Question

Although your textbook lists the rules for converting an ordinary number to scientific notation, oftentimes students remember such rules better if they put them into their own words. Pretend you are helping your 12 -year-old niece with her math homework, and write a paragraph explaining to her how to convert the ordinary number 2421 to scientific notation.

Short answer

To convert the ordinary number 2421 to scientific notation, first, find the number between 1 and 10 by placing a decimal point after the first digit, making it 2.421. Then, determine the power of 10 by counting the number of digits after the decimal point in the original number (in this case, 3). Finally, the scientific notation is formed by multiplying the number between 1 and 10 (2.421) by 10 raised to the power of 3: \(2.421 \times 10^3\).

Step-by-step solution

Understand the concept of scientific notation

Scientific notation is a way to represent large or small numbers in a simpler form using a number between 1 and 10 and a power of 10. For example, instead of writing 1000, we can write it as \(1 \times 10^3\), where 1 is the number between 1 and 10, and 3 is the exponent or power of 10.

Identify the number between 1 and 10

The first step in converting 2421 to scientific notation is to determine the suitable number between 1 and 10 that represents majority of 2421. To do this, simply place a decimal point after the first digit of 2421. So, the number between 1 and 10 would be 2.421.

Determine the power of 10

Now, we need to find out how many powers of 10 we should multiply 2.421 by to get back to the original number, 2421. Count the number of digits after the decimal point in the original number. In this case, there are three digits (4, 2, and 1), so the power of 10 would be 3.

Write the scientific notation

Finally, you can write the scientific notation of 2421 by multiplying the number between 1 and 10 (2.421) with 10 raised to the power of the number of digits after the decimal point (3). So, 2421 in scientific notation would be: \(2.421 \times 10^3\). Now you know how to convert the ordinary number 2421 to scientific notation, which is \(2.421 \times 10^3\).

Key concepts

Converting Numbers to Scientific Notation

Understanding the process of converting numbers to scientific notation simplifies dealing with extremely large or tiny values. Imagine you have the number 2421. Converting it to scientific notation involves two main steps.

Firstly, find a number between 1 and 10 by placing a decimal point after the first digit, in this case, resulting in 2.421. Think of this as resizing the number, so it's just right—not too big, not too small, but within the range of 1 to 10.

Next, count how many places you moved the decimal point to get from 2421 to 2.421. Here it's three places to the right. This tells you the 'size' of your original number in terms of powers of 10. Thus, you compactly write 2421 as the product of 2.421 (your resized number) and 10 raised to the power of 3, which expresses the 'size': 2.421 becomes the significant figures, and the exponent 3 shows how many times you'd need to multiply by 10 to get back to the original 2421.

Exponents and Powers of 10

Exponents are the shorthand for repeated multiplication of a number by itself. In the case of scientific notation, we use powers of 10 to make numbers more manageable.

The notation '10 to the power of n', written as \(10^n\), signifies that you multiply 10 by itself 'n' times. For example, \(10^3\) equals 10 multiplied by 10, and then by 10 again, for a total of 1000.

Concretely, when you convert 2421 into scientific notation as 2.421 times \(10^3\), you're saying that 2.421 has to be multiplied by 1000 (which is 10 multiplied by itself 3 times) to get back the original number. Remember, the power of 10 will be positive if you're dealing with large numbers, but it will be negative when working with very small numbers less than one.

Mathematics Education

In mathematics education, the goal is to make challenging concepts accessible. Taking scientific notation as an example, educators strive to find the right balance between procedural instructions and conceptual understanding.

By translating textbook rules into everyday language, students are more likely to understand and remember the process. By working through examples step-by-step and providing context, like comparing scientific notation to a more compact way of carrying a huge number of balloons without them floating away, students begin to appreciate the practicality of these mathematical tools in both science and daily life.

Significant Figures

Significant figures in a number are the digits that carry meaningful contributions to its precision. When converting to scientific notation, only these significant digits are used in the number that falls between 1 and 10.

For instance, the number 2421 has four significant figures: 2, 4, 2, and 1. When we write 2421 as \(2.421 \times 10^3\), we keep all the significant figures. Generally, the more significant figures a measurement has, the more precise it is considered to be. Therefore, recognizing and maintaining the correct number of significant figures is crucial in scientific measurements and calculations to ensure accuracy and convey the precision of the figures involved.