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I applaud the idea, but I think this book needs some serious work before it is suitable for your intended audience. For example, in Section 1.1 you explain how
by crntaylor 14y ago
I applaud the idea, but I think this book needs some serious work before it is suitable for your intended audience.
For example, in Section 1.1 you explain how to go about solving the equation x^2 - 4 = 45, a topic which (I would venture) most 16 year olds should be comfortable with.
Then in Section 1.2 you, out of nowhere, introduce set-theoretical ideas and notation. Even seemingly innocent notation like
{0, 1, 2, 3, ...}
to denote the natural numbers can be confusing for someone who hasn't seen it before. In this case it's clear that the dots are supposed to indicate "continue in the obvious way". But then you write
Q = { -1.5, 1/3, 22/7, 0.125, ... }
in which the dots, presumably, mean something different. Then, at the bottom of the page, you suddenly start using the notation "∈" to denote set membership, as in
if x,y ∈ N, then (x+y) ∈ N.
First, what does this weird "∈" symbol mean? Second, what does it mean to write "x,y ∈ N"? Is that x∈N and y∈N? Or is it some special object "x,y" that is ∈N? Or is it two statements, "x" on its own, and "y∈N"?
The next couple of paragraphs contain both historical and mathematical inaccuracies (can you really always divide two rational numbers to get another rational number?). I'll stop there.
Like I say, I applaud what you're trying to do. I think you have a noble aim. But if the preview is anything to go by, your textbook in its current form is falling short of that aim.
- chris_wot 14y agocan you really always divide two rational numbers to get another rational number? I'll not give a formal proof, but intuitively the answer is surely "yes"!
- crntaylor 14y agoWhat if one of them is zero?
- chris_wot 14y agoThe definition of a rational number is a number that can be expressed as the quotient of two integers where the denominator is not 0.
- DIFF 14y ago0 \in N, 5 \in N => (0/5) \in Q. Therefore 0 \in Q. QED.
- chris_wot 14y ago\in?
- bane 14y ago\in means "in the set of" Given the definition "The definition of a rational number is a number that can be expressed as the quotient of two integers where the denominator is not 0." Let N be the set of all integers and Q be the set of all rational numbers. Trivially 0 is in the set N, 5 is in the set N by the definition 0/5 is in Q. Therefore 0 is in Q. QED.
- deleted 14y ago[deleted]
- chris_wot 14y agoAh, I see! Deleted my old comment. Nice :-)
- v-yadli 14y agoWatch out for your proof. Q is constructed as a field with + and x only. The availability of the inverse operator is not well documented :-D
- Dove 14y agoYes, and it's pretty easy to prove. Multiplying any rational number by the inverse of another gives you a new rational number. (Zero doesn't have an inverse). A rational number is a number that can be represented as one integer divided by another. Take two rational numbers, A = a/b, and B = c/d. If B has an inverse, then A/B = a/b * d/c = ad/bc. ad and bc are both integers, so A/B is rational.
- crntaylor 14y agoMy point was that you can only divide one rational by another to produce a new rational if the denominator is nonzero. This is not what it says in the book, so there's potential for a newbie to get confused (and anyone reading the book in order to learn from it is, by definition, a newbie).
- Dove 14y agoOh, then that seems like a pretty pedantic worry. You can't divide by zero in any context; even children know that. It's such an obvious exception that it didn't even occur to me that's what you thought was wrong with the statement.
- DanBC 14y ago> For example, in Section 1.1 you explain how to go about solving the equation x^2 - 4 = 45, a topic which (I would venture) most 16 year olds should be comfortable with. > Then in Section 1.2 you, out of nowhere, introduce set-theoretical ideas and notation. Is there any decent "crowd editing" solution? Or at least, "crowd suggested edits" solution? Smart people know the math and the grammar and can correct those errors, or improve diagrams, or tweak the wording. People with weak math can say where they get confused; where things ramp up too quickly or where things take too long to get going. Errors in a book like this are especially dangerous for someone like me. Dunning Kruger and all that - I wont know enough to know what I don't know, or to know what I've learnt is not quite correct.
- ivan_ah 14y ago> even seemingly innocent notation like {0, 1, 2, 3, ...} I think that would be OK. There is a fine line between having to define everything and just using the math and letting the reader pick up on their own. > set membership, as in if x,y ∈ N, then (x+y) ∈ N. Yeah that was non-sense. It definitely didn't belong so early on in the book. Several people emailed me about this and it has been fixed in print/pdf versions now. More on set notation: http://cnd.mcgill.ca/~ivan/miniref/set_notation.pdf http://cnd.mcgill.ca/~ivan/miniref/set_notation.pdf