# Standard and Scientific Notation

## Understanding Scientific Notation and Standard Notation

Welcome to the intriguing world of Scientific Notation. This math concept is a game-changer when dealing with super large numbers like the number of stars in the universe or really tiny ones like the size of molecules in chemistry. It’s a helpful tool that makes working with these extreme numbers much simpler. Ready to learn how scientific notation makes big and small numbers a breeze to handle? Let’s get started!

Scientific Notation is a way of expressing large or small numbers as a product of a number between 1 and 10 and a power of ten. Standard Notation is our regular way of writing numbers.

## Scientific Notation – Real-World Application

Here's a table with real-world examples, illustrating their sizes in both standard form and scientific notation:

Item Standard Form Scientific Notation
Distance from Earth to Sun 149,600,000 km 1.496 × 108 km
Size of a Red Blood Cell 0.000007 m 7 × 10-6 m

The first row represents the vast distance from the Earth to the Sun, while the second row shows the minuscule size of a red blood cell, each expressed in a form that best suits their scale.

## Converting from Standard to Scientific Notation – Steps

Converting numbers from standard to scientific notation can be done by following these steps:

Step Direction Example
1 Move the decimal point to get a number between 1 and 10, with only one non-zero digit to its left. For 1,2345, move the decimal between the 1 and 2. 1.2345
2 Count how many places you moved the decimal. Decimal moved 4 places to the left.
3 Write the exponent based on your count. Positive for left moves (large numbers), negative for right (small numbers). 12345 in scientific notation is 1.2345 × 104

## Converting from Scientific to Standard Notation – Steps

Converting numbers from scientific to standard notation is the reverse process, with the following steps.

Step Direction Example
2 Move the decimal point based on the exponent. Right for positive exponents, left for negative. Move decimal 4 places to the right.
3 Fill in with zeros if needed, to match the number of places you moved the decimal. 3.16 becomes 31,600 (standard notation)

Check your understanding of these two conversions.

How does the exponent change when converting a small number to scientific notation?

## Standard and Scientific Notation – Exercises

Convert $50,000$ to scientific notation.
Convert $0.0005$ to scientific notation.
Convert $6.7 × 10^{2}$ to standard notation.
Convert $2.1 × 10^{-3}$ to standard notation.
A scientist is measuring a very small bacteria colony that has a size of $0.00009$ meters in diameter. To report this measurement in a scientific paper, convert this size to scientific notation.

## Standard and Scientific Notation – Summary

Key Points from This Text:

• Standard notation is the regular form of a number we typically see in the world. For example: 50,800,000.
• Scientific notation makes it easier to work with very large or small numbers.
• Scientific notation is written as a product of a decimal between 1 and 10, and a power of ten, for example: 5.08 × 107.
• The or exponent shows how many times the decimal moves, with positive for large and negative for small numbers.
Conversion Type Process
Standard to Scientific Notation Move decimal to get a number between 1-10, count moves, times 10 to the power of moves.
Scientific to Standard Notation Start with scientific notation number, move decimal based on exponent (right for positive, left for negative).

For more practice with this topic, this video is helpful for learning how to compare numbers written in scientific notation - Comparing Numbers Written in Scientific Notation

For more exciting math topics, explore our interactive problems, videos, and worksheets on our website!

## Standard and Scientific Notation – Frequently Asked Questions

What is the importance of understanding scientific notation?
What's a real-world example of a large number in scientific notation?
Is scientific notation only for positive exponents?
What is a real-world situation where scientific notation is used to represent very small numbers?
How do you identify if a number in scientific notation represents a large or a small number?
What is the difference between standard notation and scientific notation?
Why do we move the decimal point in scientific notation?
Is it possible to use scientific notation in everyday life, outside of science and math?
What are some common mistakes to avoid when converting between standard and scientific notation?
How do you write a very small number, like $0.0003$, in scientific notation?

## Standard and Scientific Notation exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Standard and Scientific Notation.
• ### 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.

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

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sofatutor Team
Standard and Scientific Notation
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