- #36

Pete Mcg

- 18

- 4

Yes. I have seen this one. It's good.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- I
- Thread starter Pete Mcg
- Start date

- #36

Pete Mcg

- 18

- 4

Yes. I have seen this one. It's good.

- #37

Pete Mcg

- 18

- 4

Thank you. You have been very helpful.

- #38

Pete Mcg

- 18

- 4

Thank you so much!It all makes sense now...Yup, that is the gist of it.

If you want to understand what mathematicians call a proof https://www.people.vcu.edu/~rhammack/BookOfProof/

It is pdf made free by the author. Happy reading.

- #39

Pete Mcg

- 18

- 4

My God! I've just had a peek at Book Of Proof and it's marvellous. Can I write you into my will?Thank you so much!It all makes sense now...

- #40

Mark44

Mentor

- 36,668

- 8,671

1 = 1 is not a definition. In general for all kinds of definitions, including words found in a dictionary, you can't define something in terms of itself.please correct me if I'm wrong. 0!=1 is a 'definition' in the same way that 1=1 or any other self evident statement is a definition and 'You do no prove a definition in Mathematics'.

The equation 1 = 1 is an example of the

Last edited:

- #41

MidgetDwarf

- 1,392

- 549

Yes .My God! I've just had a peek at Book Of Proof and it's marvellous. Can I write you into my will?

- #42

Baluncore

Science Advisor

- 12,050

- 6,169

But that is sloppy since the complete recursive definition of factorial is;

n! = n(n-1)! for n≥1 and 0!=1

Everything here is a definition.

Do not attempt to use the definition as a self referential proof of itself.

- #43

- 23,745

- 15,354

You obviously failed to see post #2.

##0! = 1## (by definition)

##n! = n(n-1)!## (for ##n \ge 1##)

- #44

- #45

- #46

Baluncore

Science Advisor

- 12,050

- 6,169

No apology is necessary in hindsight if it gets people to think. My standards are lower than yours.I apologize for not meeting your standards.

This interesting thread gave me the uncanny feeling that we were contemplating the proof of a definition, from inside a fishbowl, while looking at our different reflections in the curved wall.

Meanwhile some find the difference between antimony and an antinomy.

- #47

sysprog

- 2,613

- 1,783

Hmm ##\dots -## here's an observation not especially related to meeting standards, but at least ancillarily related to this thread topic: computationally, for e.g. card-deck-sized non-zero factorials, e.g. between euchre, poker, and pinochle or canasta sized decks, an iterative method is a little faster than recursive, but for zero, there's no comparison necessary between those methods, because we just hardcode zero factorial to 1 ##-## it's so by definition, as @PeroK said, and reasonably so, as @fresh_42, @PeroK, @Baluncore, @Mark44, and maybe others, explained.I apologize for not meeting your standards.

- #48

- #49

pbuk

Science Advisor

Homework Helper

Gold Member

- 4,029

- 2,361

Iterative (or explicit) definition:Is the definition of n! iterative or recursive ?

$$ n! = \begin{cases}

\prod\limits_{k=1}^n k, & \text{if}\ n>0 \\

1, & \text{if}\ n = 0

\end{cases}

$$

Recursive (or implicit) definition:

$$ n! = \begin{cases}

n(n-1)! , & \text{if}\ n>0 \\

1, & \text{if}\ n = 0

\end{cases}

$$

So ## 0! ## is defined identically in each case.

- #50

jbriggs444

Science Advisor

Homework Helper

- 11,573

- 6,221

Or simply $$n!=\prod\limits_{k=1}^n k$$ with the understanding that a product indexed from one to zero is the empty product which is, by definition, the neutral element for multiplication.

The meaning of the empty product was already belabored up-thread.

Edit:

In the world of computing, the Ada programming language makes explicit the notion of an empty array slice such as a[1:0]. If the initial index is one greater than the final index, the result is explicitly the empty array and no bounds checking on either index is performed.

In VAX Fortran-77 the same would work and was needed to create an empty string, but one needed to compile with bounds checking disabled.

The meaning of the empty product was already belabored up-thread.

Edit:

In the world of computing, the Ada programming language makes explicit the notion of an empty array slice such as a[1:0]. If the initial index is one greater than the final index, the result is explicitly the empty array and no bounds checking on either index is performed.

In VAX Fortran-77 the same would work and was needed to create an empty string, but one needed to compile with bounds checking disabled.

Last edited:

- #51

Pete Mcg

- 18

- 4

How does 10% of almost nothing sound?Yes .

- #52

Pete Mcg

- 18

- 4

Thank you for that. As stated previously in my original post, I'm not a Mathematician and am learning, every little bit counts and I appreciate your and everybody's input . (Learned today that you when you multiply a negative number by a negative number the result is a positive number - they didn't teach this in Maths @ school I went to so it took a bit of getting my head round it - at first I went HUH?? but it's starting to makes sense.) This a new world to me, good fun -talk about fun - this Godel incompleteness thing has got me intrigued!1 = 1 is not a definition. In general for all kinds of definitions, including words found in a dictionary, you can't define something in terms of itself.

The equation 1 = 1 is an example of thereflexiveproperty of the equality relation. This property says that any number is equal to itself. Other relations, such as < or >, do not have this property. For example, ##5 \nless 5## and ##2 \ngtr 2##.

As in learning any new 'language' one needs to understand the rules and conventions (another thing learned today was BOMDAS / PEMDAS)Even the supposedly simplest thing like 'number' has a myriad of different 'meanings' - natural numbers, rational numbers, irrational etc etc etc ...Anyway, enough of my rambling. Cheers.

- #53

- 17,645

- 18,308

A product is an area. An area is oriented. It makes a difference whether you circle it clockwise or counter-clockwise, one is noted as a positive number, the other one by a negative number. Which is which is up to you. The lines are oriented, too, up and down, left and right. Now ##(+1)\cdot (+1)## has the same orientation as ##(-1)\cdot (-1),## and ##(+1)\cdot (-1)## is of opposite orientation.they didn't teach this in Maths @ school I went to so it took a bit of getting my head round it - at first I went HUH??

- #54

jbriggs444

Science Advisor

Homework Helper

- 11,573

- 6,221

Say you have ##-2 \times (1-1)## then clearly that should evaluate as ##-2 \times 0 = 0##. If we apply the distributive law, it should be equal to ##( -2 \times 1 ) + ( -2 \times -1 )##. But if a negative times a negative is a negative, that formula evaluates as ##-2 + -2## for a result of ##-4## which is wrong.

So if a negative times a negative yields a negative, the distributive law breaks.

- #55

Mark44

Mentor

- 36,668

- 8,671

Sorry to hear that it wasn't taught at your school. My first exposure to signed-number arithmetic was in ninth grade, back when I was 14.(Learned today that you when you multiply a negative number by a negative number the result is a positive number - they didn't teach this in Maths @ school I went to so it took a bit of getting my head round it

Last edited:

- #56

pbuk

Science Advisor

Homework Helper

Gold Member

- 4,029

- 2,361

If a negative times a negative yields a negative there are no multiplicative inverses for negative numbers so we haven't even got a group.Another motivation is the distributive law: ##a \times (b+c) = a \times b + a \times c##

So if a negative times a negative yields a negative, the distributive law breaks.

Share:

- Last Post

- Replies
- 4

- Views
- 435

- Last Post

- Replies
- 1

- Views
- 394

- Replies
- 4

- Views
- 583

- Last Post

- Replies
- 6

- Views
- 427

- Last Post

- Replies
- 4

- Views
- 548

- Replies
- 12

- Views
- 776

- Last Post

- Replies
- 2

- Views
- 501

- Last Post

- Replies
- 17

- Views
- 2K

MHB
Real Zeroes

- Last Post

- Replies
- 1

- Views
- 564

- Replies
- 3

- Views
- 569