Shall we?
Let's take a look at this multiplication table
25 as the Tail
As you can see above, any number which ends with 5 if squared will have 25 as the tail.
Prefix — Yellow Colored Number
I put yellow color for the opening number sequence because it is interesting.
Let's see the 15 x 15 line.
If we isolate the first number, which is 1, and we do:
1 x (1 + 1) = 2It yields 2, the opening of 15² result, 225.
Another example 45 x 45 or 45²:
We take the first number, which is 4, and we do the same technique
4 x (4 + 1) = 2020 is the starting sequence (before the ending, 25) of the result, 2025.
Last example, the 5005 x 5005 or 5005².
500 x (500 + 1) = 250500
Which is also the first sequence of the result, 25050025.
Speed Squaring
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From the examples above, we can generalise the pattern for the prefix (opening) number sequence as this:
numberBefore5 x ( numberBefore5 + 1 )
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The suffix (tail) will always be 25.
We can attach it as the tail (suffix) of the result.
This is how you speed square for the streets, mate. 👍
Assuming... the street... only consists of the squares of that odd-5.
Odd-5: 15, 25, 35, 45, 55, ...
15 = 5 × 3
25 = 5 × 5
35 = 5 × 7
45 = 5 × 9
55 = 5 × 11
...
3, 5, 7, 9, 11, and so on are odds.
(2n - 1) in posh.
In a conversation.
Yo, sup, dude.
25!
Wut? You high?
Blimey! 13225, then!
🤦
We don't know what the conversation is about — like we listen to the telly waiting for Sadako to crawl out while we're clicking the mute/unmute button. Of a radio. Well, that's unexpected. Can we toss something to that Sadako-gate? 🤔 Once the inter-blimey-gate is opened, she crawls out, aye? Thus perhaps we can throw something into the telly. 🤔
📽️ Sadako Yamamura is that dubious specter from the film "Ring" (1998). Because Sadako can bloody scan phone number by retina match. And can call that number. But imagine if the bloke were carrying his friend's mobile phone. 🤷
VHS 📼
🧑🦱 (Watching VHS)
📞 (Ring, ring)
🧑🦱 Yellow?
👻 Nanoka-go ni shinu
🧑🦱 Oi, bazooka go boom boom
Thus the film was called that. Ring.
Watching that film was like I joined a group of people in the middle of their conversation — no background, build up, just them throwing bottles and spraying ketchup to each other. Which is good for a short attention span, twitchy gremlin as myself. 👺 (Tengu is conjured.)
Usage
Let's find the result of 75².
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The prefix sequence:
- The suffix is 25.
Then the result of 75² is 5625.
Decimal Point
Let's use 0.75.
Calculate the result of 0.75²!
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Take the 7. So the prefix number sequence:
- The suffix is always 25.
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Last step is to count how many digits after the decimal point.
We have 0.75 (two digits) squared. Then we will have 4 digits behind the decimal point.
Result of 0.75² is 0.5625.
Thank you for visiting, see you again. I hope this bit is useful. 😃
Hi Johan,
ReplyDeleteThat's some very keen thinking and a neat trick you have there! There's also another pattern: each number that we're squaring is an odd multiple of 5 (that is, 5*1, 5*3, 5*5,...), which we can write as (5*(2n-1))^2. Therefore, the prefactor must depend on which n we choose. If we pair up the first few n's with the prefactors, we find 1:0, 2:2, 3:6, 4:12, 5:20, etc. The relationship is that each n is multiplied by (n-1) (ex: 1*(1-1) = 0, 2*(2-1) = 2, 3*(3-1) = 6). Of course, the prefactors are multiples of 100, so we multiply by 100. Altogether we find that (5*(2n-1))^2 = 100*n*(n-1) + 25, which we could simplify to (2n-1)^2 = 4*n*(n-1) + 1.
Best,
Another Math Wizard
Wow that's really structured. I dig it!
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