Welcome to a video on curious calculations. In this video we will look
at a pattern when multiplying a counting number by itself with one's in every
place value. For example, 1 times 1 11 times 11 111 times 111 and so on. Let's
take a look. To begin we know that 1 times 1 is equal
to 1. 11 times 11 is equal to 121. Looking at the product of 11 times 11, notice
both factors contain exactly two ones and for the product we start at one
count up to two and then go back down to one. Following this pattern for 111 times
111, because both factors contain exactly three ones, for the product we begin with
one count up to 3 so 1 2 3 and then count back to 1 so after 3 we have 2 and
then 1. The product is 12,321. Following this pattern for the next
product, because both factors contain exactly 4 ones, the product from left to
right is 1 2 3 4 3 2 1. The product is 1,234,321.
Next both factors contain exactly 5 ones and there for the product from left to right is 1 2 3 4 5 4 3 2 1.
The product is 123,454,321.
Next both factors contain exactly 6 ones.
And therefore the product is 1 2 3 4 5 6 5 4 3 2 1. The product is 12,345,654,321.
Next both factors contain exactly 7 1 s and therefore the product is 1 2 3
4 5 6 7 6 5 4 3 2 1.
And now we're getting into products the most tech leaders cannot handle or will
convert to scientific notation. Looking at the place values here we have the
thousands millions billions and trillions the product is 1 trillion 234
billion 567 million 654 thousand 321. For the next product each factor
contains 8 ones. The product is one two three four five six seven eight seven
six five four three two one. Looking at the place values, we have the thousands
millions billions and trillions the product is 123 trillion 456 billion
seven hundred eighty seven million 654 thousand 321. For the next product both factors
contain exactly nine ones and therefore the product is one two three four five
six seven eight nine eight seven six five four three two one.
Again looking at the place values, we have the thousands millions billions
trillions quadrillions the product is twelve quadrillion 345 trillion six
hundred seventy eight billion 987,654,321. This is a product that again most
calculators cannot determine without converting to scientific notation and
losing some accuracy. Before we go you might be asking what happens if we have
two factors that contain ten ones. We can still use this pattern to help us
determine the product though it does require some extra work. Let's take a
look at one more example. Here we have our product where both factors contain
exactly ten ones, so following the pattern we start at one count up to ten
and back down to but we know we can only have a single
digit in each place value and right now we have a ten in the one billions place
value. And therefore to determine the product we do need to perform an
exchange ten one billions is equal to 1 10 billion so we place a zero in the
one billions place value perform an exchange and here we have a one in the
ten billions place value. But now notice in the ten billions place value we have
nine plus one or ten ten billions and therefore need to perform another
exchange by placing a zero in the ten billions place value and a one in the
hundred billions place value again this is because ten and ten billions equals
one 100 billion. And now we have one plus eight or nine in the hundred billions
place value, a zero when the ten billions place value a zero and the one billions
place value. And we have our final product. So we can still use the same
pattern to determine larger products but we do have to perform some exchanges
before determining the final product. Let's see if we can read this number
properly. Again we have the millions billions trillions quadrillions and
quintillions. The product is one quintillion 234 quadrillion 567 trillion
900 billion 987 million 654 thousand 321. I hope you found this interesting.