Strange notation

nyarlthotepful

My friend and I are giving maths grinds to junior cert students. We've both done maths in college but this notation has us stumped. What is the easiest way of solving this? Thanks.

x*y=x^2+2y+3

LeixlipRed

What part has you stumped? What does the question ask you to do?

nyarlthotepful

Pardon me. I was kind of getting ahead of myself. The full question is:

(viii) If x ∗ y = x 2 + 2 y + 3, find the two values of a for which a ∗ a = 6.

Fringe

This is basically defining an operation * where you take two numbers x and y such that x*y = x^2+2y+3 . You can think of it as a 2 variable function f(x,y) = x^2+2y+3. It's asking you what numbers x will give x*x = 6.

sponsoredwalk

Yeah that's right but I think questions like these are clearer when expressed
as a binary relation, I just think that the notation really helps, especially
when you are dealing with operations that are like the one you're dealing with:

* : ℝ × ℝ → ℝ | (x,y) ↦ *(x,y) = (x*y) = x²+2y+3

Which is of the exact same form as the binary relation known as addition:

+ : ℝ × ℝ → ℝ | (x,y) ↦ +(x,y) = (x+y) = (x + y)

So all you need is to find two λ such that:

* : ℝ × ℝ → ℝ | (λ,λ) ↦ *(λ,λ) = (λ*λ) = λ²+2λ+3 = 6.

MathsManiac

sponsoredwalk wrote: »

Yeah that's right but I think questions like these are clearer when expressed
as a binary relation, I just think that the notation really helps, especially
when you are dealing with operations that are like the one you're dealing with:

* : ℝ × ℝ → ℝ | (x,y) ↦ *(x,y) = (x*y) = x²+2y+3

Which is of the exact same form as the binary relation known as addition:

+ : ℝ × ℝ → ℝ | (x,y) ↦ +(x,y) = (x+y) = (x + y)

So all you need is to find two λ such that:

* : ℝ × ℝ → ℝ | (λ,λ) ↦ *(λ,λ) = (λ*λ) = λ²+2λ+3 = 6.

That's all very clear to the likes of us, but I think it would completely flummox a Junior Cycle student. Also, I don't think your last line is helpful as phrased - the definition of the binary operation should be kept separate from the equation you're being asked to solve. Your last line doesn't really make sense, as it starts off looking like it's trying to define a binary operation but then doesn't really do so.

I do think, however, that the original question could have had a little bit more explanation. I suggest it would have been better phrased as follows:

"A binary operation * is defined by the rule: x*y = x^2 + 2y + 3. Find the two values of a for which a ∗ a = 6."

sponsoredwalk

MathsManiac wrote: »

That's all very clear to the likes of us, but I think it would completely flummox a Junior Cycle student.

Welll I didn't write it for a junior cycle student I wrote it for him because:

nyarlthotepful wrote: »

We've both done maths in college

MathsManiac wrote: »

Also, I don't think your last line is helpful as phrased - the definition of the binary operation should be kept separate from the equation you're being asked to solve. Your last line doesn't really make sense, as it starts off looking like it's trying to define a binary operation but then doesn't really do so.

It's just shorthand to remind the guy who has probably seen all this in
college but didn't link it together.

LeixlipRed

"maths in college" probably means one year of service maths. Then off giving grinds to people and making it up as they go along!