Idle Musings On 201
Created Wednesday 11 September 2024
https://mathstodon.xyz/@ColinTheMathmo/113104047866813647
The other day I had a streak of 201 on one of the games I play, and that started a train of thought.
Idly I factored 201 (as one does) to get 3*67, then realised that it is nice as a difference of squares:
- 3 and 67 add to 70
- So the average of 3 and 67 is 35
- The mutual distance from 35 to each of 3 and 67 is 32
- 3 and 67 are therefore 35-32 and 35+32 respectively
- 201 = 3*67 = (35-32)*(35+32)
- 201 = 35²-32²
Interestingly, both of these squares are quite nice.
32² = 1024, because 32=2⁵ and so we get 2¹⁰ which is 1024;
35² = 30*40+25 = 1225
That's because:
35² = (35-5)*(35+5) + 5²
That's using the difference of squares in its less common form.
- a²-b² = (a-b)(a+b)
- a² = (a-b)(a+b) + b²
So we get:
201 = 1225 - 1024 ...
Which is obviously true, but we got there with some simple mental arithmetic, and using the difference of two squares in two different ways.
Which I found satisfying.
And yes, you can use this to help calibrate your beliefs about the sorts of things I find satisfying.
https://mathstodon.xyz/@ColinTheMathmo/113107783535436561
So as an example ... I don't know what 55² is, and I wouldn't bother to compute it by hand.
But I do know:
- a²-b² = (a-b)(a+b)
Adding b² to both sides:
- a² = (a-b)(a+b)+b²
Setting a=55 and b=5 we get
- 55² = (55-5)(55+5) + 5²
Suddenly it's obvious, not because of the numbers, but because of the structure.
- 55² = (55-5)(55+5) + 5²
- 55² = (50 times 60) + 25
- 55² = 3000 + 25
- 55² = 3025
The facility with numbers is an epiphenomenon.
Questions:
- I didn't know about that formula, where does b comes from?
- If I want to calculate another squared number, how do I get b?
So the starting point is the "Difference of Two Squares" formula. So we have:
a²-b² = (a-b)(a+b)
You can go ahead and check that to make sure it's valid.
Then we can switch is around to get:
a² = (a-b)(a+b) + b²
Now when we want to square a number (call it "a") we have a free choice of "b" to see if we can make it easy.
Take 62² for example. Then choosing b=2 gives us:
62² = (62-2)(62+2) + 2²
That's (60 times 64) plus 4.
Now, 60 times 64 is 60 times 60 (which is 3600) plus 4 times 60 (which is 240) so we get:
- 62² = (60 times 64) plus 4
- 62² = (3600 + 240) + 4
- 62² = 3844
... and we're done.
But remember we have a free choice for b, so we can choose something different. We can choose, for example, b=12.
(You will need to follow this through to get the full benefit, and you'll see why I claim it's not that hard.)
- 62² = (62-12)(62+12) + 12²
Now we have 50 times 74. Double 50 and halve 74 and we have 100 times 37.
- 62² = (62-12)(62+12) + 12²
- 62² = (50 times 74) + 12²
- 62² = (100 times 37) + 144
- 62² = 3700 + 144
- 62² = 3844
Nice that it's the same answer! (which is encouraging)
All this is quite straight-forward, but making a "good choice" for b comes with experience.
Practising really makes a difference.
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