Alolan_Apples
“Assorted” Collector
So I'm finally learning about the Taylor Polynomials in Calculus II, which leads to the worst part about Calculus, the Taylor Series. Strangely enough, I have a better understanding of that than I do on Limits (which is the first thing you'll learn in Calculus in general). Due to my good math skills, it will take me a while to reach the climax of mathematics.
But when does math start to get more complicated? The truth is, every concept is hard when you're first introduced to it, and it's not just limited to math. But there is a point where it begins to get harder. It's not the Taylor Series or other related subjects in Calculus. It was already hard at this point. It's also not the different integration techniques. Integration by all means is harder than differentiaton because not every function can be integrated, the same product, quotient, and chain rules used on differentiation don't apply to integration, and every function has an infinite number of antiderivatives of the same family due to the constant rule in differentiation. But even differentiation is pretty difficult as well. If you never seen derivatives, they do not make any sense at first. However, that's not when it starts to get harder. Nor is it limits. Nor is it Trigonometry. Nor is it the more advanced algebraic concepts. In fact, it goes even further than Algebra.
The correct answer is multiplication. Math begins with counting. I mean, counting small numbers (1 to 100). Then we go to addition, which involves simple counting. You can count how many you have on hand, how much you are given, and how much is in total. Once you memorize simple addition very well, you could start adding large numbers, or even add more than two numbers at once. Then there's subtraction, which at first involves simple counting as well. You can count how many you have on hand, how much is being taken away, and how much you have left. Once you memorize simple subtraction very well, you could start subtracting large numbers. Also early on in math (aside to carrying over and borrowing), you will see comparing numbers, graphs and charts, units of measurement, ratios, and even fractions. But when you get to multiplication, that's when it starts to get harder, just like how the first major event in a book or movie basically starts the plot.
The reason why multiplication is the first intensification is because it's piling another concept on top of each other. You would've handled large number addition and multiple addition at this point, but you have to memorize adding the same number over and over again. Prior to multiplication, each operator starts with counting number of items at start and number of items after a change. Multiplication doesn't begin this way. However, if you want to see counting in it, one number could be how many there are in one group, and the other number is how many groups there are if all groups had the same amount. The total is the product. Using this method, you'll also know that any number times 0 is 0, as any number times 1 is the same thing. Multiplying by 2 is easy because you're adding two numbers of the same. Multiplying by 10 is also easy because each time you do so, you add another digit by putting another 0 to the back. Multiplying by 5 is easy as well since every integer times 5 ends with a 5 or a 0, depending on if it's odd or even. But every other number, it's more difficult to do, even if you multiply two single-digit numbers. And then it gets much worse when you start multiplying two multi-digit numbers.
Multiplication also opens Pandora's Box because there is a common rule in math where every operation has an opposite (like subtraction to addition). And in that case, it would be division. It's much harder than multiplication because not only it's the inverse of an operator that doesn't involve simple counting, but also not every number is evenly divisible by the same numbers. We see the first domain restriction (aka division by zero), some dividends are prime numbers, and using long division, we have remainders. This creates a new concept - factoring. What't even worse than multiplication and division is exponents (which is basically the same as multiplication, but involving multiplication instead of addition). And then we have roots, which is even worse than division. As we go further, we start seeing more types of numbers and not just operators. Decimals, fractions, percentages, and negative numbers. Put them all in the bag, math is now much further away than simple counting and highlighting numbers.
The turning point in the progression of math is when we introduce letters to the operations. This is Algebra. The letters don't combine with the numbers, but the letters can mean anything. Math has already started to get harder beforehand, but this is the point when people start giving up in math, wich is why I called it the turning point. In fact, not everybody knows Algebra by the time they leave high school. We should all know Basic Math before starting high school, but not everybody will know Algebra after high school or even after college. Some people suggest that Algebra shouldn't be taught in high school since "it's too hard". To make matters worse, some universities are considering dropping math requirements from core requirements. Basically, they're trying to get away with requiring learning Algebra. You could blame society for becoming more math-illiterate, but whether or not that's the case, Algebra is actually that hard. Some people can't even do elementary Algebra (solving linear equations, quadratic polynomials, graphs of linear and quadratic functions) no matter how old they are.
I may understand Integral Calculus very well, but even I think multiplication is when math starts to get hard while Algebra is the turning point. Based on level of difficulty, Algebra is further from Basic Math than Calculus is from Algebra. And Calculus is further from Algebra than stuff like Differential Equations and Topology is from Calculus. I was shocked that some people can't even do Algebra in high school or college, but then again, not everybody can meet the same expectations. I don't understand literature very well. Some are just as bad at art as I am to literature. And there are well trained hard workers that work better than others with the same job, but are so bad at video games they can't even get past the first level in Super Mario Odyssey.
But when does math start to get more complicated? The truth is, every concept is hard when you're first introduced to it, and it's not just limited to math. But there is a point where it begins to get harder. It's not the Taylor Series or other related subjects in Calculus. It was already hard at this point. It's also not the different integration techniques. Integration by all means is harder than differentiaton because not every function can be integrated, the same product, quotient, and chain rules used on differentiation don't apply to integration, and every function has an infinite number of antiderivatives of the same family due to the constant rule in differentiation. But even differentiation is pretty difficult as well. If you never seen derivatives, they do not make any sense at first. However, that's not when it starts to get harder. Nor is it limits. Nor is it Trigonometry. Nor is it the more advanced algebraic concepts. In fact, it goes even further than Algebra.
The correct answer is multiplication. Math begins with counting. I mean, counting small numbers (1 to 100). Then we go to addition, which involves simple counting. You can count how many you have on hand, how much you are given, and how much is in total. Once you memorize simple addition very well, you could start adding large numbers, or even add more than two numbers at once. Then there's subtraction, which at first involves simple counting as well. You can count how many you have on hand, how much is being taken away, and how much you have left. Once you memorize simple subtraction very well, you could start subtracting large numbers. Also early on in math (aside to carrying over and borrowing), you will see comparing numbers, graphs and charts, units of measurement, ratios, and even fractions. But when you get to multiplication, that's when it starts to get harder, just like how the first major event in a book or movie basically starts the plot.
The reason why multiplication is the first intensification is because it's piling another concept on top of each other. You would've handled large number addition and multiple addition at this point, but you have to memorize adding the same number over and over again. Prior to multiplication, each operator starts with counting number of items at start and number of items after a change. Multiplication doesn't begin this way. However, if you want to see counting in it, one number could be how many there are in one group, and the other number is how many groups there are if all groups had the same amount. The total is the product. Using this method, you'll also know that any number times 0 is 0, as any number times 1 is the same thing. Multiplying by 2 is easy because you're adding two numbers of the same. Multiplying by 10 is also easy because each time you do so, you add another digit by putting another 0 to the back. Multiplying by 5 is easy as well since every integer times 5 ends with a 5 or a 0, depending on if it's odd or even. But every other number, it's more difficult to do, even if you multiply two single-digit numbers. And then it gets much worse when you start multiplying two multi-digit numbers.
Multiplication also opens Pandora's Box because there is a common rule in math where every operation has an opposite (like subtraction to addition). And in that case, it would be division. It's much harder than multiplication because not only it's the inverse of an operator that doesn't involve simple counting, but also not every number is evenly divisible by the same numbers. We see the first domain restriction (aka division by zero), some dividends are prime numbers, and using long division, we have remainders. This creates a new concept - factoring. What't even worse than multiplication and division is exponents (which is basically the same as multiplication, but involving multiplication instead of addition). And then we have roots, which is even worse than division. As we go further, we start seeing more types of numbers and not just operators. Decimals, fractions, percentages, and negative numbers. Put them all in the bag, math is now much further away than simple counting and highlighting numbers.
The turning point in the progression of math is when we introduce letters to the operations. This is Algebra. The letters don't combine with the numbers, but the letters can mean anything. Math has already started to get harder beforehand, but this is the point when people start giving up in math, wich is why I called it the turning point. In fact, not everybody knows Algebra by the time they leave high school. We should all know Basic Math before starting high school, but not everybody will know Algebra after high school or even after college. Some people suggest that Algebra shouldn't be taught in high school since "it's too hard". To make matters worse, some universities are considering dropping math requirements from core requirements. Basically, they're trying to get away with requiring learning Algebra. You could blame society for becoming more math-illiterate, but whether or not that's the case, Algebra is actually that hard. Some people can't even do elementary Algebra (solving linear equations, quadratic polynomials, graphs of linear and quadratic functions) no matter how old they are.
I may understand Integral Calculus very well, but even I think multiplication is when math starts to get hard while Algebra is the turning point. Based on level of difficulty, Algebra is further from Basic Math than Calculus is from Algebra. And Calculus is further from Algebra than stuff like Differential Equations and Topology is from Calculus. I was shocked that some people can't even do Algebra in high school or college, but then again, not everybody can meet the same expectations. I don't understand literature very well. Some are just as bad at art as I am to literature. And there are well trained hard workers that work better than others with the same job, but are so bad at video games they can't even get past the first level in Super Mario Odyssey.
