Standard and Scientific Notation
- Understanding Scientific Notation and Standard Notation
- Scientific Notation – Real-World Application
- Converting from Standard to Scientific Notation – Steps
- Converting from Scientific to Standard Notation – Steps
- Standard and Scientific Notation – Exercises
- Standard and Scientific Notation – Summary
- Standard and Scientific Notation – Frequently Asked Questions
Learning text on the topic 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 |
---|---|---|
1 | Start with the number in scientific notation. | For 3.16 × 104, start with 3.16 |
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.
Converting from Scientific to Standard Notation – Examples
Standard and Scientific Notation – Exercises
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
Standard and Scientific Notation exercise
-
Convert an ordinary number into standard form.
HintsTo 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.
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- $400 = 4\times10^{2}$
- $4,000 = 4\times10^{3}$
- $450 = 4.5\times10^{2}$
- $4,500 = 4.5\times10^{3}$
- $45,000 = 4.5\times10^{4}$
- $405 = 4.05\times10^{2}$
-
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
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.
SolutionThe 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.
HintsMultiplying 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- $10 = 10\times1 = 10$
- $100 = 10\times10 = 10^{2}$
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- $100 = 10^{2}$
- $1,000 = 10^{3}$
- $10,000 = 10^{4}$
- $100,000 = 10^{5}$
$1,000 = 10\times10\times10$
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- $1.2\times10^{-5}$
- $0.00019$
- $1.1\times10^{-3}$
- $1.8\times10^{-3}$
- $0.0099$
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