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multiplication with 11

Gini and Karl meet the very polite number 11

It was late afternoon at the Penguin Café. The snow outside fell in neat little lines, almost like... columns.

Karl stared at the chalkboard. "Why does multiplying by 11 feel both easy and mysterious?"

Gini smiled. "That's because 11 is a very polite number. It doesn't like to work alone."

Gini wrote:

1 * 11 = 11

2 * 11 = 22

3 * 11 = 33

Karl laughed. "So 11 just copies the digit!"

"For single digits," Gini said. "But wait until it meets bigger numbers."

Flip, Pip, Flipflop, and Waddle leaned in.

"Here's the rule which I call the gap trick," Gini said, drawing a number with a space in the middle.

"When you multiply a number by 11, you write the number... and leave a gap between the digits."

"To fill the gap," Karl added, "you just add the neighboring digits."

They tried:

10 * 11
→ 1 _ 0
→ 1 + 0 = 1
→ 110

12 * 11
→ 1 _ 2
→ 1 + 2 = 3
→ 132

"That's... surprisingly friendly," said Pip.

Then Gini wrote:

67 * 11
6 _ 7
6 + 7 = 13

"Oh!" Karl said. "That's too big for one gap."

"Exactly," said Gini. "So we write the 3, and carry the 1."

→ 737

They tried more:

89 * 11 → 8 _ 9 → 17 → 979

93 * 11 → 9 _ 3 → 12 → 1023

Flipflop nodded. "11 is polite, but it still expects you to handle your carry-overs."

Chef Pebble brought more cocoa just as Karl pointed at a long number.

74352 * 11

Gini explained: "We work from right to left."

write 2

add 5 + 2 = 7 → write 72

add 5 + 3 = 8 → write 872

add 3 + 4 = 7 → write 7872

add 4 + 7 = 11 → write 17872, carry 1

add carry: 7 + 1 = 8

Final answer: 817872

Karl's flippers froze mid-air. "We didn't even need paper."

Another Try: 52438 * 11

They went step by step:

write 8

3 + 8 = 11 → write 18, carry 1

3 + 4 + 1 = 8 → write 818

2 + 4 = 6 → write 6818

2 + 5 = 7 → write 76818

final digit: 5

576818

Waddle applauded softly.

Flipflop tapped the chalkboard. "This isn't magic," he said. "It's just long multiplication in disguise."

He showed them:

52438
* 11
----------
52438
+524380

"And when you add those rows," Flipflop explained, "you're already adding neighboring digits and carrying when needed."

Gini nodded. "The shortcut just lets your brain do what your pencil normally does."

Karl grinned. "So multiplying by 11 is like... fast-forward long multiplication."

Chef Pebble raised his mug. "To clever shortcuts!"

As evening settled over the Penguin Café, 11 sat quietly on the chalkboard - still polite, still helpful, and no longer mysterious at all.

So here again:

To multiply a single digit number with 11:

1*11 = 11
2*11 = 22
3*11 = 33
4*11 = 44
5*11 = 55
etc.
easy!

There is also a shortcut to multiplying bigger numbers with 11. You write the number that you want to multiply with 11 and leave a gap between the 2 digits of that number. To find what digit you need to write in the gap you just add the 2 other digits together and voila, you have your result.
Let's look at 2 digit numbers first:

10*11 : 1 __0 with __=1+0=1 --> 10*11 = 110

11*11 : 1 __1 with __=1+1=2 --> 11*11 = 121

12*11 : 1 __2 with __=1+2=3 --> 12*11 = 132

13*11 : 1 __3 with __=1+3=4 --> 13*11 = 143

14*11 : 1 __4 with __=1+4=5 --> 14*11 = 154

67 *11: 6__7 with _ = 6+7 = 13 carry-over!!! --> 67 *11 = 737

89*11: 8__9 with _= 8+9 = 17 carry-over!!! --> 979

93*11 : 9_3 with _= 9+3=12 carry-over!!! ---> 1023

You can do it in a similar way when the numbers get bigger. Just now you need to add neighbouring digits and you need to do several additions. Let's have a look:

74352 * 11

write 2
next add 5 + 2 = 7
write 72
add 5+3=8
write 872
add 3+4=7
write 7872
add 4+7=11
write 17872 remember 1 for carry over
now 7 + 1 (the carry over)=8
final result: 817872
74352 * 11 = 817872

52438 * 11

write 8
add 3 + 8 = 11
write 18 remember 1 for carry over
add 3+4=7 , 7+1 (carry over) =8
write 818
add 2+4=6
write 6818
add 2+5=7
write 76818
final result 576818
52438 * 11 = 576818

To get an idea why you can do this you have to look at long multiplication (like with the conjuring up of the numbers by magic).
example:

52438 * 11 =

		5	2	4	3	8
+	5	2	4	3	8	0
carry over	-	-	-	1	-	-
	-	-	-	-	-	-
	5	7	6	8	1	8

You can see that you are really doing the same thing when you do long multiplication. It's just that long multiplication is usually done with pen and paper while you can try to do the above method in your head.