hard shit." I think most of you can agree that in order to do well in math or anything, it is paramount for one to build a strong foundation on which more complexity can be built.
So that is indeed my plan. I'm guessing I'll spend roughly 4 hours a day adhering to this. Currently, I'm on 1 (Algebra) D (Polynomials) and I'm excited to see where I'll end up in the coming weeks and months. I'm also trying to get better at programming as well during all this. So I'm doubly excited for all the challenges that lie at my feet, just waiting for me to snuggle up and devour them.
But why am I posting this to all you mathematical folks on this subreddit? Well, I have to admit, sometimes Reddit really fuels me to take fruitful actions. And by that I mean you guys motivate me a lot. Especially on those self-improvement subs, I see countless of people getting back on their feet after months or years of depressive, suicidal, and chaotic times. In a way, I'm doing this to make myself feel in tune with the potentiality of my existence (lol Jordan Peterson fans where you at?) by confronting something hard, that could possibly be useful, disguised as fear and illusion. It's 3am here, and I'm rather tired after spending the last few hours typing this gargantuan list up. But even if the following guide can help some of you folks who are struggling to find a direction, whether it's in math or in life, then I'm happy.
It's really just fun and games at the end. To be able to sit here and do math in order to educate myself with amazing resources at my disposal, as the world just keeps innovating and progressing (cough singularity is near cough), is a real dream for some folks. And if I could elaborate on this further, I'd say that there is a meaning or purpose to be found implicit in the act of doing something worthwhile, as challenging and as exhausting as it may be. Right between the lines of aptitude and stress is where we are able to flourish and grow.
​
​

• 1. Algebra
  • A. Intro Algebra
    • Order of Operations
    • Real Numbers
    • Translating Algebraic Expressions
    • Algebraic Properties
    • Multiplying and Dividing
    • Combining Like Terms
    • Consecutive Integers
  • B. Solving Equations
    • One Step Equations Addition
    • One Step Equations Multiplications
    • Multi Step Equations
    • Variables on Both Sides
    • Linear Word Problems
    • Literal Equations
    • Absolute Value
    • Linear Inequalities
    • Absolute Value Inequalities
    • Inequality Word Problems
  • C. Exponents
    • Simplifying Exponents
    • Adding Exponents
    • Multiplying Exponents
    • Negative Exponents
    • Scientific Notation
  • D. Polynomials
    • Adding and Subtracting Polynomials
    • Multiplying Polynomials
    • FOIL Method
    • DRT Word Problems
  • E. Factoring
    • Integer Factorization
    • Greatest Common Factor
    • Difference of Squares
    • Trinomial Coefficient One
    • Trinomial Coefficient Not One
    • Factor by Grouping
    • Factoring Cubes
    • Solve by Factoring
    • Factoring Word Problems
  • F. Rationals
    • Rational Expressions
    • Multiplying Fractions
    • Dividing Fractions
    • Adding Fractions
    • Long Division
    • Proportion
    • Solving Equations
    • Percents
    • Graphing Linear Equations
    • Coordinate Plane
    • Finding Intercepts
    • Slope Formula
    • Slope Intercept Form
    • Parallel Perpendicular Lines
    • Point Slope Form
    • Graphing Linear Inequalities
  • G. Systems of Equations
    • Graphing Method
    • Substitution Method
    • Elimination Method
    • System of Inequalities
    • Linear Programming
    • Nonlinear Systems
  • H. Radicals
    • Square Root
    • Rational Exponents
    • Rationalizing
    • Multiplying Radicals
    • Conjugate Math
    • Solving Radical Equations
    • Completing the Square
    • Quadratic Formula
    • Pythagorean Theorem
  • I. Functions & Statistics
    • Relations and Functions
    • Set Notation
    • Linear Function
    • Central Tendency
    • Empirical Rule
    • Z Score
    • Line of Best Fit
    • Permutations and Combinations
    • Probability
• 2. Trigonometry (Pre-calc Part 1)
  • A. Trigonometric Functions
    • Interval Notation
    • Coterminal Angles
    • Reference Triangles
    • Reference Angles
    • Cofunctions
    • Degrees Minutes Seconds
    • Angles of Elevation and Depression
    • Applications of Right Triangles
  • B. Radian Measure
    • Degrees to Radians
    • Arc Length Formula
    • Unit Circle
    • Angular and Linear Velocity
  • C. Graphing Trig Functions
    • Graphing Sine and Cosine
    • Graphing Sine and Cosine with Period Change
    • Graphing Sine and Cosine with Phase Shift
    • Graphing Reciprocal Trig Functions
  • D. Trig Identities
    • Fundamental Trigonometric Identities
    • Steps for Verifying Trig Identities
    • Proving Trig Identities
    • Sum and Difference Identities
    • Double Angle Identities
    • Half Angle Identities
    • Product-Sum Identities
  • E. Trig Equations
    • Inverse Functions
    • Inverse Trig Functions
    • Solving Trigonometric Equations
    • Solving Trig Equations Using Inverses
  • F. Law of Sines and Cosines
    • Law of Sines
    • Law of Sine - Ambiguous Case
    • Law of Cosines
    • Heron’s Formula
  • G. Vector Applications
    • What is a Vector?
    • Displacement Vector
    • Force Vector
    • Velocity Vector
  • H. Polar Equations and Graphs
    • Polar Coordinates
    • Graphing Polar Equations
  • I. Complex Numbers
    • Complex Numbers in Standard Form
    • Complex Numbers in Polar Form
    • De Moivre’s Theorem
• 3. Math Analysis (Pre-calc Part 2)
  • A. Intro Math Analysis
    • Solving Equations
    • Solving Inequalities
    • Absolute Value Equations
  • B. Functions and Graphs
    • Transformations of Functions
    • Piecewise Functions
    • Composite Functions
    • Domain of Composite Functions
  • C. Exponentials and Logarithms
    • Exponential Equations
    • Graphing Exponential Functions
    • Logarithmic Functions
    • Solving Logarithmic Equations
    • Graphing Logarithmic Functions
    • Compound Interest Formula
    • Exponential Growth
  • D. Polynomial Function
    • Quadratic Polynomial
    • Synthetic Division
    • Zeros
    • Finding Zeros
    • Graphing Polynomial Functions
    • Polynomials Functions in Calculus
  • E. Rational Functions
    • Identifying Asymptotes
    • Graphing Rational Functions
    • Variation Equations
    • Rational Functions in Calculus
    • Partial Fractions
  • F. Conic Sections
    • Circle Conics
    • Ellipse Conics
    • Parabola Conics
    • Hyperbola Conics
    • Conics Review
    • Parametric Equations
  • G. Series and Sequences
    • Sequences
    • Geometric Sequences
    • Summation Notation
    • Geometric Series
    • Mathematical Induction
    • Binomial Theorem
• 4. Calculus 1
  • A. Limits
    • Finding Limits Graphically
    • Limit Rules
    • Squeeze Theorem
    • Limits at Infinity
    • Limits Review
    • Epsilon Delta Definition
    • Limits and Continuity
    • Limits Definition of Derivative
  • B. Derivatives
    • Definition of Derivative
    • Power Rule
    • Product Rule
    • Quotient Rule
    • Chain Rule
    • Derivative Rules
    • Derivative of Exponential Function
    • Derivatives of Logarithmic Functions
    • Trig Derivatives
    • Inverse Trig Derivatives
    • Hyperbolic Trig Derivatives
    • Derivative of Inverse Functions
    • Implicit Differentiation
    • Higher Order Derivatives
    • Logarithmic Differentiation
    • L’Hopital’s Rule
    • Particle Motion
    • Rate of Change
    • Equation of Tangent Line
    • Linear Approximation
    • Continuity and Differentiability
    • Derivatives Using Charts
    • Limit Definition of Derivative
    • Newton’s Method
    • Related Rates
  • C. Application of Derivatives
    • Application of Derivatives Lesson 1
    • Application of Derivatives Lesson 2
    • Application of Derivatives Lesson 3
    • Demand Function
    • Elasticity of Demand
  • D. Integrals (Same as Calc 2, A)
    • Riemann Sum
    • Sigma Notation
    • Integration Rules
    • Trig Integrals
    • Fundamental Theorem of Calculus
    • U Substitution
    • Arc Length Formula
    • Mean Value Theorem for Integrals
    • Particle Motion
    • Simpson’s Rule
    • Partial Fraction Decomposition
    • Integration by Parts
    • Advanced Trigonometric Integration
    • Trig Substitution
    • Improper Integrals
• 5. Calculus 2
  • A. Integrals (Same as in Calc 1)
  • B. Application of Integrals
    • Area Between Two Curves
    • Volumes with Known Cross Sections
    • Solids of Revolution - Disk and Washer Method
    • Solids of Revolution - Shell Method
    • Work and Hooke’s Law
    • Moments and Center of Mass
  • C. Differential Equations
    • Separable Differential Equations
    • Slope Fields
    • Exponential Growth and Decay
    • Euler’s Method
    • Logistic Differential Equations
  • D. Polar Functions
    • Polar Coordinates
    • Polar Graphs
    • Tangent Line in Polar Coordinates
    • Area in Polar Coordinates
  • E. Parametric and Vector Functions
    • Vectors Lesson
    • Parametric Lesson
  • F. Sequences and Series
    • Sequences and Series Intro
    • Nth Term Test
    • P Series Test
    • Geometric Series
    • Limit Comparison Test
    • Integral Test
    • Telescoping Series
    • Alternating Series
    • Ratio Test
    • Root Test
    • Sequences and Series Review
    • Radius and Interval of Convergence
    • Power Series
    • Maclaurin and Taylor Series Intro
    • Taylor Series
    • Binomial Series
    • Lagrange Error Bound
• 6. Calculus 3
  • A. Vectors and The Geometry of Space
    • Three-Dimensional Coordinate Systems
    • Vectors in 3D
    • Dot Product in 3D
    • Cross Product in 3D
    • Equations of Lines and Planes
    • Cylinders and Quadric Surfaces
    • Cylindrical and Spherical Coordinates
  • B. Vector Functions
    • Vector Functions and Space Curves
    • Derivatives and Integrals of Vector Functions
    • Arc Length and Reparameterization
    • Curvature
    • Unit Tangent Vectors
    • Motion in Space: Velocity and Acceleration
  • C. Partial Derivatives
    • Functions of Several Variables
    • Limits and Continuity
    • Partial Derivatives
    • Tangent Planes and Linear Approximations
    • Multivariable Chain Rule
    • Directional Derivatives and Gradient Vectors
    • Relative Extrema
    • Absolute Extrema
    • Lagrange Multipliers
  • D. Multiples Integrals
    • Double Integrals over Rectangles
    • Average Value and Double Integral Properties
    • Iterated Integrals
    • Double Integrals over General Regions
    • Double Integrals in Polar Coordinates
    • Applications of Double Integrals: Density, Mass and Moments of Inertia
    • Surface Area
    • Triple Integrals
    • Triple Integrals in Cylindrical Coordinates
    • Triple Integrals in Spherical Coordinates
    • Change of Variables in Multiple Integrals
  • E. Vector Calculus
    • Vector Fields
    • Line Integrals
    • Review of Line Integrals
    • The Fundamental Theorem of Line Integrals
    • Green’s Theorem
    • Curl and Divergence
    • Parametric Surfaces and Their Areas
    • Surface Integrals
    • Stoke’s Theorem
    • The Divergence Theorem
    • Vector Calculus Review
• 7. Linear Algebra
  • A. Linear Equations
    • System of Linear Equations
    • Reduced Row Echelon Form
    • Vector Equations for Matrix Algebra
    • The Matrix Equation Ax=b
    • Solution Sets of Linear Systems
    • Linear Independence
    • Linear Transformations
    • Applications of Linear Systems
  • B. Matrix Algebra
    • Matrix Operations and Determinants
    • Inverse Matrix
    • Invertible Matrix Theorem
    • Partitioned Matrices
    • Matrix Factorization
    • Applications to Computer Graphics
  • C. Determinants
    • Determinant of a Matrix
    • Cramer’s Rule
  • D. Vector Spaces
    • Vector Spaces and Subspaces
    • Null, Column, and Row Spaces
    • Linearly Independent Sets and Bases
    • Coordinate Systems and The Dimensions of a Vector Space
    • Rank
    • Change of Basis
    • Markov Chain Applications
  • E. Eigenvalues and Eigenvectors
    • Eigenvalues and Eigenvectors
    • The Characteristic Equation
    • Diagonalization
    • Eigenvectors and Linear Transformations
    • Complex Eigenvalues
  • F. Orthogonality and Least Squares
    • Inner Product, Length and Orthogonality
    • Orthogonal Sets
    • Orthogonal Projections and Decomposition
    • The Gram-Schmidt Process and QR Factorization
    • Least-Squares Problems
    • Applications to Linear Models: Least-Squares Lines
    • Inner Product Spaces
  • G. Symmetric Matrices and Quadratic Forms
    • Diagonalization of Symmetric Matrices
    • Matrix of the Quadratic Form
• 8. Differential Equations
  • A. Intro to DiffEqs
    • Ordinary Differential Equations
    • Initial Value Problems
  • B. First Order Differential Equations
    • Directional Fields
    • Separable Equations
    • Euler’s Method
    • First Order Linear Differential Equation
    • Solving Exact ODEs
    • Homogeneous Differential Equation
    • Bernoulli Differential Equation
    • Linear and Nonlinear Models
  • C. Second Order Differential Equations
    • Distinct Real Roots
    • Complex Roots
    • Repeated Roots
    • Wronskian
    • Undetermined Coefficients
    • Differential Operators
    • Variation of Parameters
    • Cauchy Euler Equation
    • Predator Prey Models and Electrical Networks
    • Mechanical Vibrations
  • D. Series Solutions
    • Power Series
    • Series Solutions
  • E. Laplace Transform
    • Laplace Transform Overview
    • Inverse Laplace Transforms
    • Initial Value Problems with Laplace Transforms
    • Translation Theorems of Laplace Transforms
  • F. Systems of Differential Equations
    • Homogeneous Linear Systems
    • Phase Plane Portraits

So I'm in my second year of undergraduate and currently I'm taking a generic intro. to chemE course (skims over things like thermodynamics, kinetics, mass balances etc). I like this class for the most part, especially the projects where we solve larger systems/simplified industrial processes.
I'm also taking the first of two semesters of organic chemistry, and it's really not my thing. I'm a pretty math oriented person (took calc II in HS and going for a math minor), and pretty much the first thing our orgo professor told us what that we will be doing absolutely no math in the course. I don't mind chemistry in general, but the course just feels like you're supposed to memorize a dozen reaction mechanisms and recount them on the exam day. Of course they say "understand" the mechanism, don't memorize, but do you really expect us to derive all the mechanisms we might need during a 75 minute exam? When I see epoxidation, ozonalysis, boronation, or free radical chlorination, these terms just mean nothing to me.
Currently I'm also taking an EE class (interest of mine, hope to do electronics process design some day), and the subjects are so much more intuitive for me. I can answer questions in class all the time without looking at the material ahead of time. It is basically an applied math class (linear signals and systems), but even the abstract concepts seem more understandable to me than the fact that something is a strong acid because it has a stable conjugate base because it has a resonance ring which you can see if you turn it 37 degrees and squint your eyes. After I had the two classes today, I really just felt like switching majors. Anyways the real question is this:
TL;DR - I hate organic chemistry, but I like essentially every other part of the chemE curriculum. Will it matter in the long run?

This is kind of a generic post,but I'm hoping you can help me better understand my current math unit. Right now in Pre-Calculus 11, we are doing radicals(specifically multiplying and dividing radicals). I understand the more basic questions such as "Multiply and simplify where possible" 7√54 * 2√6. Here are a few multiplying questions I don't understand : Simplify ~ (2√3 - √10)(√6 - 7√20). Expand and simplify ~ (5√3 - 2)² and 2(√15 - 3√5)^2. Write the conjugate of each, then multiply each pair 2√8 + √27 and -3√40 + 2√10. Here are some dividing questions I don't understand : Simplify. √96/4√3, 3√200/2√5,√27/10. Simplify by rationalizing the denominator : √32/√18 and 3√500/-√27. I don't understand anything about these questions other than what a conjugate it is and what they mean by rationalizing the denominator. Thank you very much for reading this and hopefully helping, I'm sorry for the wall of text!