writing big and small numbers with scientific notation

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scientific notation

Gini and Karl and the Case of the Really Big and Really Small Numbers

Karl burst into the snow hut wearing welding goggles, holding something that looked suspiciously like a fish can strapped to a telescope.

"Gini!" he shouted. "Come quick! I've invented a Mega-Scope! It can zoom in and out so far that we can finally see the biggest and smallest numbers in the universe!"

Gini raised an eyebrow. "Karl... last time you invented something, it launched my scarf into low Earth orbit."

"That was a feature, not a bug!" Karl insisted. "Anyway, look!"

He pointed the Mega-Scope outward into the snowy distance.

Gini peered through the lens.

"Whoa!" she cried. "Karl, I'm seeing... a thousand snowflakes all landing at once!"

Karl nodded dramatically. "Exactly! And look how nicely it fits into scientific notation:"

1000 = 10³

"That means," Karl explained, "ten multiplied by itself three times."

Gini typed quickly on her flipper-keyboard:

"Okay, let me try... a hundred snowflakes?"

100 = 10²

"And over there," she gasped, zooming out further, "a million snowflakes!"

1,000,000 = 10⁶

"And Karl, I think I just spotted... a billion snowflakes."

Karl fainted briefly from excitement.

1,000,000,000 = 10⁹

When he recovered, he puffed out his chest. "See? Scientific notation is perfect for enormous numbers. Instead of writing out a gazillion zeros, we just say 10 to the power of how many zeros we skipped!"

"Okay," said Gini, "that was fun. Let's shrink it!"

Karl flipped the Mega-Scope backwards.

The world suddenly zoomed in, and the snow beneath them looked like a vast desert.

A tiny sparkle appeared.

"What's that speck?" Gini asked.

Karl squinted. "That, my dear Gini... is the number:

0.000 000 000 043 6

Which is MUCH easier to write as:

4.36 * 10^-11

"The minus sign," Karl said, "means we're dealing with tiny, tiny numbers. The exponent tells us how far the decimal point crawled."

Gini watched the decimal point in the desert scoot left and right like a confused beetle.

"Let's try another one!" she said, tapping the zoom-in button.

4.2 * 10^-7 = 0.000 000 42

"Oh!" Gini gasped. "And look-this dust grain!"

0.000 000 005 78 = 5.78 * 10^-9

Karl whispered reverently: "Tiny numbers... nature's way of reminding us that small things matter too."

Suddenly the decimal point popped out of the sand, wearing a tiny hat and holding a picket sign.

"STOP MAKING ME MOVE SO MUCH!"

Gini burst out laughing.

Karl negotiated with it.

"Okay, okay! One last demo," he said.

He placed a small snowball in front of the decimal point.

0.00036 = 3.6 * 10^-4

The decimal point trudged four steps to the right.

Then Karl placed a negative snowball (a snowball he owed somebody?).

-0.00036 = -3.6 * 10^-4

Finally, Gini placed a giant snowball:

36,000 = 3.6 * 10⁴

And a giant negative one:

-36,000 = -3.6 * 10⁴

The decimal point sighed deeply, nodded, and went back underground.

Gini stretched her wings contentedly.

"So scientific notation is just a way to write VERY big or VERY small numbers more easily," she summarized.

"Move the decimal to create a number between 1 and 10, then multiply by 10 to the required power."

Karl grinned. "Exactly! And now... behold my newest notation invention!"

Gini frowned. "Karl, is it going to explode?"

"...possibly."

A muffled BOOM echoed in the distance.

"Ah," Karl said, "There goes test version two."

They both laughed and waddled home, ready to classify more numbers another day.

writing big numbers

1000= 10³
100 = 10²
1000000=10⁶
1000000000=10⁹

writing small numbers

0.000 000 000 043 6 = 4.36 * 10^-11
4.2 * 10^-7 = 0.000 000 42
0.000 000 005 78 = 5.78 * 10 ^-9
-0.00036 = -3.6 * 10^-4
0.00036 = 3.6 * 10^-4
36,000 = 3.6 * 10⁴
-36,000 = -3.6 * 10⁴

references

https://www.purplemath.com/modules/exponent3