Question

Suppose you solve a problem and your calculator's display reads \(0.0000000036 .\) Why not just write this down? Is there any advantage to using the scientific notation?

Short answer

Answer: Using scientific notation, such as \(3.6 \times 10^{-9}\), is advantageous when comparing numbers like the calculator display because it simplifies the representation of very large or very small numbers, making them easier to read, write, and compare. Additionally, it helps to avoid errors in calculations and data entry due to fewer digits involved, and it makes it easier to communicate and interpret numerical data in scientific, engineering, and other technical fields.

Step-by-step solution

Defining Scientific Notation

Scientific notation is a way to represent very large or very small numbers using powers of 10. A number in scientific notation is written as the product of two factors: a coefficient, which is a number between 1 and 10, and a power of 10. For example, \(3.6 \times 10^{-9}\).

Calculator Display

The calculator displays the solution of the problem as \(0.0000000036\). This number has a lot of zeros which can make it difficult to read and write or can lead to errors when transcribing or comparing to other numbers.

Converting to Scientific Notation

To convert the number into scientific notation, we will move the decimal point until we get a number between 1 and 10, which is 3.6 in this case. We moved the decimal point 9 places to the right, which means that the power of 10 will have a negative exponent of 9. So, the scientific notation of the given number is \(3.6 \times 10^{-9}\).

Advantages of Scientific Notation

There are several advantages to using scientific notation: 1. It simplifies the representation of very large or very small numbers, making them easier to read, write, and compare. 2. It helps to avoid errors in calculations and data entry due to the fewer digits involved. 3. It makes it easier to communicate and interpret numerical data in scientific, engineering, and other technical fields.

Conclusion

In summary, while writing down the calculator display as \(0.0000000036\) is not incorrect, it is advantageous to use scientific notation to represent such numbers. Converting the given number to scientific notation gives us \(3.6 \times 10^{-9}\), which is a more concise and accurate representation. This format of notation is widely adopted in scientific and engineering fields for its benefits in simplifying and improving the accuracy of calculations.

Key concepts

Powers of 10

Understanding the powers of 10 is central to scientific notation, a system that manages large numbers (like the distance between stars) or small numbers (like the width of a DNA strand) with ease. When we talk about powers of 10, we're referring to the use of exponentiation to denote how many times to multiply the number 10 by itself.

An exponent, written as a superscript number, indicates this. For instance, when we write \(10^3\), it means \(10 \times 10 \times 10\), which equals 1000. Conversely, negative exponents, such as \(10^{-3}\), represent small numbers, telling us to divide by 10 repeatedly, resulting in \(0.001\). This system is incredibly effective for representing large ranges of numbers in a simple, consistent format.

Representing Small Numbers

Representing small numbers efficiently, especially those significantly smaller than 1, can be tricky. However, using scientific notation, we can express such minute values clearly and concisely. Instead of writing a long string of zeros, we use a combination of a coefficient (a number between 1 and 10) and a negative power of 10.

For example, the number \(0.0000000036\) can be burdensome to deal with in its standard form. Converting it into scientific notation, we use \(3.6 \times 10^{-9}\), where '-9' denotes that the decimal point has moved 9 places to the left. This notation avoids the potential for errors when writing, reading, or performing calculations with very small numbers, as it minimizes the number of zeros while maintaining accuracy.

Calculator Display

Modern calculators can handle a vast range of numbers, but their displays have limitations in showing very large or very small values. To overcome this, calculators use scientific notation to display such numbers. This avoids clutter on the screen and ensures that the displayed result is readable and understood quickly by the user.

The display of \(0.0000000036\) on a calculator, if not showing in scientific notation, can be error-prone and difficult to interpret. Instead, calculators often default to scientific notation, such as \(3.6 \times 10^{-9}\), making it more manageable for both entry and interpretation. Recognizing scientific notation on calculator displays is an essential skill, especially in fields requiring precision like physics, chemistry, and engineering.