Comparing Numbers In Scientific Notation: Which Sign Is Correct?

Understanding how to compare numbers expressed in scientific notation is a fundamental skill in mathematics. It allows us to efficiently work with very large and very small numbers. In this article, we'll explore the process of comparing two numbers in scientific notation, specifically addressing the question: What sign correctly compares $9.07 imes 10^0$ and $907,000,000,000 imes 10^{-12}$? We'll break down the concepts, provide step-by-step instructions, and ensure you grasp the underlying principles. Let’s dive in and make comparing these numbers a breeze!

Understanding Scientific Notation

To effectively compare numbers, it's crucial to grasp what scientific notation is and how it works. Scientific notation is a way of expressing numbers, particularly very large or very small numbers, in a concise and standardized format. The general form of scientific notation is $a imes 10^b$, where $a$ is a number between 1 and 10 (but not including 10), and $b$ is an integer (a positive or negative whole number). This format makes it easier to compare numbers because it separates the significant digits from the magnitude (the power of 10). For instance, a large number like 3,000,000 can be written as $3 imes 10^6$, and a small number like 0.000003 can be written as $3 imes 10^{-6}$. The exponent indicates how many places the decimal point needs to be moved to get the original number. A positive exponent means the decimal point moves to the right, indicating a large number, while a negative exponent means the decimal point moves to the left, indicating a small number. Understanding this notation is the first step in confidently comparing any numbers, no matter their size. When you get comfortable with scientific notation, you'll find that working with very large and very small numbers becomes much more manageable and less prone to errors. It's a tool that simplifies complex calculations and provides a clear way to represent numerical data in various fields, from science to engineering. So, before we jump into comparing the specific numbers in our question, make sure you're comfortable with the basics of scientific notation. This foundation will make the comparison process much smoother and more intuitive.

Converting to Standard Scientific Notation

Before we compare the given numbers, we must ensure they are in standard scientific notation. Standard scientific notation means expressing a number as a product of a coefficient between 1 and 10 (excluding 10) and a power of 10. This standardized form makes comparing numbers much simpler and more straightforward. Let’s consider our two numbers: $9.07 imes 10^0$ and $907,000,000,000 imes 10^{-12}$. The first number, $9.07 imes 10^0$, is already in standard scientific notation because 9.07 is between 1 and 10, and the exponent is an integer. However, the second number, $907,000,000,000 imes 10^{-12}$, is not in standard form because 907,000,000,000 is significantly larger than 10. To convert this number to standard scientific notation, we need to move the decimal point until we have a coefficient between 1 and 10. In this case, we move the decimal point 11 places to the left, which gives us 9.07. Each move of the decimal point corresponds to a change in the exponent. Since we moved the decimal point 11 places to the left, we increase the exponent by 11. Therefore, $907,000,000,000$ becomes $9.07 imes 10^{11}$. Now, we need to incorporate the original $10^{-12}$ term. So, we have $9.07 imes 10^{11} imes 10^{-12}$. To simplify this, we use the rule of exponents that states $a^m imes a^n = a^{m+n}$. Thus, we add the exponents: $11 + (-12) = -1$. This gives us $9.07 imes 10^{-1}$. So, the second number in standard scientific notation is $9.07 imes 10^{-1}$. Now that both numbers are in standard scientific notation, we have $9.07 imes 10^0$ and $9.07 imes 10^{-1}$. This standardization is crucial because it aligns the numbers for a direct comparison, allowing us to easily see their relative magnitudes. Without this step, comparing the numbers can be confusing and lead to incorrect conclusions. The process of converting to standard scientific notation ensures we are comparing apples to apples, making the subsequent steps much more clear and straightforward. The ability to convert numbers into this form is a key skill in mathematics and science, as it simplifies complex calculations and facilitates a better understanding of numerical values.

Comparing the Numbers

With both numbers now in standard scientific notation, $9.07 imes 10^0$ and $9.07 imes 10^{-1}$, we can easily compare them. The first step in comparing numbers in scientific notation is to look at the exponents. The exponent tells us the order of magnitude of the number; in other words, how large or small the number is. A larger exponent indicates a larger number, while a smaller exponent indicates a smaller number. In our case, the exponents are 0 and -1. Since 0 is greater than -1, we know that $10^0$ is greater than $10^{-1}$. This means that $9.07 imes 10^0$ is greater than $9.07 imes 10^{-1}$. To further illustrate this, let's think about what these exponents mean in terms of decimal places. The term $10^0$ is equal to 1, so $9.07 imes 10^0$ is simply 9.07. On the other hand, $10^{-1}$ is equal to 0.1, so $9.07 imes 10^{-1}$ is equal to $9.07 imes 0.1$, which is 0.907. Comparing 9.07 and 0.907, it’s clear that 9.07 is larger. When the coefficients (the numbers multiplied by the powers of 10) are the same, as they are in this case (both are 9.07), the comparison becomes even simpler. The number with the larger exponent will always be the larger number. If the coefficients were different, we would need to consider both the coefficients and the exponents to make an accurate comparison. However, in this scenario, the exponents alone tell us the answer. This step-by-step comparison method allows us to confidently determine the relative sizes of numbers in scientific notation. Understanding how to interpret exponents and their impact on the magnitude of a number is a crucial skill in mathematics and various scientific fields. By focusing on the exponents first, we can quickly and efficiently compare even very large or very small numbers.

Determining the Correct Sign

Now that we've compared the numbers $9.07 imes 10^0$ and $9.07 imes 10^{-1}$, we can determine the correct sign to make the statement true. We found that $9.07 imes 10^0$ is greater than $9.07 imes 10^{-1}$. In mathematical notation, the "greater than" sign is represented by ">." Therefore, the correct sign to use in our statement is ">." This gives us the true statement: $9.07 imes 10^0 > 9.07 imes 10^{-1}$. To reiterate, we initially had the numbers $9.07 imes 10^0$ and $907,000,000,000 imes 10^{-12}$. After converting the second number to standard scientific notation, we had $9.07 imes 10^0$ and $9.07 imes 10^{-1}$. By comparing the exponents, we determined that $10^0$ is greater than $10^{-1}$, and thus, $9.07 imes 10^0$ is greater than $9.07 imes 10^{-1}$. The use of the ">" sign accurately reflects this relationship. Choosing the correct sign is crucial in mathematics as it correctly represents the relationship between the values being compared. The greater than sign (">" ) indicates that the value on the left is larger than the value on the right. The less than sign ("<" ) indicates that the value on the left is smaller than the value on the right, and the equals sign ("=" ) indicates that the values are the same. In this case, because the first number is larger, the greater than sign is the appropriate choice. Understanding these signs and when to use them is a fundamental skill in mathematics, essential for expressing relationships between numbers and solving equations. The ability to correctly identify and use these signs ensures clear and accurate communication of mathematical concepts. Thus, the answer to our question is that the correct sign is ">", making the statement $9.07 imes 10^0 > 9.07 imes 10^{-1}$ true.

Conclusion

In conclusion, determining the correct sign to compare $9.07 imes 10^0$ and $907,000,000,000 imes 10^{-12}$ involves understanding and applying the principles of scientific notation. We converted both numbers to standard scientific notation, compared their exponents, and identified that $9.07 imes 10^0$ is greater than $9.07 imes 10^{-1}$. Therefore, the correct sign is ">". This exercise highlights the importance of scientific notation in simplifying the comparison of very large and very small numbers. By expressing numbers in the form $a imes 10^b$, we can quickly assess their relative magnitudes by comparing the exponents. The ability to convert numbers to standard scientific notation is a valuable skill in mathematics, science, and engineering, as it facilitates accurate calculations and comparisons. Understanding the meaning of exponents and their impact on the size of a number is crucial for success in these fields. Furthermore, knowing how to use comparison signs correctly ensures clear communication of mathematical relationships. The greater than (">" ), less than ("<" ), and equals ("=" ) signs are fundamental tools for expressing how numbers relate to each other. Mastering these concepts allows us to confidently tackle a wide range of mathematical problems and applications. This detailed exploration of the comparison process not only answers the specific question but also reinforces the underlying principles of scientific notation and number comparison. By following the steps outlined—converting to standard form, comparing exponents, and choosing the correct sign—anyone can confidently compare numbers, no matter their size. To further your understanding of scientific notation, you might find helpful resources on websites like Khan Academy's Scientific Notation.