Scientific Notation: Videos & Practice Problems

Scientific notation is a method used to express very large or very small numbers in a compact form, making them easier to work with in calculations. The notation consists of three parts: a coefficient that is greater than or equal to one but less than ten, a base of ten, and an exponent which is a whole number indicating how many places the decimal point has been moved. A positive exponent means the coefficient is made larger by moving the decimal point to the right, resulting in a larger number. Conversely, a negative exponent signifies that the coefficient is made smaller by moving the decimal point to the left, yielding a smaller number. To convert from standard notation to scientific notation, adjust the decimal point so that the coefficient falls between one and ten, and then set the exponent to reflect the number of places the decimal has been moved. Understanding scientific notation is crucial for handling complex calculations in chemistry and physics, where precision and efficiency are key.

Scientific Notation Concepts

In chemistry, we often encounter extremely large or small numbers that can be cumbersome to work with. To simplify these calculations, we use scientific notation, a method that transforms these inconvenient numbers into more manageable forms. For instance, the number 6.88 × 10^{-12} is expressed in scientific notation.

In this notation, the first part, known as the coefficient, is the number 6.88, which must be equal to or greater than 1 but less than 10. The second part is the base, which is always 10 in scientific notation. Finally, the exponent (or power) indicates how many places the decimal point has moved to form the scientific notation. For example, in 10^{-12}, the exponent is -12, which signifies a very small number.

It is crucial to note that the exponent must always be a whole number, meaning it cannot be a decimal or fraction. This means values like -12, 3, or -2 are acceptable, while -12.5 or 3/2 are not. Understanding the structure of scientific notation is essential for performing calculations involving extreme values in chemistry.

Scientific Notation Example

Scientific notation is a method used to express very large or very small numbers in a compact form. It consists of three main components: a coefficient, a base, and an exponent. The coefficient must be a number between 1 and 10, the base must always be 10, and the exponent must be a whole number integer.

To determine if a scientific notation value is written correctly, consider the following examples:

In option (a), the coefficient is 1.25, which is acceptable; however, the exponent is -1/4, which is not a whole number integer. Therefore, this representation is incorrect.

In option (b), the coefficient is outside the acceptable range, as it is not between 1 and 10. Although the base is 10 and the exponent is a whole number, the incorrect coefficient disqualifies this option.

Option (c) presents a coefficient of 5, which is between 1 and 10, a base of 10, and an exponent of 3, which is a whole number integer. This representation adheres to all the rules of scientific notation, making it correct.

In option (d), while the coefficient is again acceptable, the base is incorrectly stated as 2 instead of 10. Although the exponent is correct, the incorrect base renders this option invalid.

In summary, to write scientific notation correctly, ensure that the coefficient is between 1 and 10, the base is always 10, and the exponent is a whole number integer. Adhering to these guidelines will help in accurately expressing numbers in scientific notation.

Scientific Notation to Standard Notation

Understanding how to convert from scientific notation to standard notation is essential for working with large or small numbers. Standard notation is simply the conventional way of writing numbers, while scientific notation expresses numbers in a more manageable form, especially when dealing with extreme values.

To convert from scientific notation to standard notation, focus on the exponent of the power of ten. A positive exponent indicates that you should increase the size of the coefficient. For example, in the expression \(7.17 \times 10^5\), the exponent \(5\) means you need to move the decimal point in \(7.17\) five places to the right. This results in \(717,000\), which is the standard notation for this number.

Conversely, a negative exponent signifies that the coefficient should be made smaller. For instance, in the case of \(3.25 \times 10^{-7}\), the exponent \(-7\) instructs you to move the decimal point seven places to the left. This conversion yields \(0.0000325\) in standard notation.

These conversions illustrate the utility of scientific notation, as it simplifies the representation of very large or very small numbers, making them easier to read and work with. Remember, the key to these conversions lies in the exponent: a positive exponent increases the coefficient, while a negative exponent decreases it.

Standard Notation to Scientific Notation

To convert a number from standard notation to scientific notation, it is essential to focus on the coefficient, which must be greater than or equal to 1 but less than 10. This foundational concept is crucial for understanding the conversion process.

For instance, consider the number 0.000145. To express this in scientific notation, we need to adjust the decimal point to create a coefficient of 1.45. This requires moving the decimal point 4 places to the right, resulting in:

1.45 \times 10^{-4}

Here, the exponent is negative because we moved the decimal to the right, indicating a small number.

Conversely, if we take a larger number, such as 10132500, we would move the decimal point to create a coefficient of 1.0132500. This involves moving the decimal point 7 places to the left, leading to:

1.0132500 \times 10^{7}

In this case, the exponent is positive, reflecting the movement of the decimal to the left, which signifies a larger number.

Understanding the relationship between the coefficient and the exponent is key: increasing the coefficient results in a decrease in the exponent, while decreasing the coefficient leads to an increase in the exponent. This principle is fundamental when working with scientific notation and will be further refined with the introduction of significant figures, which will help simplify the notation by eliminating unnecessary zeros.

Now that you have grasped these concepts, you can practice converting additional numbers into scientific notation to reinforce your understanding.

Here’s what students ask on this topic:

What is scientific notation?

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used by scientists, mathematicians, and engineers to handle extremely big or tiny numbers more efficiently.

In scientific notation, a number is written as the product of two parts: a coefficient and a power of 10. The coefficient must be a number greater than or equal to 1 and less than 10, and it is followed by a multiplication sign and a power of 10. This power of 10 indicates how many times the coefficient should be multiplied or divided by 10.

For example, the number 6,500,000,000 in scientific notation is written as:

$6.5\times {10}^9$

The exponent 9 indicates that the decimal point in the coefficient 6.5 has been moved 9 places to the right. Conversely, a small number like 0.00000052 would be written as:

$5.2\times {10}^{-7}$

indicating the decimal has moved 7 places to the left.

Scientific notation simplifies complex calculations, as it allows for easier multiplication and division of large numbers by manipulating the exponents of 10.

How to do scientific notation?

Scientific notation is a way to express very large or very small numbers in a compact form. Here's how you can convert a number into scientific notation:

1. Move the decimal point in the number until you have a coefficient that is greater than or equal to 1 but less than 10. Count the number of places, n, that you moved the decimal point.
2. If you moved the decimal to the left (for large numbers), your exponent will be positive, and it will be the number of places you moved the decimal. If you moved it to the right (for small numbers), your exponent will be negative.
3. Write the number as the coefficient multiplied by 10 raised to the power of the exponent (n).

For example, the number 53,000 in scientific notation is:

$5.3\times {10}^4$

because the decimal was moved 4 places to the left to get the coefficient 5.3. Conversely, 0.0053 would be:

$5.3\times {10}^{-3}$

because the decimal was moved 3 places to the right.

How do you write a number in scientific notation?

Writing a number in scientific notation involves expressing it as the product of two factors: a decimal number and a power of ten. Here's how you do it step by step:

1. Move the decimal point in the number to the right of its first non-zero digit. Count the number of places you moved the decimal point; this will be the exponent of the power of ten.
2. The decimal number (now between 1 and 10) is the coefficient.
3. If you moved the decimal to the left to reach the coefficient, the exponent is positive. If you moved it to the right, the exponent is negative.
4. Write the scientific notation as the coefficient followed by "x 10" raised to the power of the exponent.

For example, to write 0.00052 in scientific notation:

• Move the decimal to the right of the first non-zero digit (5), which gives you 5.2.
• You moved the decimal 4 places to the right, so the exponent is -4.
• The scientific notation is $5.2x{10}^{-4}$.

For a large number, like 85,000:

• Move the decimal to the left to get 8.5.
• You moved the decimal 4 places to the left, so the exponent is +4.
• The scientific notation is $8.5x{10}^4$.