Doesn’t x also equal -3?
Uhm, actually 🤓☝️!
Afaik sqrt only returns positive numbers, but if you’re searching for X you should do more logic, as both -3 and 3 squared is 9, but sqrt(9) is just 3.
If I’m wrong please correct me, caz I don’t really know how to properly write this down in a proof, so I might be wrong here. :p
(ps: I fact checked with wolfram, but I still donno how to split the equation formally)
x^2 = 9
<=>
|x| = sqrt(9)
would be correct. That way you get both 3 and -3 for x.
That’s the way your math teacher would do it. So the correct version of the statement in the picture is: “if x^2 = 9 then abs(x) = 3”
You’re correct. The square root operator only returns the principal root (the positive one).
So if x^2 = 9 then x = ±√9 = ±3
That’s why in something like the quadratic formula we all had to memorize in school its got a “plus or minus” in it: -b ± √…(etc)
-3 feels like cheating.
Middle school math memes
-3 = 3
Adding 3 on both sides
3-3=3+3
0 = 6
1•0 = 6
1 = 6/0
1 = inf
Multiplying e^(iπ) on both sides,
e^(iπ) = - inf
iπ = ln|-inf|
π = ln|-inf| ÷ i
I gotta say second half of that goes over my head, but I raise my hat to you
This only ever got handed down to us as gospel. Is there a compelling reason why we should accept that (-3) × (-3) = 9?
You can look at multiplication as a shorthand for repeated addition, so, for example:
3x3=0 + 3 + 3 + 3 = 9
In other words we have three lots of three. The zero will be handy later…
Next consider:
-3x3 = 0 + -3 + -3 + -3 = -9
Here we have three lots of minus three. So what happens if we instead have minus three lots of three? Instead of adding the threes, we subtract them:
3x-3 = 0 - 3 - 3 - 3 = -9
Finally, what if we want minus three lots of minus three? Subtracting a negative number is the equivalent of adding the positive value:
-3x-3 = 0 - -3 - -3 - -3 = 0 + 3 + 3 + 3 = 9
Do let me know if some of that isn’t clear.
i think this is a really clean explanation of why (-3) * (-3) should equal 9. i wanted to point out that with a little more work, it’s possible to see why (-3) * (-3) must equal 9. and this is basically a consequence of the distributive law:
0 = 0 * (-3)
= (3 + -3) * (-3)
= 3 * (-3) + (-3) * (-3)
= -9 + (-3) * (-3).
the first equality uses 0 * anything = 0. the second equality uses (3 + -3) = 0. the third equality uses the distribute law, and the fourth equality uses 3 * (-3) = -9, which was shown in the previous comment.
so, by adding 9 to both sides, we get:
9 = 9 - 9 + (-3) * (-3).
in other words, 9 = (-3) * (-3). this basically says that if we want the distribute law to be true, then we need to have (-3) * (-3) = 9.
it’s also worth mentioning that this is a specific instance of a proof that shows (-a) * (-b) = a * b is true for arbitrary rings. (a ring is basically a fancy name for a structure with addition and distribute multiplication.) so, any time you want to have any kind of multiplication that satisfies the distribute law, you need (-a) * (-b) = a * b.
in particular, (-A) * (-B) = A * B is also true when A and B are matrices. and you can prove this using the same argument that was used above.
Here’s another example:
A) -3 × (-3 + 3) = ?
You can solve this by figuring out the brackets first. -3 × 0 = 0
You can also solve this using the distributive property of multiplication, rewriting the equation as
A) -3 × (-3 + 3) = 0
(-3 × -3) + (-3 × 3) = 0
(-3 × -3) - 9 = 0
(-3 × -3) = 9
If (-3 × -3) didn’t equal 9 then you’d get different answers to equation A depending on what method you used to solve it.
