Power of Ten and Scientific Notation

Power of ten is a method for writing very large and very small numbers in a clear way.

Instead of writing all the zeros, we can use powers of ten and make the numbers short and easy to read.

 

 

Large numbers

When the exponent is positive, we get large numbers. The exponent indicates how many zeros stand after the 1:

 

$$ \large 10^2 = 100 $$

$$ \large 10^3 = 1,000 $$

$$ \large 10^6 = 1,000,000 $$

 

Example:

 

The distance to the Moon is about \(384,000,000 \,\text{m} = 3.84 \cdot 10^8 \,\text{m}\).

 

In the same way, one can write the number of stars in the Milky Way. It is estimated to be about \(100,000,000,000 = 1 \cdot 10^{11}\) stars.

 

 

Small numbers

Negative exponents are used to write small numbers. The exponent shows how many decimal places the number shifts:

 

$$ \large 10^{-1} = 0.1 $$

$$ \large 10^{-2} = 0.01 $$

$$ \large 10^{-4} = 0.0001 $$

 

Examples:

 

A human hair is typically \(0.0001\,\text{m} = 1 \cdot 10^{-4}\,\text{m}\) thick.

An atom’s radius can be about \(0.00000000005\,\text{m} = 5 \cdot 10^{-11}\,\text{m}\).

 

 

Scientific notation

When we write large or small numbers in this way, it is called scientific notation. The form looks like this:

 

$$ \large a \cdot 10^b $$

 

Here \(a\) is a number between 1 and 10, and \(b\) is an integer. Requiring \(a\) to be between 1 and 10 ensures that the number is written as short as possible.

 

Example:

 

$$ \large 384,000,000 = 3.84 \cdot 10^8 $$

 

 

Rules

The usual rules of exponents also apply to powers of ten:

 

$$ \large 10^m \cdot 10^n = 10^{m+n} $$

$$ \large \frac{10^m}{10^n} = 10^{m-n} $$

$$ \large (10^m)^n = 10^{m \cdot n} $$

 

 

Notation on calculators

Many calculators and computer programs use E (or EE) instead of 10 in scientific notation. The exponent is written directly after the E:

 

$$ \large 3.84 \cdot 10^8 = 3.84E8 $$