1. Rational Numbers

**Scientific notation** is a way of representing very large or very small numbers, without needing to write lots of zeros. Consider the mass of the sun, which is approximately 1,988,000,000,000,000,000,000,000,000,000 \text{ kg}. That's a very large number. How do scientists deal with numbers so large?

This applet shows how a decimal number is written in scientific notation.

- Click the "Generate a random number" button to explore different numbers.
- Check the box to see what each number looks like written in scientific notation.

Explore 10 random numbers and how the power of ten relates to the number in the tens position.

- What types of numbers have positive or negative powers of ten?
Describe the relationship between the place value of the digits and the number in scientific notation.

- Can you come up with a rule for taking a very large number in scientific notation? What about a very small number?

Going back to the example about the mass of the sun:

The sun has a mass of approximately 1.988\times 10^{30} \text{ kg} , which is much easier to write than 1,988,000,000,000,000,000,000,000,000,000 kg.

The mass of an atom of Uranium (one of the heaviest elements) is approximately 3.95\times 10^{-22} g. That is 0.000\,000\,000\,000\,000\,000\,000\,395 g.

In scientific notation, numbers are written in the form:

\displaystyle a \times 10^{n}

\bm{a}

decimal number greater than or equal to 1 but less than 10

\bm{n}

positive or negative integer

A negative exponent indicates how many factors of ten smaller than a the value is.

A positive exponent indicates how many factors of ten larger than a the value is.

We can follow these steps in writing numbers in scientific notation.

Move the decimal point to the left or right so that it is right after the first non-zero digit (from 1to 9).

Where's the decimal point in 2,680,000? Because it's a whole number, the decimal point is understood to be at the end of the number: 2,680,000.

The first non-zero number is 2. If we move the decimal point 6 places from the end of the number to the right of the 2, we will get 2.68. We don’t need the extra zeroes. The number 2.68 is between 1 and 10 as we wanted.

Multiply by 10 to the power of the number of places the decimal moved.

We moved 6 places to the left so we have 10^{6}.

Standard form | Product form | Scientific notation |
---|---|---|

2,680,000 | 2.68\times 1,000,000 | 2.68\times 10^{6} |

Remember, we're not actually "moving" the decimal point. We're adjusting the place value. Then to balance that out, we have to multiply by the correct power of 10.

What value should go in the space to make the expression on the right in scientific notation form? 300 = ⬚ \times 10^{2}

Worked Solution

Express 0.07 in scientific notation.

Worked Solution

Express 3.66\times 10^{-6} as a decimal number.

Worked Solution

Scientists recorded the diameter of Mercury as 4.879 \times 10^{3}\text{ km}. Write the diameter of mercury in standard form.

Worked Solution

Idea summary

A number is written in **scientific notation** if it has the form a \times 10^{n} where a is greater than or equal to 1 and less than 10, and n is an integer.

Standard form | Product form | Scientific notation |
---|---|---|

2,680,000 | 2.68\times 1,000,000 | 2.68\times 10^{6} |

0.000794 | 7.94\times \dfrac{1}{10,000} | 7.94\times 10^{-4} |