0
Mathematicians
Often, to completely understand the importance of a mathematical theory, it's essential to know about the person who developed it. This category is where all questions about historically significant mathematicians should be asked.
6,570 Questions
It is 10200.
What is the Cauchy Kovalevskaya theorem?
The Cauchy kovalevskaya theorem tells us about solutions to systems of differential equations. If we look at m equations in n dimension, with coefficient that are analytic function, we can know about the existence of solutions using this theorem.
If the exterior angle of a regular polygon is 108 degrees how many sides does the polygon have?
That's true if the interior angles are 108 degrees, but a regular polygon cannot have exterior angles of 108 degrees.
70 feet below sea level.
The total number of rolls, R, is
R = 5p + 3
What is the quotient of a number and -5 increased by one is at most 7.?
It is the equation:
n/(-5) + 1 <= 7
0
A can be 0 or 9.
A + b = 0, 9, 18
(9,0)(8,1)(7,2)(6,3)(5,4)(4,5)(3,6)(2,7)(1,8)(0,9)(0,0)(9,9)
-------------------------------------------
To be divisible by 9, the sum 6 + 3 + 1 + 8 + 7 + 2 + A + B = 27 + A + B must be divisible by 9
→ A + B must be a multiple of 9.
→ (A, B) must be (0, 0), (0, 9), (1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1), (9, 0)
Note that (0, 0) must not be forgotten as 27 + 0 + 0 = 27 = 3 × 9.
What irrational number has the decimal representation that begins 0.232323232?
23/99, which is rational.
What fraction of whole numbers is equal to 17100 and that has a denominator that is a power of 10?
17100 = 17100/1
= 17100/1 × 10/10 = 171000/10
= 17100/1 × 10²/10² = 1710000/100
= 17100/1 × 10³/10³ = 17100000/1000
...
= 17100/1 × 10ⁿ/10ⁿ
There is no upper limit to this sequence: n = 0, 1, 2, 3, ....
These numbers can be put in a one-to-one relationship with the counting numbers 1, 2, 3, ...
The counting numbers can also be put in a one-to-one relationship with the whole numbers
Therefore the sequence of fractions can be put in a one-to-one relationship with the whole numbers.
Therefore the fraction of whole numbers which are in the sequence is 1.
-----------
Let's try another approach:
As 10 to some power is positive, there is no way we can change a negative number to make it positive by dividing by 10 to some power, therefore the solution numbers are all positive.
As negative whole numbers make up half of the whole numbers, the fraction of whole numbers which are equivalent to 17100 when divided by a power of 10 is at most ½.
The fraction of the ½ of the whole numbers which are positive can now be considered:
After the nth whole number has been found which matches the criteria that is it equivalent to 17100 when it is divided by a power of 10, in total there are n numbers matching out of a total of 17100 × 10ⁿ⁻³ numbers (the value of the nth number); thus:
The first number which meets the criteria is 171/10⁻² → the fraction is 1/171
The second number to meet the criteria is 1710/10⁻¹ → the fraction is 2/1710
The third number to meet the criteria is 17100/10⁰ → the fraction is 3/17100
The fourth number to match the criteria is 171000/10 → the fraction is 4/171000
The fifth number to match the criteria is 1710000/10²→ the fraction is 5/1710000
→ For the nth match, the fraction is: n/(17100 × 10ⁿ⁻³) = (1/17100) × n/10ⁿ⁻³
This gives us a sequence of fractions:
1/17100 × 1/10⁻², 1/17100 × 2/10⁻¹, 1/17100 × 3/10⁰, 1/17100 × 4/10¹, 1/17100 × 5/10², ...
= 100000/17100000, 20000/17100000, 3000/17100000, 400/17100000, 35/17100000, 6/17100000, ....
Consider terms n and n+1:
U{n} = (1/17100) × n/10ⁿ⁻³ = (1/17100) × 10 × n/10ⁿ⁻²
U{n+1} = (1/17100) × (n+1)/10ⁿ⁻²
U{n} - U{n+1} = (1/17100) × 10 × n/10ⁿ⁻² - (1/17100) × (n+1)/10ⁿ⁻²
= (10n - (n+1))/(17100 × 10ⁿ⁻²)
= (9n - 1)/(17100 × 10ⁿ⁻²)
As n ≥ 1,
9n - 1 ≥ 9×1 - 1 = 8
→ term n - term n+1 ≥ 8 > 0
→ term n is larger than term n+1
for all n ≥ 1
Each of these terms is less than 1, and each term of the sequence is smaller than the previous one and so as n increases the value of the each term (the fraction of whole numbers which meet the criteria) tends towards 0.
→ The fraction of all whole numbers which equate to 17100 when divided by a power of 10 is as near enough to zero as make no odds.
ie the fraction is so small it is effectively none of the whole numbers.
-----------------------
This is a problem of dealing with the infinite.
What are integers less than or equal to 2?
The integers less than or equal to 2 are: {2, 1, 0, -1, -2, -3, ...}
