Write The Numbers In Scientific Notation. 673.5

Writing Numbers in Scientific Notation: A Comprehensive Guide

Scientific notation is a powerful tool used to represent very large or very small numbers concisely. It's essential in various scientific fields, engineering, and even everyday calculations involving extremely large or tiny quantities. This article provides a comprehensive guide to understanding and applying scientific notation, using the number 673.5 as our primary example, and expanding to encompass a wide range of numbers and scenarios. We'll delve into the underlying principles, practical applications, and frequently asked questions to solidify your understanding.

Understanding Scientific Notation

Scientific notation expresses a number as a product of a coefficient and a power of 10. The coefficient is always a number between 1 (inclusive) and 10 (exclusive), while the exponent of 10 indicates the order of magnitude. The general form is:

a x 10b

Where:

'a' is the coefficient (1 ≤ a < 10)
'b' is the exponent (an integer)

For example, the number 673.5 can be written in scientific notation by following these steps:

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

Identify the Decimal Point: Locate the decimal point in the number. In 673.5, the decimal point is implicitly at the end: 673.5.
Move the Decimal Point: Move the decimal point to the left until you obtain a number between 1 and 10. In this case, we move the decimal point two places to the left, resulting in 6.735.
Determine the Exponent: The exponent (b) is the number of places you moved the decimal point. Since we moved it two places to the left, the exponent is +2. If we moved the decimal point to the right, the exponent would be negative.
Write in Scientific Notation: Combine the coefficient and the exponent to express the number in scientific notation. Thus, 673.5 in scientific notation is 6.735 x 102.

Practical Applications of Scientific Notation

The real power of scientific notation lies in its ability to simplify calculations and represent extremely large or small numbers efficiently. Here are some examples:

Astronomy: Representing distances between stars and planets. For example, the distance to the sun is approximately 1.496 x 1011 meters. Writing this number in standard form would be cumbersome and difficult to comprehend.
Chemistry: Dealing with Avogadro's number (6.022 x 1023), which represents the number of atoms or molecules in one mole of a substance.
Physics: Handling extremely small quantities like the charge of an electron (1.602 x 10-19 Coulombs).
Computer Science: Representing large data sizes (e.g., gigabytes, terabytes).

Working with Scientific Notation: Multiplication and Division

Scientific notation simplifies multiplication and division significantly:

Multiplication: To multiply numbers in scientific notation, multiply the coefficients and add the exponents.

For example: (2.5 x 103) x (4 x 102) = (2.5 x 4) x 10(3+2) = 10 x 105 = 1 x 106

Division: To divide numbers in scientific notation, divide the coefficients and subtract the exponents.

For example: (8 x 106) / (2 x 103) = (8/2) x 10(6-3) = 4 x 103

Working with Scientific Notation: Addition and Subtraction

Addition and subtraction require a little more care. Before performing the operation, ensure that both numbers have the same exponent. Then, add or subtract the coefficients while keeping the exponent the same.

For example: Add 2.5 x 103 and 3 x 102

Convert both numbers to have the same exponent: 2.5 x 103 and 0.3 x 103.
Add the coefficients: 2.5 + 0.3 = 2.8
The result is: 2.8 x 103.

Converting from Standard Notation to Scientific Notation: Examples

Let's practice converting some more numbers into scientific notation:

0.000005: Move the decimal point six places to the right. The coefficient is 5, and the exponent is -6. Therefore, the scientific notation is 5 x 10-6.
450,000,000: Move the decimal point eight places to the left. The coefficient is 4.5, and the exponent is +8. Therefore, the scientific notation is 4.5 x 108.
12.0004: Move the decimal point one place to the left. The coefficient is 1.20004, and the exponent is +1. Therefore, the scientific notation is 1.20004 x 101.
0.0003256: Move the decimal point four places to the right. The coefficient is 3.256, and the exponent is -4. Therefore, the scientific notation is 3.256 x 10-4.

Converting from Scientific Notation to Standard Notation

To convert from scientific notation to standard notation, simply move the decimal point according to the exponent:

3.7 x 104: Move the decimal point four places to the right: 37,000
2.1 x 10-3: Move the decimal point three places to the left: 0.0021
9.02 x 100: The exponent is 0, so the decimal point doesn't move: 9.02

Significant Figures in Scientific Notation

When working with scientific notation, it's crucial to consider significant figures. Significant figures represent the precision of a measurement. The number of significant figures in a number written in scientific notation is determined by the number of digits in the coefficient.

For example:

2.5 x 103 has two significant figures.
1.200 x 10-2 has four significant figures.
4.0 x 106 has two significant figures.

Frequently Asked Questions (FAQ)

Q: What if the coefficient is already between 1 and 10?

A: If the number is already in the range of 1 to 10, its scientific notation is simply the number itself multiplied by 100. For example, 5.2 would be 5.2 x 100.

Q: Can the exponent be zero?

A: Yes, an exponent of zero means the number is already in standard form (between 1 and 10).

Q: What if I have a number with leading zeros but no digits before the leading zeros?

A: For example, 0.0004. You would move the decimal point four places to the right. The exponent is -4 and the coefficient is 4. The scientific notation is 4 x 10-4.

Q: Why is scientific notation important?

A: Scientific notation is important because it allows us to handle very large or very small numbers efficiently, making calculations and comparisons much easier. It provides a standard, concise way of representing these numbers that is universally understood in scientific and technical fields.

Q: How do I handle numbers with multiple decimal places in the coefficient?

A: Keep all the digits after the decimal point in the coefficient. The number of digits after the decimal point reflects the precision of the measurement.

Conclusion

Scientific notation is an indispensable tool for expressing and manipulating very large and very small numbers. Mastering its principles and applications will significantly enhance your ability to perform calculations and comprehend data across diverse scientific and engineering fields. By understanding the systematic approach to converting between standard and scientific notation, along with the rules for mathematical operations, you can confidently navigate the world of vast and minuscule quantities. Remember the core principle: find a coefficient between 1 and 10, and use the exponent of 10 to reflect the magnitude of the original number. With practice, converting numbers to and from scientific notation will become second nature.