Question

Is the number \(12.38 \times 10^{2}\) in scientific notation? Explain.

Short answer

No, the number \(12.38 \times 10^{2}\) is not in scientific notation because the factor is greater than 10.

Step-by-step solution

Understanding Scientific Notation

The scientific notation of a number is written as one non-zero digit followed by a decimal point and all the other significant digits of the original number, which is then multiplied by 10 raised to an exponent. The exponent corresponds to the number of places the decimal has been moved from the original number. In other words, a number is said to be in scientific notation if it is in the form \(a \times 10^{n}\), where \(1 \leq a < 10\) and \(n\) is an integer.

Comparing the Given Number with the Rules of Scientific Notation

We are given the number \(12.38 \times 10^{2}\). To be in scientific notation, the factor in front of \(10^{2}\), which in this case is 12.38, should be a number greater than or equal to 1 and less than 10. Here the factor 12.38 is greater than 10, so the number is not in scientific notation.

Key concepts

Algebra

Algebra forms the foundation for understanding scientific notation and other mathematical concepts. It is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it's about finding the unknown or putting real-life variables into equations and then solving them.

When algebraic expressions contain exponents, as they often do in scientific notation, it's important to know the rules that govern these operations. For instance, to state a number in scientific notation, we use a base of 10 raised to a power, and this base-power combination simplifies data presentation, especially for very large or small numbers.

Relevance in Scientific Notation

In the context of scientific notation, algebra teaches us to adjust the exponents and coefficients to maintain the equality of an equation when we convert a number into or out of scientific notation. If you're working with the number 12.38 and need it in scientific notation, algebraic principles would guide you in dividing 12.38 by 10 to shift the decimal point, which in turn adjusts the exponent to reflect this change.

Exponents

Exponents are a critical element of scientific notation and are used to represent how many times a number, known as the base, is multiplied by itself. An exponent is written as a small number to the upper right of the base. For example, in the expression 10 raised to the power of 2, written as \(10^2\), 10 is the base and 2 is the exponent, indicating that 10 should be multiplied by itself once (since \(10^1 = 10\) and \(10^2 = 10 \times 10 = 100\)).

Role in Scientific Notation

Scientific notation requires that the exponent reflects how far the decimal point is moved to convert a number into a single-digit number times a power of 10. In the given exercise, \(10^2\) suggests that the decimal point in the original number has been moved two places to the right. Understanding exponents is imperative for interpreting scientific notation and for executing operations such as multiplying and dividing numbers in scientific notation correctly.

Significant Digits

Significant digits, also referred to as significant figures, are the digits in a number that carry meaning toward its precision. They include all non-zero numbers, any zeroes between non-zero digits, and trailing zeroes in a decimal number. For example, in the number 0.0045070, the significant digits are 45070.

In the realm of scientific notation, significant digits play a vital role. The number we convert into scientific notation should begin with a non-zero digit, followed by the appropriate number of significant digits. The given exercise presents the number \(12.38 \times 10^2\), where 12.38 contains four significant digits.

Importance for Scientific Notation

For a number to be correctly formatted in scientific notation, it must contain all the significant digits from the original number but be represented in a way that the value starts with a single-digit number from 1 to 9 inclusive, followed by a decimal point, and then any remaining significant digits. This requirement helps in expressing the value precisely while making it easy to read and write, especially when dealing with extremely large or small scales.