Standard and Scientific Notation

Standard and Scientific Notation exercise

• Convert an ordinary number into standard form.

  Hints

  To convert from an ordinary number to standard form:

  • Rewrite the number as a number between $1$ and $10$

  • Count how many jumps it would be to get the number as it was

  • The power of $10$ is the number of jumps needed.

  For example,

  If we write a number between $1$ and $10$ we have to multiply by $10$ a number of times to get it back to its original number.

  For example, $2,000$ can be written as $2$ but it is not equal unless we multiply by $1,000$.

  We write $2,000 = 2\times1,000 = 2\times10^{3}$

  Solution

  • $400 = 4\times10^{2}$
  • $4,000 = 4\times10^{3}$
  • $450 = 4.5\times10^{2}$
  • $4,500 = 4.5\times10^{3}$
  • $45,000 = 4.5\times10^{4}$
  • $405 = 4.05\times10^{2}$

  To convert from an ordinary number to standard form, we need to use multiplication. $4,000$ can be written as $4$ but it is not equal unless we multiply by $1,000$. We write $4,000 = 4\times1,000 = 4\times10^{3}$

• Convert an ordinary number into standard form.

  Hints

  • Place the decimal point after the first non-zero digit
  • Count the places moved from the decimal’s original location

  For example,

  If we have a very small number and write it as a number between $1$ and $10$, then the powers will be negative in order to make it very small again.

  Solution

  The correct answer is $6.3\times10^{-4}$

  • Place the decimal point after the first non-zero digit $6.3$
  • Count the places moved from the decimal’s original location $-4$

• Convert the standard form into ordinary numbers.

  Hints

  $1.7\times10^{2}$ means the number is multiplied by $100$.

  The number moves $2$ places to the left.

  $1.7\times10^{-2}$ means the number is divided by $100$.

  The number moves $2$ places to the right.

  Solution

  • $5.8\times10^{2} = 580$
  • $5.8\times10^{3} = 5,800$
  • $5.8\times10^{-2} = 0.058$
  • $5.8\times10^{-1} = 0.58$

• Find the ordinary number.

  Hints

  Multiplying by $10^{6}$ means multiply by $10$ six times.

  Moving the number to the left.

  Check out the example pattern to help you.

  • $3.21\times10^{1} = 32.1$
  • $3.21\times10^{2} = 321$
  • $3.21\times10^{3} = 3,210$

  Solution

  $9.43\times10^{6} = 9,430.000$

  We multiply $9.43$ by $10$ six times.

  Move the number to the left, six places making it bigger.

• Find the powers of ten.

  Hints

  • $10 = 10\times1 = 10$
  • $100 = 10\times10 = 10^{2}$

  Continue with this to find the solutions.

  The power is the number of times we multiply by $10$.

  For example , if we multiply a number by $10$ six times, we can write it as $10^{6}$

  Solution

  • $100 = 10^{2}$
  • $1,000 = 10^{3}$
  • $10,000 = 10^{4}$
  • $100,000 = 10^{5}$

  We split the number into factors of $10$

  $1,000 = 10\times10\times10$

  We can write it as $10^{3}$

• Mixed standard form and ordinary numbers

  Hints

  • Change the two ordinary numbers into standard form to make them easier to compare
  • We place the decimal point after the first non-zero digit
  • Count the jumps back to the original location.

  Write all the standard form numbers underneath each other to help compare.

  For example,

  Solution

  1. $1.2\times10^{-5}$
  2. $0.00019$
  3. $1.1\times10^{-3}$
  4. $1.8\times10^{-3}$
  5. $0.0099$

  Change the two ordinary numbers into standard form by placing the decimal point after the first non-zero digit and counting the jumps back to the original location.