As a non-mathematician, (but I plan to go to re-take elementary school maths at the Khan Academy soon), I enjoyed these cartoons, and I can tell they’re clever, but didn’t understand more than one of them. Still hoping for someone to pick up the thread on my question of multiplying negative numbers. I feel once I understand that, I can trust you lot, and may eventually *get* some of these cartoons!
I just found it! It’s on the post “The Catchy Nonsense of ‘Two Negatives Make a Positive.'” I attempted an answer there, which I doubt will be satisfactory, in part because I find that curious people rarely find the answers to this question satisfactory!
multiplication distributes over addition.
a(b+1) = ab + a
(-a) is the additive inverse of a
(a + (-a)) = 0
And multiplication by 0 always gives 0.
a(0) = 0
So:
a(b + (-b)) = ab + (-ab) = a(0) = 0
Not to bad:
but what about
-a(b + (-b)) = -ab + (-a)(-b) = 0
and (-a)(-b) must equal (ab)
And now for a completely different:
multiplication by (-1) completely flips the number-line. Everything that was on the right side of zero flips to the left side of zero, and everything on the left side of 0 flips to the right side of zero.
And now for a stupid one:
consider the function f(x) = kx with k not equal to 0.
This function is “injective” that is
if f(a) = f(b) then a=b
Injective functions are sometimes called 1-1 functions. For every output there is a unique input.
and if k is an integer, and a is an integer than f(a) must be an integer.
if k = -1,We map every positive integer to its corresponding negative integer. And every negative integer has a positive integer mapping to it. So where else are the negative numbers supposed to go! The function is still injective so $f(a) = -a$ must be mapping negative integers to positive integers.
Kat, Somebody down below tried to give you a bunch of algebraic proof of the “Multiplying two negatives gives a positive”. I bet that wasn’t what you were looking for. As a former theoretical mathematician that decided almost 50 years ago to go back to the “real world”, let’s try some real world examples instead. Double negatives in English mean a positive, as in “I didn’t not lie”. Think of negatives as the “opposite” of a positive and multiplication as repetitive addition. As in 3 x 3 = 3 + 3 + 3. If I add “-7” apples to a basket, that really means I took 7 apples out of the basket. If I did that 3 times, there would be 27 less apples in the basket or -7 x 3 = -21. Now think about what would happen if I “undid” removing 7 apples 3 times then I would have 21 MORE apples, -7 x -3 = 21. Does it make more sense to you now?
In reference to large numbers: I think I’ve encountered numbers as large as 5 a couple of times—I seem to recall some fiddly epsilon-over-five proof in an undergraduate course on functional analysis that I took once. But “7”? What the hell is a “7”?
The joke is that pushing or pulling on the door actually has nothing to do with Euclidean or non-Euclidean space! The mathematician is just trying to cover for a dumb mistake (that we’ve all made too many times…) by using fancy terms to imply that it’s a mistake that an intelligent person would make.
As a non-mathematician, (but I plan to go to re-take elementary school maths at the Khan Academy soon), I enjoyed these cartoons, and I can tell they’re clever, but didn’t understand more than one of them. Still hoping for someone to pick up the thread on my question of multiplying negative numbers. I feel once I understand that, I can trust you lot, and may eventually *get* some of these cartoons!
can you repost your question? don’t know where you originally posted it…
I just found it! It’s on the post “The Catchy Nonsense of ‘Two Negatives Make a Positive.'” I attempted an answer there, which I doubt will be satisfactory, in part because I find that curious people rarely find the answers to this question satisfactory!
Let take a crack at multiplying negatives…
From the axioms of arithmattic.
multiplication distributes over addition.
a(b+1) = ab + a
(-a) is the additive inverse of a
(a + (-a)) = 0
And multiplication by 0 always gives 0.
a(0) = 0
So:
a(b + (-b)) = ab + (-ab) = a(0) = 0
Not to bad:
but what about
-a(b + (-b)) = -ab + (-a)(-b) = 0
and (-a)(-b) must equal (ab)
And now for a completely different:
multiplication by (-1) completely flips the number-line. Everything that was on the right side of zero flips to the left side of zero, and everything on the left side of 0 flips to the right side of zero.
And now for a stupid one:
consider the function f(x) = kx with k not equal to 0.
This function is “injective” that is
if f(a) = f(b) then a=b
Injective functions are sometimes called 1-1 functions. For every output there is a unique input.
and if k is an integer, and a is an integer than f(a) must be an integer.
if k = -1,We map every positive integer to its corresponding negative integer. And every negative integer has a positive integer mapping to it. So where else are the negative numbers supposed to go! The function is still injective so $f(a) = -a$ must be mapping negative integers to positive integers.
Kat, Somebody down below tried to give you a bunch of algebraic proof of the “Multiplying two negatives gives a positive”. I bet that wasn’t what you were looking for. As a former theoretical mathematician that decided almost 50 years ago to go back to the “real world”, let’s try some real world examples instead. Double negatives in English mean a positive, as in “I didn’t not lie”. Think of negatives as the “opposite” of a positive and multiplication as repetitive addition. As in 3 x 3 = 3 + 3 + 3. If I add “-7” apples to a basket, that really means I took 7 apples out of the basket. If I did that 3 times, there would be 27 less apples in the basket or -7 x 3 = -21. Now think about what would happen if I “undid” removing 7 apples 3 times then I would have 21 MORE apples, -7 x -3 = 21. Does it make more sense to you now?
When you multiply negative numbers, the result is positive
Just sayin’, the common conception of what philosophers do is even more skewed than the conception of what mathematicians do.
In reference to large numbers: I think I’ve encountered numbers as large as 5 a couple of times—I seem to recall some fiddly epsilon-over-five proof in an undergraduate course on functional analysis that I took once. But “7”? What the hell is a “7”?
Nice comics. Could anyone explain the “pull” door joke?
The joke is that pushing or pulling on the door actually has nothing to do with Euclidean or non-Euclidean space! The mathematician is just trying to cover for a dumb mistake (that we’ve all made too many times…) by using fancy terms to imply that it’s a mistake that an intelligent person would make.
Lol, that is really funny! I like the peer review process joke too. The slowness of peer review is a big insider joke for mathematicians.
Reblogged this on Singapore Maths Tuition.
Great! (Your comics are always linked in the Iowa State University weekly math email)
these comics are so funny and cool!
Please can I share this with my students?