Math is hard.
That may sound strange coming from an astrophysicist with a degree in math, but in some ways, it’s very true. Yes, it gets easier with practice, but only up to a point. I can attest from experience that some parts of calculus are still hard, even when you’re learning differential equations. Some parts of algebra are still hard, even when you’re learning calculus–indeed, it’s not uncommon for the algebra to be harder than the calculus.
And no matter how much math you know, one of the hardest parts is still keeping your plus and minus signs straight. Case in point: a recently-discovered document penned by Albert Einstein in 1931. The section above shows two equations that should be well-known to anyone who has studied cosmology, describing the shape and expansion of the universe. but in the top one, he initially wrote some number–it’s hard to tell which one, but it was a positive number–and had to write a -3 over top of it. And that’s not all. About 20% of Einstein’s papers include various mathematical mistakes. It turns out that even being Albert Einstein does not make you immune to the dangers of plus and minus signs.
The interesting thing is that Einstein’s entire idea behind this draft paper was wrong. By 1931, we already knew that the universe was expanding and therefore had a beginning. But Einstein didn’t like the idea of a beginning. He preferred the idea, which was current in his younger years, that the universe had always been here. He preferred this idea so much that when he realized his theory of general relativity predicted an expanding universe, he added a cosmological constant in 1917 to make his steady-state model work. He later called this move his “greatest blunder”, but, ironically, it turned out to be right for a different reason. We need the cosmological constant to explain why the universe is expanding faster than it should–you may also know of it as dark energy.
But in 1931, Einstein tried another approach to make the steady-state model work. He suggested that the universe was, indeed, expanding, but that new matter was always being created to fill in the space and keep everything from getting too spread out, the same idea that was proposed by astronomer Fred Hoyle, to much criticism, 20 years later. Einstein thought his equations could be made to work like this as is, but it turned out that they couldn’t. Amazingly, it was that same mistake with the minus sign that led him to realize that his idea didn’t fit his equations and wouldn’t work after all. The draft paper was abandoned long before it could get a chance to be published.
So I guess the moral is, don’t worry if the math is hard. Just remember to check your work.