What is Harder Than Calculus?

Ah, calculus. The subject that tends to send shivers down the spines of many students—myself included, back in the day. I still remember the first time I encountered derivatives and integrals. I think my brain actually hurt trying to wrap my head around the concept of limits. But here’s the thing: For many people, calculus, while challenging, is far from the hardest thing they’ve ever encountered in their academic careers. So, what’s harder than calculus?

Well, this is where it gets interesting.

Complex Analysis: The "Next Level"

If you’ve survived calculus, and I mean really survived—not just scraped through—you might be ready for something that takes those comfy derivatives and integrals and gives them a nice, mind-bending twist. Enter complex analysis. It’s like calculus on steroids, but with imaginary numbers. Yeah, you read that right. Imaginary numbers are the stars of the show in complex analysis.

In this field, you’ll dive deep into functions that deal with complex numbers, which aren’t just real numbers (like 1, 2, or even -5), but numbers involving the square root of -1, denoted by the symbol “i.” Trust me, these numbers don’t just pop up out of nowhere; they’re part of some seriously abstract and difficult concepts, making them much harder than the “real” calculus that you’re probably familiar with.

Don’t just take my word for it—professors sometimes joke that if you think calculus was hard, wait until you see complex analysis. If you’ve ever taken a course in it, you probably know what I mean. It’s all about understanding functions that go way beyond what you think you know.

Differential Equations: The Real Brain-Teasers

Next on the list: differential equations. Sure, calculus introduces derivatives and integrals, but differential equations take things a step further. Essentially, you're tasked with finding unknown functions that satisfy certain conditions, based on their derivatives. These equations are used everywhere in the real world—from physics to economics—and they can be really tricky to solve.

Imagine you're trying to figure out the behavior of a population of rabbits over time. The solution to this problem involves differential equations that help model the growth rate, accounting for all sorts of external factors like food supply, predation, etc. It’s fascinating, but it’s also one of those things that requires an advanced level of understanding. Just when you think you’ve got it figured out, a new, more complex problem pops up, and you're back to square one.

It’s like this never-ending puzzle. And when you start solving them, you realize that the methods you learned in basic calculus only get you so far. At some point, you’ll need to learn more sophisticated techniques, some of which can make your head spin. It's challenging, and for some, it's downright daunting.

Abstract Algebra: It’s Like Learning a New Language

If you're already breaking a sweat thinking about complex numbers and derivatives, let's introduce abstract algebra into the conversation. This is where math starts to look like something else entirely. Forget the friendly number crunching you’re used to in calculus—abstract algebra is all about the structure of sets and the operations that you can perform on them. It’s like learning a new language, and in some ways, it really is.

You’ll be introduced to groups, rings, fields, and other concepts that sound like they're straight out of a science fiction novel. But the kicker is that these concepts are crucial for higher-level math and theoretical physics. So, while it might seem esoteric at first, it’s foundational for a lot of advanced work in both mathematics and science.

And let's not even get started on Galois theory... once that beast rears its head, you’ll wish you could just go back to calculus. Seriously, if you’re looking for something that’ll make you question your existence as a math lover, abstract algebra might just be your answer.

Topology: The Mind-Bender

Okay, I can hear some of you already: “But what about topology? Isn’t that the real challenge?” Well, yes and no. Topology is often seen as one of the more abstract branches of mathematics—and it’s often mentioned in the same breath as calculus, complex analysis, and abstract algebra when people talk about tough subjects.

In topology, you’re dealing with spaces and shapes that can be deformed and stretched, but not torn. The concept of “continuous deformation” is key here, and it's harder to wrap your head around than it sounds. It’s like trying to figure out how you can turn a donut into a coffee mug without cutting or gluing anything. Mind-boggling, right? And that's just the tip of the iceberg.

Topology’s beauty comes from its generality and its ability to describe objects in ways that are completely different from what you might be used to in the more familiar world of Euclidean geometry. But, boy, it can be tough to get your head around.

So, What’s the Verdict?

Now, I know what you're thinking: "Okay, I get it. There are a lot of things harder than calculus, but is it all worth it?" Well, it depends on what you're after. If you’re fascinated by abstract concepts and enjoy challenging your mind, then these subjects can be incredibly rewarding. But, if you’re more into practical applications, then maybe you’ll want to stick with calculus for a while longer.

Ultimately, the hardest subject for you will depend on your interests, your background, and your mathematical inclinations. Personally, I think abstract algebra and topology take the crown in terms of mental gymnastics. But that’s just me—others might find differential equations or complex analysis more difficult.

One thing’s for sure though: No matter which one you tackle, you’ll come out the other side feeling like a mathematical superhero (or at least a very, very tired one).

So, are you up for the challenge? I can almost guarantee that if you survive calculus, these next levels of math will be tough, but you’ll feel a massive sense of accomplishment once you start to understand them. Or at least, you'll probably appreciate the simplicity of calculus once you're done.

Is 165 cm normal for a 15 year old?

The predicted height for a female, based on your parents heights, is 155 to 165cm. Most 15 year old girls are nearly done growing. I was too. It's a very normal height for a girl.

Is 160 cm too tall for a 12 year old?

How Tall Should a 12 Year Old Be? We can only speak to national average heights here in North America, whereby, a 12 year old girl would be between 137 cm to 162 cm tall (4-1/2 to 5-1/3 feet). A 12 year old boy should be between 137 cm to 160 cm tall (4-1/2 to 5-1/4 feet).

How tall is a average 15 year old?

Average Height to Weight for Teenage Boys - 13 to 20 Years

How to get taller at 18?

Staying physically active is even more essential from childhood to grow and improve overall health. But taking it up even in adulthood can help you add a few inches to your height. Strength-building exercises, yoga, jumping rope, and biking all can help to increase your flexibility and grow a few inches taller.

Is 5.7 a good height for a 15 year old boy?

Generally speaking, the average height for 15 year olds girls is 62.9 inches (or 159.7 cm). On the other hand, teen boys at the age of 15 have a much higher average height, which is 67.0 inches (or 170.1 cm).

Can you grow between 16 and 18?

Most girls stop growing taller by age 14 or 15. However, after their early teenage growth spurt, boys continue gaining height at a gradual pace until around 18. Note that some kids will stop growing earlier and others may keep growing a year or two more.

Can you grow 1 cm after 17?

Even with a healthy diet, most people's height won't increase after age 18 to 20. The graph below shows the rate of growth from birth to age 20. As you can see, the growth lines fall to zero between ages 18 and 20 ( 7 , 8 ). The reason why your height stops increasing is your bones, specifically your growth plates.