Chapter 1: Problem 43
What is the advantage of using scientific notation over decimal notation?
Short Answer
Expert verified
Scientific notation offers clearer magnitude interpretation, easier calculations, and saves space.
Step by step solution
01
Introduce Scientific and Decimal Notation
Scientific notation is a way of expressing numbers that are too large or too small in a compact form using powers of ten, such as \(6.02 \times 10^{23}\). In contrast, decimal notation expresses numbers in their standard form, such as 602,000,000,000,000,000,000,000.
02
Discuss Ease of Interpretation
Scientific notation provides a clear way to read and understand very large or very small numbers by converting them into a form that quickly reveals their magnitude and significant figures. For example, \(3.2 \times 10^{-4}\) quickly shows the number is small and consists of three significant figures.
03
Explain Computational Convenience
Calculations, especially multiplication and division, can be easier and less error-prone when using scientific notation because you can directly add or subtract the exponents. For example, multiplying \(2 \times 10^5\) by \(3 \times 10^3\) becomes simpler as you calculate \(6 \times 10^8\) by adding the exponents.
04
Highlight Space Efficiency
Scientific notation reduces the number of digits needed to write extremely large or small numbers, saving space on paper or screen. For instance, \(1.67 \times 10^{-27}\) requires fewer digits than 0.00000000000000000000000000167.
Over 22 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Decimal Notation
Decimal notation is the standard way of writing numbers using the base ten system, which most of us are familiar with. In decimal notation, each digit in a number has a specific place, such as units, tens, hundreds, and so forth. This system is intuitive because it aligns with how we typically count daily.
For example, the number 1500 in decimal notation clearly shows one thousand, five hundred units without any need for further symbols or characters. However, the bigger or smaller the number becomes, the longer and more complicated it can get to write and understand.
Decimal notation can lead to overly lengthy numbers when dealing with very large figures like the distance between stars or very small figures like the size of an atom. That's where scientific notation comes in handy to simplify these expressions.
For example, the number 1500 in decimal notation clearly shows one thousand, five hundred units without any need for further symbols or characters. However, the bigger or smaller the number becomes, the longer and more complicated it can get to write and understand.
Decimal notation can lead to overly lengthy numbers when dealing with very large figures like the distance between stars or very small figures like the size of an atom. That's where scientific notation comes in handy to simplify these expressions.
Significant Figures
Significant figures are the digits in a number that carry meaning contributing to its precision. They play a crucial role in scientific notation and are essential when measuring and reporting scientific data.
In decimal notation, significant figures can sometimes be buried and not immediately clear. For instance, the number 0.005430 has four significant figures: 5, 4, 3, and the last 0. Leading zeros are just placeholders. In scientific notation, this number can be expressed as 5.430 × 10^{-3}, making the significant figures explicit.
This is advantageous because it avoids ambiguity. When numbers are presented in scientific notation, it becomes apparent which digits are significant, allowing for more precise calculations and clearer communication of scientific information.
In decimal notation, significant figures can sometimes be buried and not immediately clear. For instance, the number 0.005430 has four significant figures: 5, 4, 3, and the last 0. Leading zeros are just placeholders. In scientific notation, this number can be expressed as 5.430 × 10^{-3}, making the significant figures explicit.
This is advantageous because it avoids ambiguity. When numbers are presented in scientific notation, it becomes apparent which digits are significant, allowing for more precise calculations and clearer communication of scientific information.
Powers of Ten
The concept of powers of ten is integral to understanding scientific notation. A power of ten notation expresses the number of times ten is multiplied by itself. For example, 10^3 means 10 multiplied by 10, then again by 10, resulting in 1000.
Using powers of ten makes it simpler to express both very large and very small numbers. If you're talking about the mass of the Earth or distances in outer space, it avoids the clutter of zeros that would occur in decimal notation. Thus, 5.97 × 10^{24} is much easier to read than writing the entire number out in decimal form.
These small exponents make it easy to identify the magnitude of the number, simplifying calculations and comparisons. The efficiency and clarity of using powers of ten are pivotal to scientific communication and education.
Using powers of ten makes it simpler to express both very large and very small numbers. If you're talking about the mass of the Earth or distances in outer space, it avoids the clutter of zeros that would occur in decimal notation. Thus, 5.97 × 10^{24} is much easier to read than writing the entire number out in decimal form.
These small exponents make it easy to identify the magnitude of the number, simplifying calculations and comparisons. The efficiency and clarity of using powers of ten are pivotal to scientific communication and education.
Computational Convenience
One of the significant benefits of scientific notation is its computational convenience. Performing arithmetic operations, especially multiplication and division, is much easier with scientific notation.
When numbers are in scientific notation, you only need to deal with the coefficients independently and add or subtract the exponents. For instance, multiplying 4 × 10^6 with 2 × 10^3 requires you to just multiply the coefficients (4 and 2) to get 8, and then add the exponents (6 and 3) to get 9, resulting in 8 × 10^9.
This simplification reduces errors and saves time, especially in fields that demand precision and efficiency, such as physics and engineering. By handling complex calculations more swiftly, scientific notation enhances accuracy and productivity.
When numbers are in scientific notation, you only need to deal with the coefficients independently and add or subtract the exponents. For instance, multiplying 4 × 10^6 with 2 × 10^3 requires you to just multiply the coefficients (4 and 2) to get 8, and then add the exponents (6 and 3) to get 9, resulting in 8 × 10^9.
This simplification reduces errors and saves time, especially in fields that demand precision and efficiency, such as physics and engineering. By handling complex calculations more swiftly, scientific notation enhances accuracy and productivity.
