The number 3:

• Take any number: 9
• Double it: 9 * 2 = 18
• Add 6: 18 + 6 = 24
• Take half of it: 24 / 2 = 12
• Take away the number your started with: 12 - 9 = 3

The number 1089:

• Take any 3-digit number where, from left to right, it descends: 321
• Inverse the digits: 123
• Subtract the two: 321 - 123 = 198
• Inverse this result: 891
• Add these two numbers: 198 + 891 = 1089

The number 11880:

• Take any 4-digit number where, from left to right, it descends: 4321
• Inverse the digits: 1234
• Subtract the two: 4321 - 1234 = 3087
• Inverse this result: 7803
• Add these two numbers: 3087 + 7803 = 11880

The numbers 3, 6, and 9:

• Take a circle, 360 degrees. Any half angle from 360 degrees, the sum of its halved numbers is a factor of 9
• 3 + 6 + 0 = 9
• Divide 360 by 2 = 180. 1 + 8 + 0 = 9
• Divide 180 by 2 = 90. 9 + 0 = 9
• Divide 90 by 2 = 45. 4 + 5 = 9
• Divide 45 by 2 = 22.5. 2 + 2 + 5 = 9
• Divide 22.5 by 2 = 11.25. 1 + 1 + 2 + 5 = 9
• Divide 11.25 by 2 = 5.625. 5 + 6 + 2 + 5 = 18
• Divide 5.625 by 2 = 2.8125. 2 + 8 + 1 + 2 + 5 = 18
• Divide 2.8125 by 2 = 1.40625. 1 + 4 + 0 + 6 + 2 + 5 = 18