Algebra I, geometry, algebra 2

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student (e.g., no complex numbers in Algebra 1, no linear programming). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

That's not The Correct Way handed down from God.
Chapter 10 is factoring, which has the same problem (some problems don't have solutions) that the quadratic formula has (chapter 13), until i is introduced.

i is very much a teaser in AoPS Algebra 1. Algebra 2 reintroduces i and covers it in a lot more detail.

I know

Still, one of the last writing problems in Algebra 1 was factoring z^4+1=0 using elementary means. Which I thought was really cool.

do you mean z^4 - 1 = 0? That can be factored using alg1 and introduces imaginary numbers. z^4 + 1 is much harder to factor.

Nope, I meant z^4+1=0. (Now that I think about it, it was actually z^4 + 4 = 0, which gives nice integer solutions.) Don't use the polar form, though. Set z=a+bi and see where this gets you.

Setting z = a+bi will work, but it's a quite lengthy to work through the algebra (expanding by binomial theorem, forming two separate equations and solving each of the cases). Much simpler is to rewrite the expression as the square of a binomial by completing the square, i.e:

z^4 + 4 = (z^2 + 2)^2 - 4z^2

Now apply difference of squares to turn the above into a product of two quadratics. The four roots are easily extracted with the quadratic formula, two for each of the quadratics. (Bonus for students who have studied polar/exponential form of complex numbers: Plot the roots of z^4 + 4 in the complex plane, and also plot the roots of z^4 - 4... what interesting observation can you make? Can you explain how they are related to each other and exactly why this is the case?)

That's un-completing a square while completing another square.

Much simpler is to use the already complete square, and go directly to difference of squares, analogy to the more elementary z^2-4=0:

z^4 + 4
= z^4 - -4
= (z^2)^2 - (2i)^2
= (z^2 + 2i)(z^2 - 2i)

z^2 + ±2i = 0
(a + bi)^2 = 0 + ±2i
a^2 - b^2 + 2abi = 0 + ±2i

real: a^2 = b^2, so a = ±b
imaginary: ab=±1
z = ±1 + ±1i

All 4 roots solved simultaneously and symmetrically,
no guessing a square to complete,
no trinomial quadratics to solve,
no fractions or division or subtraction,
no integers besides {0,1,2} after immediately removing the 4s from the problem.

Yes, that's how my 5th grader did it. Also remember that the problem was posed after introducing complex numbers but before exhaustive treatment of the quadratics. I love how AoPS includes so many deep problems in their curriculum.

By contrast, and this is true, when they took Algebra 1 in 6th, their Algebra teacher refused to include any kind of derivation of the quadratic formula because it's not part of Virginia's SOL requirements (and thus optional in their mind). Going back to the topic of this thread, this is why we cannot do without supplementation.

Bruh, my DS derived the quadratic formula in 7th grade. This is a teacher issue, not a public school issue. No, kids do not need supplementation. We can do without it.

How does one deal with this issue, whether it's a teacher issue or a public issue, without supplementation? Do you replace them with a better teacher? If so, how?

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student (e.g., no complex numbers in Algebra 1, no linear programming). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

That's not The Correct Way handed down from God.
Chapter 10 is factoring, which has the same problem (some problems don't have solutions) that the quadratic formula has (chapter 13), until i is introduced.

i is very much a teaser in AoPS Algebra 1. Algebra 2 reintroduces i and covers it in a lot more detail.

I know

Still, one of the last writing problems in Algebra 1 was factoring z^4+1=0 using elementary means. Which I thought was really cool.

do you mean z^4 - 1 = 0? That can be factored using alg1 and introduces imaginary numbers. z^4 + 1 is much harder to factor.

Nope, I meant z^4+1=0. (Now that I think about it, it was actually z^4 + 4 = 0, which gives nice integer solutions.) Don't use the polar form, though. Set z=a+bi and see where this gets you.

Setting z = a+bi will work, but it's a quite lengthy to work through the algebra (expanding by binomial theorem, forming two separate equations and solving each of the cases). Much simpler is to rewrite the expression as the square of a binomial by completing the square, i.e:

z^4 + 4 = (z^2 + 2)^2 - 4z^2

Now apply difference of squares to turn the above into a product of two quadratics. The four roots are easily extracted with the quadratic formula, two for each of the quadratics. (Bonus for students who have studied polar/exponential form of complex numbers: Plot the roots of z^4 + 4 in the complex plane, and also plot the roots of z^4 - 4... what interesting observation can you make? Can you explain how they are related to each other and exactly why this is the case?)

That's un-completing a square while completing another square.

Much simpler is to use the already complete square, and go directly to difference of squares, analogy to the more elementary z^2-4=0:

z^4 + 4
= z^4 - -4
= (z^2)^2 - (2i)^2
= (z^2 + 2i)(z^2 - 2i)

z^2 + ±2i = 0
(a + bi)^2 = 0 + ±2i
a^2 - b^2 + 2abi = 0 + ±2i

real: a^2 = b^2, so a = ±b
imaginary: ab=±1
z = ±1 + ±1i

All 4 roots solved simultaneously and symmetrically,
no guessing a square to complete,
no trinomial quadratics to solve,
no fractions or division or subtraction,
no integers besides {0,1,2} after immediately removing the 4s from the problem.

Yes, that's how my 5th grader did it. Also remember that the problem was posed after introducing complex numbers but before exhaustive treatment of the quadratics. I love how AoPS includes so many deep problems in their curriculum.

By contrast, and this is true, when they took Algebra 1 in 6th, their Algebra teacher refused to include any kind of derivation of the quadratic formula because it's not part of Virginia's SOL requirements (and thus optional in their mind). Going back to the topic of this thread, this is why we cannot do without supplementation.

Bruh, my DS derived the quadratic formula in 7th grade. This is a teacher issue, not a public school issue. No, kids do not need supplementation. We can do without it.

How does one deal with this issue, whether it's a teacher issue or a public issue, without supplementation? Do you replace them with a better teacher? If so, how?

Whether students learn these things in 6th grade, 7th grade, 8th grade, 9th grade or whenever, it's all good.
It's not a contest. You won't earn more money from knowing about imaginary numbers at X age than at Y age.

Fwiw, the more you supplement, the less the teachers will teach.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student (e.g., no complex numbers in Algebra 1, no linear programming). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

Complex numbers have never been part of Algebra I. If you want to pay money to have someone teach your DC ahead, fine, but it's not "abridged" or "appropriate".

On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems.

Can you name one such book?

Stewart, College Algebra
Openstax, College Algebra

All college algebra books, I’m not aware of a single one that doesn’t use complex numbers in the treatment of quadratics.

The division between algebra, geometry and pre calculus is also quite arbitrary, with many areas of overlap. Of course arithmetic, equations and systems belong in algebra, shapes, angles, lines belong to geometry, and vectors and matrices traditionally taught in precalculus. But functions are thought in both algebra and precalculus, analytical geometry are taught in algebra, geometry and precalculus, etc. trigonometry is another one that can be taught alone, or almost any high school math class.

I found it somewhat amusing when posters decree that complex number are taught in Algebra 2. What exactly is Algebra 2? Then why are they taught again in precalculus. There’s an apt analogy that math is a spiral, it’s up to the individual how fast one goes around and the breath and depth of the material studied.

Those are college algebra books, not elementary algebra books. Those are two different math classes.

It’s the same content, my kids school is using Stewart as textbook for Algebra 1 and 2, although they skip a lot of sections.

No it's not.
https://www.knewton.com/wp-content/uploads/2020/04/alta-Elementary-Algebra-v2-TOC.pdf
https://openstax.org/details/books/elementary-algebra-2e

Notice the lack of imaginary numbers.

It is the same content, some books introduce concepts earlier, some omit them etc. an honors class will go more in depth, introduce more topics etc. as you correctly point some books include complex numbers some don’t. Both Elementary Algebra and College Algebra books from openstax are one semester introductory courses that have some differences.

Essentially the student should choose based on interest and capability, if you can handle go for the more difficult version. If not the lighter version is good too.

It's clearly not the same content, as elementary algebra does not include complex numbers even though you (or the person whose point you're arguing) claimed otherwise.
"On the other hand elementary algebra textbooks for college are better structured and start with numbers (including complex) and arithmetic, then move to equations and then systems" -> unproven and now demonstrably false.

Lol, I’d be curious to know what your math background is. Based on your deep paragraph analysis in guessing you majored in English (or worse!).

For college, all algebra is actually remedial and is called either simply Algebra, or Elementary Algebra to differentiate from Linear Algebra, which is a typical sophomore year math course. Some community colleges break Algebra in two semesters and call it Elementary and Intermediate, but that’s because they see more students with large deficiencies that need to be plugged before credit courses starting with precalculus. There’s no credit for any Algebra class. Basically if it has numbers, equations and systems, it is (Elementary) Algebra.

Seriously, you’re out of your depth here nit picking on semantics.

If we’re talking on supplementation for a more advanced student, it matters little to me how some random book is organized. I’ll simply go for the most complete and exhaustive treatment the student can handle. For quadratics that will include complex numbers. For linear graph it will include all the forms: standard, slope intercept, two points, point slope, parametric. I’m sure some of the books you listed omit one or more, and some will include all. As I said I’ll choose the curriculum that goes in more detail.

You’re free to pick whatever you like for your child.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student (e.g., no complex numbers in Algebra 1, no linear programming). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

That's not The Correct Way handed down from God.
Chapter 10 is factoring, which has the same problem (some problems don't have solutions) that the quadratic formula has (chapter 13), until i is introduced.

i is very much a teaser in AoPS Algebra 1. Algebra 2 reintroduces i and covers it in a lot more detail.

I know

Still, one of the last writing problems in Algebra 1 was factoring z^4+1=0 using elementary means. Which I thought was really cool.

do you mean z^4 - 1 = 0? That can be factored using alg1 and introduces imaginary numbers. z^4 + 1 is much harder to factor.

Nope, I meant z^4+1=0. (Now that I think about it, it was actually z^4 + 4 = 0, which gives nice integer solutions.) Don't use the polar form, though. Set z=a+bi and see where this gets you.

Setting z = a+bi will work, but it's a quite lengthy to work through the algebra (expanding by binomial theorem, forming two separate equations and solving each of the cases). Much simpler is to rewrite the expression as the square of a binomial by completing the square, i.e:

z^4 + 4 = (z^2 + 2)^2 - 4z^2

Now apply difference of squares to turn the above into a product of two quadratics. The four roots are easily extracted with the quadratic formula, two for each of the quadratics. (Bonus for students who have studied polar/exponential form of complex numbers: Plot the roots of z^4 + 4 in the complex plane, and also plot the roots of z^4 - 4... what interesting observation can you make? Can you explain how they are related to each other and exactly why this is the case?)

That's un-completing a square while completing another square.

Much simpler is to use the already complete square, and go directly to difference of squares, analogy to the more elementary z^2-4=0:

z^4 + 4
= z^4 - -4
= (z^2)^2 - (2i)^2
= (z^2 + 2i)(z^2 - 2i)

z^2 + ±2i = 0
(a + bi)^2 = 0 + ±2i
a^2 - b^2 + 2abi = 0 + ±2i

real: a^2 = b^2, so a = ±b
imaginary: ab=±1
z = ±1 + ±1i

All 4 roots solved simultaneously and symmetrically,
no guessing a square to complete,
no trinomial quadratics to solve,
no fractions or division or subtraction,
no integers besides {0,1,2} after immediately removing the 4s from the problem.

Yes, that's how my 5th grader did it. Also remember that the problem was posed after introducing complex numbers but before exhaustive treatment of the quadratics. I love how AoPS includes so many deep problems in their curriculum.

By contrast, and this is true, when they took Algebra 1 in 6th, their Algebra teacher refused to include any kind of derivation of the quadratic formula because it's not part of Virginia's SOL requirements (and thus optional in their mind). Going back to the topic of this thread, this is why we cannot do without supplementation.

Bruh, my DS derived the quadratic formula in 7th grade. This is a teacher issue, not a public school issue. No, kids do not need supplementation. We can do without it.

How does one deal with this issue, whether it's a teacher issue or a public issue, without supplementation? Do you replace them with a better teacher? If so, how?

Whether students learn these things in 6th grade, 7th grade, 8th grade, 9th grade or whenever, it's all good.
It's not a contest. You won't earn more money from knowing about imaginary numbers at X age than at Y age.

Fwiw, the more you supplement, the less the teachers will teach.

What a dumb comment! Of course it matters when the student leans these things, because they are the building blocks for more advanced coursework. That will enable the student to enroll in a more lucrative major (like engineering) and literally earn more money.

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student (e.g., no complex numbers in Algebra 1, no linear programming). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

That's not The Correct Way handed down from God.
Chapter 10 is factoring, which has the same problem (some problems don't have solutions) that the quadratic formula has (chapter 13), until i is introduced.

i is very much a teaser in AoPS Algebra 1. Algebra 2 reintroduces i and covers it in a lot more detail.

I know

Still, one of the last writing problems in Algebra 1 was factoring z^4+1=0 using elementary means. Which I thought was really cool.

do you mean z^4 - 1 = 0? That can be factored using alg1 and introduces imaginary numbers. z^4 + 1 is much harder to factor.

Nope, I meant z^4+1=0. (Now that I think about it, it was actually z^4 + 4 = 0, which gives nice integer solutions.) Don't use the polar form, though. Set z=a+bi and see where this gets you.

Setting z = a+bi will work, but it's a quite lengthy to work through the algebra (expanding by binomial theorem, forming two separate equations and solving each of the cases). Much simpler is to rewrite the expression as the square of a binomial by completing the square, i.e:

z^4 + 4 = (z^2 + 2)^2 - 4z^2

Now apply difference of squares to turn the above into a product of two quadratics. The four roots are easily extracted with the quadratic formula, two for each of the quadratics. (Bonus for students who have studied polar/exponential form of complex numbers: Plot the roots of z^4 + 4 in the complex plane, and also plot the roots of z^4 - 4... what interesting observation can you make? Can you explain how they are related to each other and exactly why this is the case?)

That's un-completing a square while completing another square.

Much simpler is to use the already complete square, and go directly to difference of squares, analogy to the more elementary z^2-4=0:

z^4 + 4
= z^4 - -4
= (z^2)^2 - (2i)^2
= (z^2 + 2i)(z^2 - 2i)

z^2 + ±2i = 0
(a + bi)^2 = 0 + ±2i
a^2 - b^2 + 2abi = 0 + ±2i

real: a^2 = b^2, so a = ±b
imaginary: ab=±1
z = ±1 + ±1i

All 4 roots solved simultaneously and symmetrically,
no guessing a square to complete,
no trinomial quadratics to solve,
no fractions or division or subtraction,
no integers besides {0,1,2} after immediately removing the 4s from the problem.

Yes, that's how my 5th grader did it. Also remember that the problem was posed after introducing complex numbers but before exhaustive treatment of the quadratics. I love how AoPS includes so many deep problems in their curriculum.

By contrast, and this is true, when they took Algebra 1 in 6th, their Algebra teacher refused to include any kind of derivation of the quadratic formula because it's not part of Virginia's SOL requirements (and thus optional in their mind). Going back to the topic of this thread, this is why we cannot do without supplementation.

Bruh, my DS derived the quadratic formula in 7th grade. This is a teacher issue, not a public school issue. No, kids do not need supplementation. We can do without it.

How does one deal with this issue, whether it's a teacher issue or a public issue, without supplementation? Do you replace them with a better teacher? If so, how?

Whether students learn these things in 6th grade, 7th grade, 8th grade, 9th grade or whenever, it's all good.
It's not a contest. You won't earn more money from knowing about imaginary numbers at X age than at Y age.

Fwiw, the more you supplement, the less the teachers will teach.

What a dumb comment! Of course it matters when the student leans these things, because they are the building blocks for more advanced coursework. That will enable the student to enroll in a more lucrative major (like engineering) and literally earn more money.

That's not how college works. Or engineering.

- majored in BSEE

[Anonymous]

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

pettifogger wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:

Anonymous wrote:Yes, you should complement. Have your child enroll in RSM or AoPS either concurrently or shortly before. That's what we did. School instruction is insufficient in multiple ways. First, students aren't doing any problem solving in school (all they do is textbook worksheets and SOL prep); second, they don't do any mathematical writing in school; third, the school curriculum is abridged for an advanced student (e.g., no complex numbers in Algebra 1, no linear programming). Fourth, school math is much less fun.

In short, if your child is gifted and interested in math then you cannot rely on the school curriculum.

Is this a joke? Why would there be complex numbers in Algebra I? You sound nutty.

When covering math at a level appropriate for mathematically gifted students you introduce complex numbers before the quadratic formula. For instance, AoPS's textbook does so in Chapter 12.

That's not The Correct Way handed down from God.
Chapter 10 is factoring, which has the same problem (some problems don't have solutions) that the quadratic formula has (chapter 13), until i is introduced.

i is very much a teaser in AoPS Algebra 1. Algebra 2 reintroduces i and covers it in a lot more detail.

I know

Still, one of the last writing problems in Algebra 1 was factoring z^4+1=0 using elementary means. Which I thought was really cool.

do you mean z^4 - 1 = 0? That can be factored using alg1 and introduces imaginary numbers. z^4 + 1 is much harder to factor.

Nope, I meant z^4+1=0. (Now that I think about it, it was actually z^4 + 4 = 0, which gives nice integer solutions.) Don't use the polar form, though. Set z=a+bi and see where this gets you.

Setting z = a+bi will work, but it's a quite lengthy to work through the algebra (expanding by binomial theorem, forming two separate equations and solving each of the cases). Much simpler is to rewrite the expression as the square of a binomial by completing the square, i.e:

z^4 + 4 = (z^2 + 2)^2 - 4z^2

Now apply difference of squares to turn the above into a product of two quadratics. The four roots are easily extracted with the quadratic formula, two for each of the quadratics. (Bonus for students who have studied polar/exponential form of complex numbers: Plot the roots of z^4 + 4 in the complex plane, and also plot the roots of z^4 - 4... what interesting observation can you make? Can you explain how they are related to each other and exactly why this is the case?)

That's un-completing a square while completing another square.

Much simpler is to use the already complete square, and go directly to difference of squares, analogy to the more elementary z^2-4=0:

z^4 + 4
= z^4 - -4
= (z^2)^2 - (2i)^2
= (z^2 + 2i)(z^2 - 2i)

z^2 + ±2i = 0
(a + bi)^2 = 0 + ±2i
a^2 - b^2 + 2abi = 0 + ±2i

real: a^2 = b^2, so a = ±b
imaginary: ab=±1
z = ±1 + ±1i

All 4 roots solved simultaneously and symmetrically,
no guessing a square to complete,
no trinomial quadratics to solve,
no fractions or division or subtraction,
no integers besides {0,1,2} after immediately removing the 4s from the problem.

Yes, that's how my 5th grader did it. Also remember that the problem was posed after introducing complex numbers but before exhaustive treatment of the quadratics. I love how AoPS includes so many deep problems in their curriculum.

By contrast, and this is true, when they took Algebra 1 in 6th, their Algebra teacher refused to include any kind of derivation of the quadratic formula because it's not part of Virginia's SOL requirements (and thus optional in their mind). Going back to the topic of this thread, this is why we cannot do without supplementation.

Bruh, my DS derived the quadratic formula in 7th grade. This is a teacher issue, not a public school issue. No, kids do not need supplementation. We can do without it.

How does one deal with this issue, whether it's a teacher issue or a public issue, without supplementation? Do you replace them with a better teacher? If so, how?

Whether students learn these things in 6th grade, 7th grade, 8th grade, 9th grade or whenever, it's all good.
It's not a contest. You won't earn more money from knowing about imaginary numbers at X age than at Y age.

Fwiw, the more you supplement, the less the teachers will teach.

What a dumb comment! Of course it matters when the student leans these things, because they are the building blocks for more advanced coursework. That will enable the student to enroll in a more lucrative major (like engineering) and literally earn more money.

That's not how college works. Or engineering.

- majored in BSEE

I sincerely hope my kids won’t enroll in the same program you graduated from.

Of course knowing imaginary numbers alone is not making a difference, we are talking about aggregate knowledge, and that particular topic as one of the many check points.