In arithmetic, you’ll have memorized the next guidelines of indicators.
constructive × constructive = constructive
constructive × adverse = adverse
adverse × constructive = adverse
adverse × adverse = constructive
The primary three guidelines make sense to most college students. However what’s up with the final rule? Why is that adverse × adverse = constructive?
If you happen to had been ever confused by this rule, don’t fear, you might be in good firm. Many nice mathematicians have been confused by adverse numbers, together with Pascal, Euler, Descartes and De Morgan. Damaging numbers had been thought of absurd or unattainable at the same time as late because the nineteenth century.
So why precisely is adverse × adverse = constructive, and who first found this? I give just a few causes within the video beneath.
Why is a adverse occasions a adverse a constructive?
0:00 intro
1:20 sample
4:34 scaling
6:32 velocity
8:15 money owed and discovery
10:55 textbook
Or preserve studying.
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Why is a adverse occasions a adverse a constructive?
(Just about all posts are transcribed rapidly after I make the movies for them–please let me know if there are any typos/errors and I’ll right them, thanks).
Sample
Think about you’re a mathematician a whole bunch of years in the past, and also you produce the next desk of merchandise.
2×3 = 6
2×2 = 4
2×1 = 2
2×0 = 0
In every new line, the second quantity decreases by 1, and the product decreases by 2. We are able to proceed the sample into adverse numbers.
2×(-1) = -2
2×(-2) = -4
2×(-3) = -6
However now let’s see what occurs as we lower the primary quantity by 1. We all know 1 occasions any quantity is similar quantity, and 0 occasions any quantity is 0, so we will calculate:
2×(-3) = -6
1×(-3) = -3
0×(-3) = 0
In every new line, the product will increase by 3. So if we lower the primary quantity by 1, the brand new product has to extend by 3. Thus we’ve got:
(-1)×(-3) = 3
(-2)×(-3) = 6
And there you might have it! A adverse occasions a adverse leads to a constructive quantity!
This isn’t a rigorous proof. Nevertheless, that is exactly the form of reasoning that constructed the inspiration of arithmetic, and I believe we want extra of this type of exploration.
Quantity line
Begin with a quantity line. If you happen to settle for that multiplying a constructive by a adverse is a adverse, then multiplying the non-negative actual numbers by -1 will flip all of the numbers to the opposite facet of 0.
However then should you multiply the non-negative numbers of the second quantity line by -1, that must originate from the adverse numbers of the primary quantity line.
So if we full the diagram we’ve got one thing like this:
And now look: the adverse numbers within the first quantity line, multiplied by a adverse quantity (-1), will find yourself because the constructive numbers within the second quantity line. A adverse occasions a adverse is a constructive!
Velocity
By definition velocity v is a change in displacement x over a change in time t.
v = (Δx)/(Δt)
v(Δt) = Δx
Think about you had footage of a automobile driving in reverse which might be a adverse velocity v. As we transfer ahead in time we’ve got Δt > 0. The place does the automobile transfer if we play the video footage? Clearly the automobile can be shifting backwards (Δx < 0). So we’ve got a adverse occasions a constructive is a adverse:
v(Δt) = Δx
(-)(+) = (-)
What occurs should you rewind the footage? The automobile’s velocity is unchanged and nonetheless adverse (-), however the time is shifting in reverse (-), and naturally this leads to the automobile apparently shifting in the other way which is the ahead course (+). And like magic this demonstrates {that a} adverse occasions a adverse is a constructive!
v(Δt) = Δx
(-)(-) = (+)
Money owed
What occurs if I offer you 3 notes of $20? This corresponds to a achieve of $60, as a constructive occasions a constructive is a constructive:
3(20) = 60
What occurs if I take away 3 notes of $20 notes? This corresponds to a lack of $60, as a adverse occasions a constructive is a adverse:
-3(20) = -60
What occurs if I offer you 3 money owed of $20? This corresponds to a lack of $60, as a constructive occasions a adverse is a adverse:
-3(20) = -60
However now what if I take away 3 of your $20 money owed? This corresponds to a achieve of $60, and this illustrates {that a} adverse occasions a adverse is a constructive:
-3(-20) = 60
This was basically the reason given by the mathematician Brahmagupta round 628 CE. It’s wonderful he devised the principles of multiplication a whole bunch of years earlier than different mathematicians even accepted the concept of adverse numbers. One would possibly say we owe an incredible DEBT to Brahmagupta for this concept!
Algebra
That is the reason taught in colleges. It could be unsatisfying to many college students, however it’s nonetheless good to know. Think about we wish a quantity system that obeys two totally different guidelines.
zero product
a × 0 = 0
distributive property
a(b + c) = ab + ac
If we wish these to be true, it have to be the case {that a} adverse occasions a adverse is a constructive. Begin with a, b > 0. Then we’ve got:
–a(0) = 0
–a(b – b) = 0
–ab + (-a)(-b) = 0
ab + –ab + (-a)(-b) = 0 + ab
(-a)(-b) = ab
The product of two adverse numbers –a and –b is the same as the product of two constructive numbers ab, which is after all constructive. Due to this fact a adverse occasions a adverse is a constructive!
References
sample
https://math.stackexchange.com/questions/9933/why-is-negative-times-negative-positive
quantity line
polymathematic
https://www.youtube.com/watch?v=rpI78oh2liM
https://www.desmos.com/calculator/rsqgblsqte
velocity
EpsilonDelta
https://www.youtube.com/watch?v=4nG49xTTjIA
dr sean
https://www.youtube.com/watch?v=x_xxxvCJjBo
debt
https://hsm.stackexchange.com/questions/2631/historically-how-did-people-define-multiplication-for-negative-numbers
https://www.reddit.com/r/askmath/feedback/iernsb/why_does_multiplying_two_negative_numbers_make_a/
Brahmagupta
https://en.wikipedia.org/wiki/Brahmagupta
quotes on adverse
https://www.meaningfulmaths.nt.edu.au/mmws/nz/websites/default/information/2021-02/babylonian-mathematics2-1.pdf
https://nirajyadav.in/historical-quotes-on-negative-numbers-and-acceptance-by-brahmagupta/
discovermath
https://www.youtube.com/watch?v=qe_PnidC9IE
Khan Academy
https://www.youtube.com/watch?v=rK4sXm_MPWo
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