Standard Form Lecture Notes

Introduction to Standard Form

  • Definition: Standard form is a more convenient method for writing numbers that are exceptionally large or exceptionally small.

  • Required Format: To express a number in standard form, it must be written in the following mathematical format:

    • A×10BA \times 10^B

  • Criteria for Variables:

    • A: This must be a whole number between 11 and 1010.

    • B: This must be a whole number, which can be either positive or negative.

Conversion Examples and Techniques

  • Conversion Examples:

    • Large Numbers:

      • 15700=1.57×10415700 = 1.57 \times 10^4

      • 200=2.00×102200 = 2.00 \times 10^2

    • Small Numbers:

      • 0.00729=7.29×10−30.00729 = 7.29 \times 10^{-3}

      • 0.000059=5.9×10−50.000059 = 5.9 \times 10^{-5}

  • Instructional Hints for Conversion:

    • For Large Numbers: Count the digits from the right to the first one. For example, in the number 56005600, there are 55. Standard form is utilized because 103=10×10×1010^3 = 10 \times 10 \times 10.

    • For Small Numbers: Count the decimal places. For example, with the number 0.00250.0025, the standard form utilizes the principle that 10−3=110×110×11010^{-3} = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10}.

Mathematical Rules for Manipulating Standard Form

Your teacher will assist in using a calculator to process standard form, but there are two fundamental rules of indices that apply:

  • Rule 1: Multiplication:

    • General Formula: ya×yb=y(a+b)y^a \times y^b = y^{(a+b)}

    • Algebraic Example: 2a2×3a3=6a(2+3)=6a52a^2 \times 3a^3 = 6a^{(2+3)} = 6a^5

    • Standard Form Application: 2×103×6×104=(6×2)×10(3+4)=12×1072 \times 10^3 \times 6 \times 10^4 = (6 \times 2) \times 10^{(3+4)} = 12 \times 10^7

  • Rule 2: Division:

    • General Formula: ya÷yb=y(a−b)y^a \div y^b = y^{(a-b)}

    • Algebraic Example: 2a6÷4a2=8a(6−2)=8a42a^6 \div 4a^2 = 8a^{(6-2)} = 8a^4

Addition and Subtraction of Standard Form Numbers

  • Methodology: If you need to add two numbers expressed in standard form, the standard procedure is as follows:

    1. Convert both numbers into ordinary (decimal) form.

    2. Perform the addition or subtraction.

    3. Convert the final result back into standard form.

  • Example Calculation:

    • Problem: 3×104+5×1023 \times 10^4 + 5 \times 10^2

    • Step 1 (Ordinary Form): 30 000+50030\,000 + 500

    • Step 2 (Sum): 30 50030\,500

    • Step 3 (Re-convert): 3.05×1043.05 \times 10^4

Practice Exercises

Part 1: Expressing in Standard Form

Ref: Mathematics for Caribbean Schools

  1. 9 000 0009\,000\,000

  2. 600600

  3. 89 00089\,000

  4. 5555

  5. 0.2450.245

  6. 0.000980.00098

Part 2: Multiplication and Division Exercises

Use your calculator or the mathematical rules provided above.

  1. (1.8×104)×(1.2×105)(1.8 \times 10^4) \times (1.2 \times 10^5)

  2. (9.6×102)÷(3×10−3)(9.6 \times 10^2) \div (3 \times 10^{-3})

  3. (5×102)×(8×105)(5 \times 10^2) \times (8 \times 10^5)

  4. (4.8×107)÷(8×103)(4.8 \times 10^7) \div (8 \times 10^3)

Part 3: Addition and Subtraction Exercises

Use your calculator or the method of converting to ordinary form.

  1. (7.5×103)+(1.4×105)(7.5 \times 10^3) + (1.4 \times 10^5)

  2. 3.4×103+6.2×1033.4 \times 10^3 + 6.2 \times 10^3

  3. 9.37×104−6.51×1049.37 \times 10^4 - 6.51 \times 10^4

Authorship and Credits

  • Contributors: T. Harding, A. Lovell & D. Whitehall