hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)
Why is 9 such a unique number?
The number 9 has fascinated mathematicians for centuries because it has some genuinely unusual mathematical properties.
1/ Its multiples always reduce back to 9.
Add the digits of any multiple of 9 until you get a single digit, and you’ll always end up with 9.
9 → 9
18 → 1 + 8 = 9
72 → 7 + 2 = 9
1,161 → 1 + 1 + 6 + 1 = 9
2/ The famous finger trick.
You can multiply 9 by numbers 1 through 10 using your hands.
3/ The multiplication pattern.
09, 18, 27, 36, 45, 54, 63, 72, 81, 90.
The tens digit increases.
The ones digit decreases.
Each pair of digits adds up to 9.
4/ The divisibility test.
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
4,356 → 4 + 3 + 5 + 6 = 18, so 4,356 is divisible by 9.
5/ It’s the largest single-digit number.
Nine represents the end of one cycle before numbers move into two digits, which is why many cultures associate it with completion.
6/ Since 9 = 10 − 1, it makes mental math simple.
9 × 47 = (10 × 47) − 47 = 470 − 47 = 423.
The reason this works is that multiplying by 10 gives you 10 copies of the number. To get 9 copies, you simply remove 1 copy. Since 1 copy is 1 out of 10.
People often confuse percentages with percentiles, but they’re completely different.
A percentage tells you how much out of 100.
Example:
You scored 80% on a test.
That simply means you got 80 out of every 100 marks.
A percentile tells you how you performed compared to everyone else.
Example:
You scored in the 80th percentile.
That means you performed better than 80% of the people who took the test. It does not mean you scored 80%.
Another example:
- Student A scores 95% but is only in the 70th percentile because many students scored similarly high.
- Student B scores 75% but is in the 95th percentile because the test was very difficult and almost everyone scored lower.
Percentage = your score.
Percentile = your rank relative to others.
Same word root but different meanings.
I pray for righteous friendships
Friendships free from strife, envy and the corruption of this world
Friendships that’ll sharpen one other
Friendships that’ll challenge us to be closer to God
Friendships that we’ll outdo each other in humility and love.
math is so beautiful. these people predicted that the crew would come all the way from space and land in the water at exactly 8:07 PM EST and that's the exact same time - down to the minute - that they splashed down.
My God will make all things new
He will wipe away all tears
He will make sense of the evil we see in this world.
May God comfort the families of the victims of the recent killings.
God have mercy.