Reading and Writing Scientific Notation

Reading and Writing Scientific Notation exercise

• Explain how to write numbers in scientific notation.

  Hints

  The power $10^a$ represents a product:

  $10^a=\underbrace{10\times 10 \times... \times 10}_{a \text{ times}}$.

  $a$ can either be positive or negative. An example for a number written in scientific notation with positive $a$ is $1.23\times 10^4=1,230$.

  The following number is not written in scientific notation:

  Solution

  Scientific notation makes really big and really small numbers easy to read and easy to interpret.

  $~$

  Any number can be written in the form $n\times 10 ^a$, where:

  • $n$ is the coefficient.
  • $10$ is the base.
  • $a$ is the exponent.

  $~$

  The coefficient $n$ must be greater than or equal to $1$ and less than $10$.

• Write the given numbers in scientific notation.

  Hints

  Any number can be written in scientific notation;

  i.e. $n\times 10^a$, where we must have that $1\le n<10$.

  Keep in mind:

  • $a$ is negative for a small number.
  • $a$ is positive for a big number.

  The following is an example for a scientific notation:

  Solution

  The number of bacteria is given by $5,120,000,000,000,000,000,000,000,000,000$.

  We scoot the decimal from right to left until we get a number which is greater than or equal to $1$ and less than $10$. In this case, we get $n=5.12$... we moved the decimal $30$ to the left!

  With this information we can write our large number in scientific notation: $5.12\times 10^{30}$.

  $~$

  Similarly, we can write $0.0000000000000546$ in scientific notation.

  We move the decimal point $14$ times to the right until we reach the first nonzero number, namely $5$. The coefficient is thus $n=5.46$ and the exponent is $a=-14$. Why $-14$? Well, we moved the decimal point to the right 14 times, meaning that we divided by $10$... $14$ times. Thus we get $a=-14$.

  So we have that $0.0000000000000546=5.46\times 10^{-14}$.

• Examine how the numbers are written in scientific notation.

  Hints

  Any number can be written in scientific notation;

  i.e. $n\times 10^a$, where we must have that $1\le n<10$.

  An example of a number being converted into scientific notation is $1000000=1\times 10^6$.

  The coefficient for a big number is given by the leading numbers not equal zero, with a decimal point after the first number.

  For instance, with $314100000$ we have $n=3.141$.

  For the exponent $a$ we either count how often we move the decimal point to the left or we count the positions after the leading $1$ until we reach the end of the number or a decimal point.

  Solution

  Numbers written in scientific notation are of the form

  $n\times 10^a$, with $1\le n<10$.

  $~$

  For big numbers:

  • $a>0$ can be found by the scooting of the decimal point to the left.
  • The first numbers not equal zero with a decimal point after the first number is $n$.

  For example:

  • $14120000000=1.412\times 10^{10}$.
  • $141.2=1.412\times 10^2$

  $~$

  For small numbers

  • $a<0$ can be found by the numbers of scooting the decimal point to the right.
  • The first numbers not zero after the decimal point with a decimal point behind the leading number is $n$.

  For example:

  • $0.001412=1.412\times 10^{-3}$
  • $0.00000001412=1.412\times 10^{-8}$

• Write the following numbers in scientific notation.

  Hints

  In the scientific notation, $n\times 10^a$, the exponent $a$ must be negative for small numbers.

  An example of a number written in scientific notation is $0.000001=1\times 10^{-6}$.

  The coefficient of $0.000003141$ when written in scientific notation is $n=3.141$.

  Solution

  A number written in scientific notation looks like $n\times 10^a$, where $1\le n<10$.

  $~$

  Let's figure out how to write the weights of Dorothy's objects in scientific notation:

  • Grain of sand: $0.0000000007645=7.645\times 10^{-10}$
  • Hair: $0.00001036=1.036\times 10^{-5}$
  • Fingernail: $0.000441=4.41\times 10^{-4}$
  • Ant: $0.0176=1.76\times 10^{-2}$

• Describe the definition of scientific notation.

  Hints

  In the following picture, we see $t$ is raised by the power of $a$.

  $t$ is the base and $a$ is the exponent.

  Solution

  Scientific notation allows us to write very big or very small numbers in an easy to read and easy to understand way.

  A number written in scientific notation looks like $n\times 10^a$, where:

  • The coefficient $n$ must be greater than or equal to $1$ and less than $10$.
  • The sign of the exponent $a$ is positive when a number is very big, or negative when a number is very small.

• Determine the standard form of the number given in scientific notation.

  Hints

  An example of a big number written in scientific notation is $2.25\times 10^4=22500$.

  For a big number, the exponent represents how much you have to move the decimal point to the right.

  Let's look at the example above once again:

  $\begin{array}{rcl} 2.25\times 10^4=2.25\times 10\times 10\times 10\times 10&=&22.5\times 10\times 10\times\\ & =&225\times 10\times 10\\ &=&2250\times 10\\ &=&22500 \end{array}$

  Solution

  Let's see how to convert scientific notation into a big (or small) number through the following example: $2.25\times 10^4$.

  We already know that $10$ raised by $4$ can be written in the following way:

  $2.25\times 10\times 10\times 10\times 10$.

  Step by step, we have:

  $\begin{array}{rcl} 2.25\times 10\times 10\times 10\times 10&=&22.5\times 10\times 10\times\\ & =&225\times 10\times 10\\ &=&2250\times 10\\ &=&22500 \end{array}$

  $~$

  In other words, we scoot the decimal point four times to the right. If we reach the number given by the coefficient and we still have to move the decimal point, we write down zeros.

  $~$

  So, for the given distances above we get:

  • Earth to Saturn: $5.448\times 10^11=544800000000$
  • Neptun to Sun: $1.479\times 10^{13}=14790000000000$
  • Mercury to Sun: $1.903\times 10^{11}=190300000000$
  • Jupiter to Earth: $3.171\times 10^{12}=3171000000000$