Scientific Notation

Introduction to Scientific Notation

The number 10,000,000,000,000 (ten trillion) is a large number that can be tedious and confusing to write repeatedly in standard notation.
To address this, scientific notation was developed as a concise method to express both very large and very small numbers.

Definition: Scientific notation expresses numbers in the form of a product of a number (greater than or equal to 1 but less than 10) and a power of ten.
Mathematical Expression: It is represented as: ext{a} imes 10^n where:
- a is a number such that 1 ≤ a < 10
- n is an integer (which can be positive or negative)
Example: The number 10,000,000,000,000 in scientific notation can be written as:
1 imes 10^{13}

Converting from scientific notation to standard notation involves multiplying the value of a by 10 raised to the power of n.
- For instance:
  - Convert 1 imes 10^{13} back to standard notation:
  - This equals 1 followed by 13 zeros:
    10,000,000,000,000
Converting from standard notation to scientific notation involves determining the appropriate a and n such that the number falls between 1 and 10, then expressing it accordingly.
- For example, the standard number 6,400 can be converted to scientific notation:
  - It is written as:
    6.4 imes 10^{3}

To compare numbers written in scientific notation, one must look at the value of n (the exponent).
If two numbers have the same exponent, compare their a values. The larger a value will be the larger number.
If the exponents are different, the number with the larger exponent will be larger.

Here are a few additional examples of numbers written in correct scientific notation:
1. 6.4 imes 10^{3}
2. 5.248 imes 10^{5}
3. 8 imes 10^{-9}

To determine if a number is correctly written in scientific notation, it must adhere to the following criteria:
- The leading coefficient must be between 1 and 10.
- The base must be 10.
Examples:
- A. 3.2 imes 10^{-2}
- Correct: 3.2 is between 1 and 10.
- B. 0.67 imes 10^{13}
- Incorrect: 0.67 is not between 1 and 10.
- C. 11.2 imes 10^{3}
- Incorrect: 11.2 is not between 1 and 10.

By the end of this lesson, students should be able to define scientific notation, convert numbers to and from scientific notation, and effectively compare numbers expressed in scientific notation.