Write The Numbers In Scientific Notation. 673.5

Expressing numbers in scientific notation is a fundamental concept in mathematics and science, allowing us to represent very large or very small numbers in a concise and manageable way. Scientific notation, also known as standard form, is particularly useful in fields like physics, chemistry, astronomy, and computer science, where dealing with extreme values is common. This article provides a comprehensive guide on how to write numbers in scientific notation, focusing on the specific example of converting the number 673.5. We'll delve into the principles, step-by-step methods, practical applications, and address frequently asked questions to ensure a thorough understanding of the topic.

Understanding Scientific Notation

Scientific notation is a way of expressing numbers as the product of two factors: a coefficient and a power of 10. The coefficient (also called the significand or mantissa) is a number greater than or equal to 1 and less than 10, and the power of 10 indicates how many places the decimal point must be moved to convert the number back to its original form.

The general form of scientific notation is:

a × 10^b

Where:

• a is the coefficient, such that 1 ≤ |a| < 10
• b is the exponent, which is an integer

Why Use Scientific Notation?

1. Conciseness: Scientific notation simplifies the representation of very large or very small numbers. For example, the number 300,000,000 can be written as 3 × 10^8, which is much more compact.
2. Ease of Comparison: It makes it easier to compare the magnitudes of different numbers. Comparing 3 × 10^8 and 5 × 10^6 is straightforward, while comparing 300,000,000 and 5,000,000 requires more effort.
3. Significant Figures: It clearly indicates the number of significant figures in a measurement. For example, writing 3.00 × 10^8 indicates that the number has three significant figures.
4. Mathematical Operations: It simplifies mathematical operations, especially multiplication and division. For example, multiplying (2 × 10^3) by (3 × 10^4) is easier than multiplying 2,000 by 30,000.

Steps to Write 673.5 in Scientific Notation

Converting the number 673.5 into scientific notation involves the following steps:

Step 1: Identify the Decimal Point

The number 673.5 has a decimal point between the 3 and the 5. We need to move this decimal point to obtain a coefficient between 1 and 10.

Step 2: Determine the Coefficient

To get a coefficient between 1 and 10, we move the decimal point to the left until we have a number that meets this criterion. In the case of 673.5, we move the decimal point two places to the left, resulting in 6.735. This number is between 1 and 10, so it is our coefficient.

Step 3: Determine the Exponent

The exponent is determined by the number of places we moved the decimal point. Since we moved the decimal point two places to the left, the exponent will be positive 2.

Step 4: Write in Scientific Notation

Combine the coefficient and the exponent to write the number in scientific notation.

6.  735 × 10^2

So, the number 673.5 in scientific notation is 6.735 × 10^2.

Detailed Explanation and Examples

To further illustrate the process, let's consider additional examples and scenarios.

Example 1: Converting a Larger Number

Convert 12,500 to scientific notation:

1. Identify the Decimal Point: The decimal point is implicitly at the end of the number (12,500.).
2. Determine the Coefficient: Move the decimal point four places to the left to get 1.25.
3. Determine the Exponent: Since we moved the decimal point four places to the left, the exponent is 4.
4. Write in Scientific Notation:

1.  25 × 10^4

Example 2: Converting a Smaller Number

Convert 0.00456 to scientific notation:

1. Identify the Decimal Point: The decimal point is between the two zeros (0.00456).
2. Determine the Coefficient: Move the decimal point three places to the right to get 4.56.
3. Determine the Exponent: Since we moved the decimal point three places to the right, the exponent is -3.
4. Write in Scientific Notation:

5.  56 × 10^-3

Example 3: Number Already Between 1 and 10

Convert 5.2 to scientific notation:

1. Identify the Decimal Point: The decimal point is already in the correct position (5.2).
2. Determine the Coefficient: The number is already between 1 and 10, so the coefficient is 5.2.
3. Determine the Exponent: Since we didn't move the decimal point, the exponent is 0.
4. Write in Scientific Notation:

6.  2 × 10^0

Practical Applications of Scientific Notation

Scientific notation is widely used in various fields. Here are some examples:

Physics

• 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.
• Gravitational Constant: The gravitational constant G is approximately 0.000000000066743 N(m/kg)^2. In scientific notation, this is 6.6743 × 10^-11 N(m/kg)^2.

Chemistry

• 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.
• Atomic Mass Unit: The atomic mass unit (amu) is approximately 0.00000000000000000000000000166053906660 kg. In scientific notation, this is 1.66053906660 × 10^-27 kg.

Astronomy

• Distance to the Sun: The average distance from the Earth to the Sun is approximately 149,600,000,000 meters. In scientific notation, this is 1.496 × 10^11 m.
• Mass of the Earth: The mass of the Earth is approximately 5,972,000,000,000,000,000,000,000 kg. In scientific notation, this is 5.972 × 10^24 kg.

Computer Science

• Storage Capacity: A terabyte (TB) is approximately 1,000,000,000,000 bytes. In scientific notation, this is 1 × 10^12 bytes.
• Processor Speed: Processor speeds are often measured in gigahertz (GHz), where 1 GHz is 1,000,000,000 Hz. In scientific notation, this is 1 × 10^9 Hz.

Rules for Significant Figures in Scientific Notation

Significant figures are the digits in a number that are known with certainty plus one final digit that is uncertain or estimated. When converting numbers to scientific notation, it’s important to maintain the correct number of significant figures.

1. Non-Zero Digits: All non-zero digits are significant. For example, in 456.7, all four digits are significant.
2. Zeros Between Non-Zero Digits: Zeros between non-zero digits are significant. For example, in 2005, all four digits are significant.
3. Leading Zeros: Leading zeros are not significant. For example, in 0.0034, only the 3 and 4 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 1.200, all four 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 1200, the 1 and 2 are significant, but the zeros are ambiguous.

When expressing numbers in scientific notation, the number of digits in the coefficient indicates the number of significant figures. For example:

• 3. 0 × 10^3 has two significant figures.
• 4. 00 × 10^3 has three significant figures.
• 5. 000 × 10^3 has four significant figures.

Common Mistakes and How to Avoid Them

1. Incorrect Coefficient: Ensure the coefficient is always between 1 and 10. If the coefficient is less than 1 or greater than or equal to 10, adjust the decimal point and the exponent accordingly.
2. Incorrect Exponent: Double-check the direction and number of places you moved the decimal point. 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 Sign of the Exponent: Always include the sign (+ or -) of the exponent, especially for numbers less than 1.
4. Rounding Errors: When rounding numbers to a specific number of significant figures, ensure you follow the standard rounding rules.
5. Misunderstanding Significant Figures: Pay close attention to the rules for significant figures to accurately represent the precision of your measurements.

Advanced Topics and Considerations

Normalized Scientific Notation

In some contexts, scientific notation is normalized, meaning that the coefficient is always written with only one non-zero digit to the left of the decimal point. This is the most common and preferred form of scientific notation.

Engineering Notation

Engineering notation is similar to scientific notation, but the exponent is always a multiple of 3. This is often used in engineering applications to align with common units like milli (10^-3), micro (10^-6), kilo (10^3), and mega (10^6). For example, the number 47,000 would be written as 47 × 10^3 in engineering notation, rather than 4.7 × 10^4 in scientific notation.

Scientific Notation in Calculators and Software

Most calculators and software programs can display numbers in scientific notation. The notation may vary depending on the device or program. Common notations include:

• E Notation: For example, 6.735 × 10^2 might be displayed as 6.735E2.
• EE Notation: Some calculators use EE to represent "times ten to the power of." For example, to enter 6.735 × 10^2, you might type 6.735 EE 2.

Conclusion

Writing numbers in scientific notation is a crucial skill for anyone working with very large or very small numbers. It provides a concise, clear, and standardized way to represent numerical values, making them easier to compare, manipulate, and understand. By following the steps outlined in this article, you can confidently convert any number to scientific notation, understand its applications, and avoid common mistakes. Whether you are a student, scientist, engineer, or simply someone interested in mathematics, mastering scientific notation will undoubtedly enhance your quantitative literacy and problem-solving abilities.