# How does scientific notation work?

Mar 28, 2014

Scientific notation is the way that scientists handle large numbers and small numbers. For example, instead of writing 0.000 000 0056, they write 5.6 × 10⁻⁹.

We can think of 5.6 × 10⁻⁹ as the product of two numbers: 5.6 (the digits) and 10⁻⁹ (the power of 10).

Here are some examples of scientific notation.

1000 = 1 × 10³; 7354 = 7.354 × 10³

100 = 1 × 10²; 482 = 4.82 × 10²

10 = 1 × 10¹; 89 = 8.9 × 10¹

1 = 1 × 10⁰; 6 = 6 × 10⁰

1/10 = 0.1 = 1 × 10⁻¹; 0.32 = 3.2 × 10⁻¹

1/100 = 0.01 = 1 × 10⁻²; 0.053 = 5.3 × 10⁻²

1/1000 = 0.001 = 1 × 10⁻³; 0.0078 = 7.8 × 10⁻³

The exponent of 10 is the number of places we must shift the decimal point to get the scientific notation.

Each place the decimal moves to the left increases the exponent by 1.

Each place the decimal point moves to the right decreases the exponent by 1.

EXAMPLE:

Write the following numbers in scientific notation: 1001;
6 926 300 000; -392; 0.000 000 13; -0.0038

Solution:

1.001 × 10³; 6.9263 × 10⁹; -3.92 × 10²; 1.3 × 10⁻⁷; -3.8 × 10⁻³

Scientific notation is a shorter way to write really large or really small numbers. Numbers in scientific notation will include a number (a) multiplied by 10 to a power (b). The number a must be at least 1 and less than 10.
a x ${10}^{b}$

For large numbers the power of ten will be positive.

The number 2,130,000,000 can be written in scientific notation as follows:
2,130,000,000 = 2.13 x ${10}^{9}$
The decimal would follow the final 0 in the number on the left, you need to move the decimal 9 spots over to the left to produce the value 2.13

For very small numbers the power of ten will be negative.
The number 0.00000000213 will be written in scientific notation as follows:
0.00000000213 = 2.13 x ${10}^{-} 9$
The decimal need to move 9 spots over to the right to produce the value of 2.13

Make sure to watch the following video to learn how to avoid a common calculator error when working with scientific notation.