Write The Numbers In Scientific Notation 673.5

Scientific notation, a cornerstone of mathematics and science, provides a concise and standardized way to express very large or very small numbers. Instead of writing out a number with numerous digits, we represent it as a product of a number between 1 and 10 and a power of 10. This method simplifies calculations, enhances readability, and is universally recognized, making it an indispensable tool in fields ranging from astrophysics to microbiology. Understanding and applying scientific notation is fundamental for anyone dealing with quantitative data. This comprehensive guide will walk you through the process of converting the number 673.5 into scientific notation, explain the underlying principles, and illustrate why this notation is so powerful.

Understanding Scientific Notation

Scientific notation, also known as standard form, is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is particularly useful in scientific and mathematical contexts where dealing with extremely large or small values is common.

The General Form

The general form of scientific notation is:

a × 10^b

Where:

• a is a number between 1 and 10 (1 ≤ |a| < 10), known as the coefficient or significand.
• 10 is the base, which is always 10 in scientific notation.
• b is an integer exponent, which can be positive, negative, or zero.

Key Components Explained

1. Coefficient (a):

   • The coefficient is a real number that captures the significant digits of the original number.
   • It must be greater than or equal to 1 and less than 10.
   • The coefficient provides the precision of the number in scientific notation.

2. Base (10):

   • The base is always 10 in scientific notation because it is based on the decimal system.
   • Using base 10 allows for easy scaling of the number by powers of ten.

3. Exponent (b):

   • The exponent is an integer that determines how many places the decimal point must be moved to convert the number back to its original form.
   • A positive exponent indicates that the original number is larger than the coefficient, while a negative exponent indicates that the original number is smaller than the coefficient.
   • An exponent of 0 means the number is already between 1 and 10, so it does not need to be scaled.

Why Use Scientific Notation?

1. Conciseness:

   • Scientific notation allows very large and very small numbers to be written in a compact and readable form.
   • For example, the number 6,000,000,000 can be written as 6 × 10^9.

2. Ease of Calculation:

   • Scientific notation simplifies arithmetic operations, especially multiplication and division.
   • When multiplying numbers in scientific notation, you multiply the coefficients and add the exponents.
   • When dividing numbers in scientific notation, you divide the coefficients and subtract the exponents.

3. Standardization:

   • Scientific notation provides a standard format for expressing numbers, making it easier to compare and communicate values across different contexts.
   • It is widely used in scientific publications, textbooks, and technical documents.

4. Precision:

   • Scientific notation allows you to easily indicate the number of significant digits in a measurement.
   • This is crucial in scientific contexts where the precision of a measurement is important.

Converting 673.5 to Scientific Notation: Step-by-Step Guide

Converting a number to scientific notation involves expressing it as the product of a number between 1 and 10 and a power of 10. Here’s how to convert 673.5 into scientific notation:

Step 1: Identify the Decimal Point

The first step is to locate the decimal point in the number. In the number 673.5, the decimal point is between the 3 and the 5.

Step 2: Move the Decimal Point

Next, you need to move the decimal point so that there is only one non-zero digit to the left of it. In this case, you need to move the decimal point two places to the left to get the number 6.735.

673.5  ->  6.735

Step 3: Determine the Exponent

The exponent is determined by the number of places you moved the decimal point. Since you moved the decimal point two places to the left, the exponent will be positive 2. If you had moved the decimal point to the right, the exponent would be negative.

Step 4: Write in Scientific Notation

Now, write the number in scientific notation using the format a × 10^b, where a is the number you obtained by moving the decimal point (which is between 1 and 10), and b is the exponent you determined.

In this case, a = 6.735 and b = 2. Therefore, the scientific notation for 673.5 is:

6.735 × 10^2

Step 5: Verification

To verify that you have correctly converted the number, you can convert it back from scientific notation to decimal form. To do this, move the decimal point in 6.735 two places to the right:

6.735 × 10^2 = 6.735 × 100 = 673.5

Since we obtained the original number, the conversion is correct.

Examples of Converting Different Numbers to Scientific Notation

To further illustrate the process, let's convert a few more numbers to scientific notation.

Example 1: Converting 1250 to Scientific Notation

1. Identify the Decimal Point: The decimal point is at the end of the number: 1250.
2. Move the Decimal Point: Move the decimal point three places to the left to get 1.250.
3. Determine the Exponent: Since you moved the decimal point three places to the left, the exponent is 3.
4. Write in Scientific Notation: 1.250 × 10^3 or 1.25 × 10^3 (since trailing zeros after the decimal point are usually dropped).
5. Verification: 1.25 × 10^3 = 1.25 × 1000 = 1250.

Example 2: Converting 0.0045 to Scientific Notation

1. Identify the Decimal Point: The decimal point is at the beginning of the number: 0.0045.
2. Move the Decimal Point: Move the decimal point three places to the right to get 4.5.
3. Determine the Exponent: Since you moved the decimal point three places to the right, the exponent is -3.
4. Write in Scientific Notation: 4.5 × 10^-3.
5. Verification: 4.5 × 10^-3 = 4.5 × 0.001 = 0.0045.

Example 3: Converting 98765 to Scientific Notation

1. Identify the Decimal Point: The decimal point is at the end of the number: 98765.
2. Move the Decimal Point: Move the decimal point four places to the left to get 9.8765.
3. Determine the Exponent: Since you moved the decimal point four places to the left, the exponent is 4.
4. Write in Scientific Notation: 9.8765 × 10^4.
5. Verification: 9.8765 × 10^4 = 9.8765 × 10000 = 98765.

The Importance of Significant Digits in Scientific Notation

Significant digits play a crucial role in scientific notation, as they indicate the precision of a measurement. When converting a number to scientific notation, it is important to retain the correct number of significant digits.

Identifying Significant Digits

1. Non-zero Digits: All non-zero digits are significant. For example, in the number 345.6, all five digits are significant.
2. Zeros Between Non-zero Digits: Zeros between non-zero digits are significant. For example, in the number 2007, all four digits are significant.
3. Leading Zeros: Leading zeros are not significant. For example, in the number 0.0045, only the 4 and 5 are significant.
4. Trailing Zeros in a Number with a Decimal Point: Trailing zeros in a number with a decimal point are significant. For example, in the number 12.300, all five digits are significant.
5. Trailing Zeros in a Number without a Decimal Point: Trailing zeros in a number without a decimal point are generally not significant unless otherwise indicated. For example, in the number 1200, it is unclear whether the zeros are significant. To indicate that they are significant, you can use scientific notation (e.g., 1.200 × 10^3).

Retaining Significant Digits When Converting to Scientific Notation

When converting a number to scientific notation, retain all significant digits in the coefficient. For example, if the number 1234.5 has five significant digits, then in scientific notation, it should be written as 1.2345 × 10^3.

Rounding

If you need to reduce the number of significant digits, you must round the number appropriately. For example, if you want to express 1234.5 with only three significant digits, you would round it to 1230, which in scientific notation is 1.23 × 10^3.

Common Mistakes to Avoid When Using Scientific Notation

Using scientific notation correctly involves understanding the basic principles and avoiding common errors. Here are some common mistakes to watch out for:

1. Incorrect Coefficient: The coefficient must be between 1 and 10 (1 ≤ |a| < 10). A common mistake is to have a coefficient that is either less than 1 or greater than or equal to 10.
2. Incorrect Exponent: The exponent should reflect the number of places the decimal point was moved. Make sure to count the places correctly and remember that moving the decimal point to the left results in a positive exponent, while moving it to the right results in a negative exponent.
3. Forgetting the Base 10: The base is always 10 in scientific notation. Forgetting to include the base or using a different base is a common error.
4. Incorrect Significant Digits: Retaining the correct number of significant digits is crucial. Make sure to include all significant digits in the coefficient and round appropriately if necessary.
5. Misunderstanding Negative Exponents: A negative exponent indicates a number between 0 and 1, not a negative number. For example, 5 × 10^-3 is 0.005, not -5000.
6. Confusion with Engineering Notation: Engineering notation is similar to scientific notation, but the exponent must be a multiple of 3. Confusing engineering notation with scientific notation can lead to errors.
7. Not Verifying the Conversion: Always verify the conversion by converting the number back from scientific notation to decimal form to ensure that you obtain the original number.

Practical Applications of Scientific Notation

Scientific notation is used extensively in various fields, making it an essential tool for scientists, engineers, and mathematicians. Here are some practical applications:

Physics

In physics, scientific notation is used to express very large and very small quantities, such as the speed of light, the mass of an electron, and the gravitational constant.

• Speed of Light: The speed of light in a vacuum is approximately 299,792,458 meters per second. In scientific notation, this is 2.99792458 × 10^8 m/s.
• Mass of an Electron: The mass of an electron is approximately 0.000000000000000000000000000000910938356 kilograms. In scientific notation, this is 9.10938356 × 10^-31 kg.
• Gravitational Constant: The gravitational constant (G) is approximately 0.0000000000667430 newton meters squared per kilogram squared. In scientific notation, this is 6.67430 × 10^-11 N(m/kg)^2.

Chemistry

In chemistry, scientific notation is used to express the number of atoms or molecules in a sample, the concentrations of solutions, and other quantities.

• Avogadro's Number: Avogadro's number is approximately 602,214,076,000,000,000,000,000. In scientific notation, this is 6.02214076 × 10^23.
• Molarity: A solution with a molarity of 0.0005 M can be expressed as 5 × 10^-4 M.
• Planck's Constant: Planck's constant is approximately 0.0000000000000000000000000000662607015 joule-seconds. In scientific notation, this is 6.62607015 × 10^-34 Js.

Astronomy

In astronomy, scientific notation is used to express the vast distances between celestial objects, the masses of stars and planets, and other astronomical quantities.

• Distance to the Nearest Star: The distance to Proxima Centauri, the nearest star to our Sun, is approximately 40,200,000,000,000 kilometers. In scientific notation, this is 4.02 × 10^13 km.
• Mass of the Sun: The mass of the Sun is approximately 1,989,000,000,000,000,000,000,000,000,000 kilograms. In scientific notation, this is 1.989 × 10^30 kg.
• Size of the Universe: The estimated number of stars in the observable universe is 1,000,000,000,000,000,000,000. In scientific notation, this is 1 × 10^21.

Engineering

In engineering, scientific notation is used to express very large or very small dimensions, forces, and other parameters.

• Young's Modulus of Steel: Young's modulus of steel is approximately 200,000,000,000 pascals. In scientific notation, this is 2 × 10^11 Pa.
• Size of a Nanoparticle: The size of a nanoparticle might be 0.000000005 meters. In scientific notation, this is 5 × 10^-9 m.
• Electrical Resistance: The electrical resistance in a circuit might be 0.00000012 ohms. In scientific notation, this is 1.2 × 10^-7 ohms.

Biology

In biology, scientific notation is used to express the sizes of cells, the concentrations of molecules, and other biological quantities.

• Size of a Bacterium: The size of a bacterium might be 0.000001 meters. In scientific notation, this is 1 × 10^-6 m.
• Concentration of DNA: The concentration of DNA in a solution might be 0.00000001 moles per liter. In scientific notation, this is 1 × 10^-8 M.
• Number of Cells in a Human Body: The estimated number of cells in a human body is 37,200,000,000,000. In scientific notation, this is 3.72 × 10^13.

Conclusion

Converting the number 673.5 into scientific notation is a straightforward process that highlights the utility of this notation. By moving the decimal point to obtain a coefficient between 1 and 10 and adjusting the exponent accordingly, we express 673.5 as 6.735 × 10^2. This method not only simplifies the representation of numbers but also facilitates calculations and comparisons in various scientific and mathematical contexts. Scientific notation is an invaluable tool, and mastering its use enhances your ability to work with quantitative data effectively. Understanding and correctly applying scientific notation is a fundamental skill that empowers you to tackle complex problems and communicate numerical information with precision and clarity.