Homework Help: Bored in math class again, so I made this

1. Oct 25, 2005

Blahness

If B - A = 1,
Then
A^2 + A + B = B^2.

Good for figuring out an exponent next to one you know.
Example:
You know 50^2 is 2500, but need 49^2.

B^2(2500) - B(50) - A(49) = B^2(2401).

Make sense?
Useful, useless, w/e?

2. Oct 25, 2005

TD

Sure:

$$a^2 + a + b = b^2 \Leftrightarrow a^2 - b^2 = - \left( {a + b} \right) \Leftrightarrow \left( {a - b} \right)\left( {a + b} \right) = - \left( {a + b} \right) \Leftrightarrow \left( {a - b} \right) = - 1 \Leftrightarrow b - a = 1$$

I suppose so, but I doubt it's "new"

3. Oct 25, 2005

Blahness

Probably not, but I need to know that the crap I think up in math class isn't junk! ^_^
Ty ^_^

4. Oct 25, 2005

Galileo

Knowing and applying 'tricks' like these is usually how I am able to multiply big numbersin my head. Note: Big means 2 digits.
For example, using (a+b)(a-b)=a^2-b^2, calculating 47 times 53 is easy:
$$47 \cdot 53=(50+3)(50-3)=50^2-3^2=2500-9=2491$$

5. Oct 25, 2005

Kamataat

To find roots near 50, use (50+/-x)^2 = 2500 +/- 100x + x^2. In other words, to find 47^2 just subtract 3 from 25 to get 22 and square 3 to get 09, so 47^2=2209.

Read it from one of Feynman's books.

- Kamataat

6. Oct 25, 2005

masudr

Unfortunately, these sorts of tricks alone will not get you very far.

7. Oct 26, 2005

Galileo

They always work like a charm for me. Then again, I really suck at mental calculations without 'tricks'.

To square a number a ending with a five quickly:
Write $a = 10b+5$. (division by 10 with remainder 5). b is simply the number you get after dropping the 5 mentally.
$$a^2=(10b+5)^2=100b^2+100b+25=100b(b+1)+25$$
So you simply take b, multiply with the next integer and glue 25 at the end.

25^2: 2 times 3 is 6. 'add' 25 to get 625
85^2: 8 times 9 is 72. 'add' 25 to get 7225
etc.

8. Oct 26, 2005

quasar987

I think this trick given to Feynman by Hans Bethe while they were at Los Alamos!

9. Oct 26, 2005

MathematicalPhysicist

mental calculations as you coined it really depend on your memory.

10. Oct 26, 2005

Cosmo16

Another one I have found is this.

5^2= 25 0=n
15^2= 225 2=n
25^2= 625 6=n
35^2=1225 12=n
45^2=2025 20=n
55^2=3025 30=n

How would I express that algebracly?

11. Oct 26, 2005

shmoe

Try expanding (m*10+5)^2

12. Oct 26, 2005

Kamataat

Yes, it was indeed!

- Kamataat

13. Oct 26, 2005

Robokapp

My achievements in math: a^2=(a+1)*(a-1)+1 for {a>N/a>0}

in other words 49^2=50*48

so 49^2=2400

14. Oct 26, 2005

roger

15. Oct 31, 2005

Werg22

A power has the same factors as its rational roots.

16. Nov 1, 2005

Blahness

49^2 = 2401.

Anyway, if it's a^2=(a+1)*(a-1)+1
That becomes
49^2=(50)*(48)+1
reduces to
2401 = 2401.

Just clearing that up. ^_^

Much more simplistic version of my equation, nice Robo! ^_^''