Lesson 1-5: Scientific Notation

Property

A number is expressed in scientific notation when it is of the form:
a x 10^n
where "a" is greater than or equal to 1 and less than 10, and "n" is an integer. Scientific notation is a useful way of writing very large or very small numbers.

Examples

• For a large number like 4,000, we write it as 4 x 1000, which becomes 4 x 10^3 in scientific notation.
• For a small number like 0.004, we write it as 4 x (1/1000), which becomes 4 x 10^-3 in scientific notation.
• The population of the world, over 6,850,000,000, can be written more simply as 6.85 x 10^9.

Explanation

Think of scientific notation as a compact, secret code for huge or tiny numbers. The first number (the coefficient) holds the most important, significant digits, while the power of 10 acts as an instruction manual, telling you exactly how many places to move the decimal point to see the number's true size.

Property

How to Convert Scientific Notation to Decimal Form

1. Determine the exponent, $n$, on the factor 10.
2. Move the decimal $n$ places, adding zeros if needed.
   • If the exponent is positive, move the decimal point $n$ places to the right.
   • If the exponent is negative, move the decimal point $|n|$ places to the left.

Examples

• Convert $4.5 \times 10^5$ to decimal form. The exponent is positive 5, so move the decimal 5 places to the right to get 450,000.

• Convert $7.1 \times 10^{-3}$ to decimal form. The exponent is negative 3, so move the decimal 3 places to the left to get 0.0071.