Standard Form

Understanding Standard Form

  • Standard form, also known as scientific notation, is a method of writing very large or very small numbers by using the powers of 10.
  • A number is in standard form if it is expressed as a × 10^n, where 1 ≤ a < 10 (a is known as the mantissa) and n is an integer (known as the exponent).

Writing Numbers in Standard Form

  • To write a number in standard form, first move the decimal point to the right of the first non-zero digit. This gives you the mantissa.
  • The exponent is the number of places you moved the decimal point. If the original number was less than 1, the exponent is negative. If the original number was greater than 1, the exponent is positive.
  • For example, 700 can be written as 7 × 10^2 and 0.005 can be written as 5 × 10^-3.

Reading and Interpreting Standard Form

  • To read a number in standard form, the mantissa tells you the digits of the number, and the exponent tells you where to place the decimal point.
  • The exponent suggests how many spaces to move decimal point, if exponent is positive move to the right, otherwise, move to the left for negative exponent.
  • For example, the number 4.5 × 10^3 means “4.5 moved 3 places to the right”, which is equal to 4500.

Multiplying and Dividing in Standard Form

  • To multiply two numbers in standard form, you multiply the mantissae together and add the exponents together.
  • To divide two numbers in standard form, you divide the mantissae and subtract the exponents.
  • For example, (3 × 10^4) x (2 × 10^3) = 6 × 10^7 and (1.2 × 10^5) ÷ (3 × 10^3) = 4 × 10^1

Practice Problems

  • Problem: Write 8000 in standard form.
    • Solution: 8 × 10^3
  • Problem: What number does 4 × 10^-2 represent?
    • Solution: 0.04
  • Problem: Multiply (5 × 10^3) by (4 × 10^-2).
    • Solution: 20 × 10^1 = 2 × 10^2 = 200

Final Notes

  • Standard form is also useful while performing computations, and helps to simplify the process.
  • Understanding and mastering standard form is essential for solving problems in maths and science, especially where large or small numbers are prevalent.
  • Despite useful for computations, this form has slightly distorts the size perception of figures. Therefore, when analysing or comparing figures, it’s important to fully convert them to ordinary numbers.