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 of Scientific Notation

  • 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: extaimes10next{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:
    1imes10131 imes 10^{13}

Converting Between Scientific Notation and Standard Notation

  • Converting from scientific notation to standard notation involves multiplying the value of a by 10 raised to the power of n.
    • For instance:
      • Convert 1imes10131 imes 10^{13} back to standard notation:
      • This equals 1 followed by 13 zeros:
        10,000,000,000,00010,000,000,000,000
  • 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:
        6.4imes1036.4 imes 10^{3}

Comparing Numbers in Scientific Notation

  • 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.

Examples of Scientific Notation

  • Here are a few additional examples of numbers written in correct scientific notation:
    1. 6.4imes1036.4 imes 10^{3}
    2. 5.248imes1055.248 imes 10^{5}
    3. 8imes10−98 imes 10^{-9}

Evaluating Correctness of Scientific Notation

  • 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. 3.2imes10−23.2 imes 10^{-2}
    • Correct: 3.2 is between 1 and 10.
    • B. 0.67imes10130.67 imes 10^{13}
    • Incorrect: 0.67 is not between 1 and 10.
    • C. 11.2imes10311.2 imes 10^{3}
    • Incorrect: 11.2 is not between 1 and 10.

Conclusion

  • 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.