Scientific Notation

• What is Scientific Notation?
• Scientific Notation Definition
• Rules of Scientific Notation
• How to do Scientific Notation Operations?
• Practice Problems
• Frequently Asked Questions

What is Scientific Notation?

Scientific notation is a way of writing very large or very small numbers concisely. It is particularly useful in scientific and mathematical contexts where dealing with numbers of extreme magnitudes is common. In scientific notation, a number is expressed as the product of a coefficient and a power of `10`. 

Example:

1. `602,000,000,000,000,000,000,000` can be written in scientific notation as `6.02 × 10^23`.
2. \( 0.000001 \) can written in scientific notation with a negative exponent as \( 1.0 \times 10^{-6} \).

Scientific Notation Definition

The scientific notation involves expressing a number as the product of a coefficient and a power of `10`. Mathematically, it can be represented as:

where:

• \( a \) is the coefficient (a number greater than or equal to `1` and less than `10`).
• \( n \) is the exponent, which denotes the power of `10` by which \( a \) is multiplied.

Example `1`: Express the distance from Earth to the Sun, which is approximately `93,000,000` miles, in scientific notation.

Solution:

To express `93,000,000` miles in scientific notation, we follow the scientific notation rule: \( a \times 10^n \).

1. Identify the coefficient \( a \): Move the decimal point `7` places to the left to form the coefficient a, that is, `9.3`.
2. Determine the exponent \( n \): Count the number of places we moved the decimal point to the left to obtain a number between `1` and `10`. In this case, we moved the decimal point `7` places to the left to get `9.3`, so the exponent is \( 10^7 \).
3. Write the number in scientific notation: \( 9.3 \times 10^7 \).

The distance from Earth to the Sun, `93,000,000` miles, can be expressed in scientific notation as \( 9.3 \times 10^7 \). This format simplifies the representation of the large distance, making it easier to work within astronomical calculations and comparisons.

Example `2`: Convert the diameter of a cell, which is approximately `0.000001` meters to scientific notation.

Solution: 

To convert `0.000001` meters in scientific notation, we follow the scientific notation rule: \( a \times 10^n \).

1. Identify the coefficient \( a \): Move the decimal point `6` places to the right to form the coefficient a, that is, `1`.
2. Determine the exponent \( n \): Count the number of places we moved the decimal point to the right to obtain a number between `1` and `10`. In this case, we moved the decimal point `6` places to the right to get `1`, so the exponent is \( 10^{-6} \).
3. Write the number in scientific notation: \( 1 \times 10^{-6} \).

The diameter of the cell `0.000001` meters, can be expressed in scientific notation as \( 1 \times 10^{-6} \). This format simplifies the representation of the small values, making it convenient for the researchers to deal with calculations.

Rules of Scientific Notation

The rules of scientific notation provide guidelines for expressing numbers in a concise and standardized format. Here are the key rules:

1. Coefficient: The coefficient \( a \) must be greater than or equal to 1 and less than `10`.
2. Base: The base of the power term should always be `10`.
3. Exponent: The exponent \( n \) indicates the power of `10` by which the coefficient is multiplied. It can be positive or negative.
   1. Positive Exponent: If the original number is greater than or equal to `1`, the exponent is positive.
      Example:  \( 3.14 \times 10^4 \) represents the number `31,400`.
   2. Negative Exponent: If the original number is less than `1`, the exponent is negative.
      Example:  \( 6.02 \times 10^{-5} \) represents the number `0.0000602`.
4. Decimal Placement: The decimal point is placed after the first digit of the coefficient.
   Example: \( 4.5 \times 10^6 \), the decimal point is after the `4`.
5. Zero as Coefficient: Zero cannot be used as the coefficient in scientific notation.

How to do Scientific Notation Operations?

Performing operations with numbers written in scientific notation involves basic arithmetic operations such as addition, subtraction, multiplication, and division. Here's how to perform each operation:

`1`. Adding and Subtracting with Scientific Notation:

• Ensure that both numbers have the same exponent.
• If the exponents are different, adjust one or both numbers to have the same exponent by moving the decimal point and changing the exponent accordingly.
• After adjusting the exponents, add or subtract the coefficients while keeping the exponent the same.

Example `1`: Add \(2.5 \times 10^6 + 3.7 \times 10^6\) and give the sum in scientific notation.

Solution:

Since both numbers have the same exponent (\(10^6\)), we can directly add their coefficients: 
\(2.5 + 3.7 = 6.2\)

The result is \(6.2 \times 10^6\).

Example `2`: Subtract two numbers \(5.6 \times 10^8 - 4.2 \times 10^7\) and give the answer in scientific notation.

Solution:

To subtract these numbers, we need to make their exponents the same. 

Let's rewrite \(4.2 \times 10^7\) with an exponent of \(10^8\) :

\(5.6 \times 10^8 - 0.42 \times 10^8\)

Now, we can subtract the coefficients:
\(5.6 - 0.42 = 5.18\)

The result is \(5.18 \times 10^8\).

`2`. Multiplying in Scientific Notation

• Multiply the coefficients together.
• Add the exponents together to get the exponent of the result.
• If necessary, adjust the result to ensure that the coefficient is between `1` and `10`.

Example `1`: Multiply \((1.5 \times 10^5) \times (3 \times 10^2)\) and give the product in scientific notation. 

Solution:

Multiply the coefficients:
\(1.5 \times 3 = 4.5\)

Add the exponents:
\(5 + 2 = 7\)

The result is \(4.5 \times 10^7\). Since the coefficient \(4.5\) is already between `1` and `10`, there is no need to adjust it.

Example `2`: Multiply \((6 \times 10^4) \times (8 \times 10^3)\) and give the product in scientific notation.

Solution:

Multiply the coefficients:
\(6 \times 8 = 48\)

Add the exponents:
\(4 + 3 = 7\)

The result is \(48 \times 10^7\). However, the coefficient \(48\) is not between `1` and `10`. To adjust it, we can rewrite \(48 \times 10^7\) as \(4.8 \times 10^8\).

Thus, the final result after adjustment is \(4.8 \times 10^8\).

`3`. Dividing in Scientific Notation

• Divide the coefficients.
• Subtract the exponent of the divisor from the exponent of the dividend to get the exponent of the result.
• If necessary, adjust the result to ensure that the coefficient is between `1` and `10`.

Example `1`: Divide \((7.2 \times 10^6) \div (3 \times 10^2)\) and give the answer in scientific notation.

Solution:

Divide the coefficients:
\( \frac{7.2}{3} = 2.4\)

Subtract the exponent of the divisor from the exponent of the dividend:
\(6 - 2 = 4\)

The result is \(2.4 \times 10^4\). Since the coefficient \(2.4\) is already between `1` and `10`, there is no need to adjust it.

Example `2`: Divide \((96 \times 10^7) \div (6 \times 10^3)\) and give the answer in scientific notation.

Solution:

Divide the coefficients:
\( \frac{96}{6} = 16\)

Subtract the exponent of the divisor from the exponent of the dividend:
\(7 - 3 = 4\)

The result is \(16 \times 10^4\). However, the coefficient \(16\) is not between `1` and `10`. To adjust it, we can rewrite \(16 \times 10^4\) as \(1.6 \times 10^5\).

Thus, the final result after adjustment is \(1.6 \times 10^5\).

`4`. Comparing in Scientific Notation

• Compare the exponents first. The number with the larger exponent is greater if the coefficients are the same.
• If the exponents are equal, compare the coefficients.

Example `1`: Compare \(3.5 \times 10^7\) and \(3.5 \times 10^6\).

Solution:

Since the coefficients are the same (\(3.5\)), we compare the exponents:

• Exponent of the first number: \(7\)
• Exponent of the second number: \(6\)

The number with the larger exponent is greater. Therefore, \(3.5 \times 10^7\) is greater than \(3.5 \times 10^6\).

Example `2`: Compare \(4.2 \times 10^4\) and \(6.8 \times 10^4\).

Solution:

Since the exponents are the same (\(10^4\)), we compare the coefficients:

• Coefficient of the first number: \(4.2\)
• Coefficient of the second number: \(6.8\)

The number with the larger coefficient is greater. Therefore, \(6.8 \times 10^4\) is greater than \(4.2 \times 10^4\).

`5`. Converting Between Scientific Notation and Standard Form

• To convert a number from scientific notation to standard form, multiply the coefficient by `10` raised to the power of the exponent.
• To convert a number from standard form to scientific notation, move the decimal point to obtain a coefficient between `1` and `10`, and count the number of places moved to determine the exponent.

Practice Problems

Q`1`. Express `0.000025` in scientific notation.

1.    \( 2.5 \times 10^{-5} \)
2.    \( 25 \times 10^{-6} \)
3.    \( 2.5 \times 10^{-6} \)
4.    \( 25 \times 10^{-5} \)

Answer: a

Q`2`. Write `9,300,000` in scientific notation.

1. \( 9.3 \times 10^7 \)
2. \( 93 \times 10^5 \)
3. \( 930 \times 10^4 \)
4. \( 9.3 \times 10^6 \)

Answer: d

Q`3`. The distance to the nearest star, Proxima Centauri, is approximately `25,000,000,000,000` miles. Express this distance in scientific notation.

1. \(2.5 \times 10^{13}\) miles
2. \(2.5 \times 10^{14}\) miles
3. \(2.5 \times 10^{15}\) miles
4. \(2.5 \times 10^{16}\) miles

Answer: a

Q`4`. The mass of a grain of sand is approximately \(0.0000000003\) kilograms. Express this mass in scientific notation.

1. \(3 \times 10^{-10}\) kilograms
2. \(3 \times 10^{-9}\) kilograms
3. \(3 \times 10^{-8}\) kilograms
4. \(3 \times 10^{-7}\) kilograms

Answer: a

Q`5`. Convert \( 6.8 \times 10^3 \) to standard form.

1. `6,800`
2. `68,000`
3. `680`
4. `0.0068`

Answer: a

Q`6`. What is \( 1.25 \times 10^{-2} \) in standard form?

1. `0.0125`
2. `125`
3. `0.125`
4. `12.5`

 Answer: a

Frequently Asked Questions

Q`1`. What is the scientific notation used for?

Answer: Scientific notation is used to express very large or very small numbers in a concise and standardized format, making them easier to work with in scientific and mathematical calculations.

Q`2`. How do you convert a number into scientific notation?

Answer: To convert a number into scientific notation, write it as a product of a coefficient (greater than or equal to 1 and less than `10`) and a power of `10`, adjusting the exponent to place the decimal point correctly.

Q`3`. What are the `3` parts of a scientific notation?

Answer: The three main parts of a scientific notation are coefficient, base, and exponent.

Q`4`. Can any number be expressed in scientific notation?

Answer: Yes, any number, whether it's extremely large or small, can be expressed in scientific notation by appropriately adjusting the coefficient and exponent.

Q`5`. Why is scientific notation important?

Answer: Scientific notation is important because it simplifies the representation of large or small numbers, making them more manageable in calculations and facilitating comparisons between numbers with vastly different magnitudes.

Q`6`. How do you multiply or divide numbers in scientific notation?

Answer: To multiply or divide numbers in scientific notation, multiply or divide the coefficients and add or subtract the exponents accordingly, ensuring that the result remains in proper scientific notation format.