r/Algebra

I started with x from 1 to 7.

X^2

I squared the x, yielding 1,4,9,16,25,36,49 I then subtracted each x^2 term from the previous (I.e., 9-4=5, 16-9=7, etc.) I then subtracted those differences a second time (I.e., 5-3=2, 7-5=2, etc.)

On the second time subtraction pass, all the differences were the same. (2)

X^3

I again started with x from 1 to 7, then cubed x. Then I made subtraction passes, much like the above. At the third time subtraction pass, all differences were the same. (6)

X^4

I yet again started with x from 1 to 7, and then raised to the 4th power. At the fourth time subtraction pass, all differences were the same. (24)

I’m wondering about the pattern here: Why does x^2 have all differences the same on the second pass… x^3 has all differences the same on the third pass… x^4 has all differences the same on the fourth pass…

Seems like the exponent is equal to the number of subtraction passes. I imagine if you used, say, x^17, it would take you 17 passes until all the differences are the same?

Why is that?

Solve for m.

-7+4m+10 = 15 - 2m

This stuff is probably super easy im just starting to learn it and really want to get it right as well as get tips and tricks for when it gets harder can anyone help me

I have the algebra 2 regents exam coming in a few weeks, although im able to coast through most tests with 70s I genuinely have no clue what the topics are and how to actually do them. How can I learn the subject in the next few weeks?

My understanding is because the 2 has no variable, when simplifying it stays on its own.

Is the answer 4ab / 2 or...? How can I explain this?

Thanks.

(ALG 1) i have a major test tomorrow that if i dont pass, i will have to either retake over the summer (which is not an option because im going on vacation overseas with my family) or retake algebra next year. i am absolute shit at almost all the last units in algebra (exponents, polynomials, factoring ect) and never learned them properly. would i be able to get at least 50% in this 50 question test just by relying on desmos?

I come from humanities background so maths especially algebra has been considerably tough for me.

I began working on my algebra 20 days ago starting with linear and then quadratic equations. It took me 10 days just to understand the relations of the formulas that exist in quadratic equations and then another 10 days to understand its placement in questions . I followed many lectures went back and forth to understand what i was missing but i still couldn’t solve questions even if i was able to understand why and how the questions were being solved.

After such an immense failure i went to a new chapter,functions this time and boy oh boy was it even messier than quadratics.

If you relate with me and have conquered this problem then let me know how .

for example 37 und 73, they are both prime numbers & 37 mirrored is 73. 37 is the 12th prime number and 73 is the 21st prime number. 12 mirrored is 21.

i‘m looking for another set (or a proof there doesnt exist another pair).

i couldnt find something in the internet. thanks in advance!

Imagine we want to build a theory of reality — not a theory within the universe, like quantum mechanics or general relativity, but a theory of the universe itself, of what it is at its most fundamental level.

It is an outsized ambition, of course. But we can begin with a more modest question: if the universe were truly fundamental — and not a mere consequence of something deeper — what minimal properties would it have to possess? Three, only three, seem unavoidable.

1. The universe must be quantum

Not in the sense of "obeying the Schrödinger and Dirac equations," but in the most basic sense: there must be states (ways things can be), observables (things we can measure), and a notion of positive probability that does not depend on an external structure.

In all known formulations of quantum theory, these three things live comfortably in a Hilbert space — a mathematical object that describes the system from the outside. But if the universe is fundamental, it cannot have an "outside." We must be able to speak of states and observables within the universe itself, without postulating an external Hilbert space.

This is not a technical whim. It is the difference between describing an aquarium and describing the ocean. The aquarium can be represented as a subsystem of the laboratory; the ocean is not inside anything. If we want a theory of the ocean, the mathematics must contain the fish, the waves, and the measurements within the same object.

1. The universe cannot be inside something else

This is the hardest requirement, and also the most fascinating.

If the universe were a subsystem of a larger structure, that larger structure would be the truly fundamental one. We would have to find it, and our theory would be only a provisional description of a piece of reality. To avoid this infinite regress, we require that the fundamental structure cannot be represented as a subsystem of anything.

In mathematical language, this means two things. First, the universe cannot be a sum of independent, non-interacting parts — because then each part would be as fundamental as the whole. Second, and crucially, the universe cannot be a mere algebra of operators on a Hilbert space. Why? Because if it were, the Hilbert space would be the true fundamental arena, and the universe would be just a set of observables within it. The "aquarium" again.

Now something astonishing happens. The mathematics of structures that unify states and observables — the so-called Jordan algebras — was completely classified in 1934 by Jordan, von Neumann, and Wigner. They discovered that there are many possible quantum algebras, but only one cannot be represented as operators on a Hilbert space. It is called the Albert algebra, J3(O), and it is the most eccentric and beautiful object in the theory of non-associative algebras.

The conclusion is as unexpected as it is rigorous: the requirement that the universe not be inside something else literally forces a single possible mathematical structure. It is not an aesthetic choice; it is a classification theorem.

1. The universe is happening

A photograph is not a universe. For there to be physics, the structure describing reality must evolve. But how?

If the evolution were perfectly reversible — a unitary group, as in ordinary quantum mechanics — that would require the existence of an associative algebra where the unitary operators live. But the only structure that survived the first two requirements is precisely non-associative. Therefore, a reversible dynamics is not compatible with an autonomous universe.

What remains? An irreversible process, a Markov semigroup that cannot be extended to a group. A universe that spontaneously forgets its initial conditions and converges toward an internal equilibrium. A universe with an arrow of time that depends neither on an external observer nor on special initial conditions — simply because the mathematical structure admits nothing else.

The arrow of time, in this perspective, is not a mystery. It is a consequence of autonomy.

The fixed point

What we have just walked through is not a philosophical opinion. It is the narrative version of a theorem.

Three simple ideas — the universe is quantum, the universe is not inside something else, the universe is happening — converge to a fixed point. Among all the mathematical structures that could describe a fundamental physical reality, only one simultaneously satisfies the three requirements. That structure is the Albert algebra, J3(O). It has dimension 27, it is deeply non-associative, and it contains within itself the symmetry groups that physicists recognize as those of the Standard Model and of gravity.

The question "why this algebra?" now has an answer that is not "because it works." It is "because the alternatives are subsystems, not universalities."

What to do with this?

This is where the thing becomes truly interesting. If the fundamental structure is fixed by three conceptual requirements, then physics does not begin with an arbitrary choice of algebra. It begins with a choice of principles. The algebra is not the starting point; it is the destination.

And the rest — the particles, the forces, the geometry of spacetime, the mysterious acceleration of galaxies — emerges as a consequence of taking the dynamical implications of that algebra seriously. Not as an act of faith, but as a work programme.

If this path is correct, then the question "what is the universe made of?" has an unexpected answer: of autonomy, of temporality, and of the strangest and most beautiful algebraic structure that twentieth-century mathematics discovered. Reality is not a mechanism. It is a consequence of being impossible to be anything else.

Answer: 20πb^(3) +12πb^(2)

Question was: The formula for the volume of a right circular cylinder is V=πr^(2) h. If r=2b and h=5b+3 what is the volume of the cylinder in terms of b.