Question

If you move the decimal place to the left to convert a number to scientific notation, will the power of 10 be positive or negative?

Short answer

If you move the decimal place to the left to convert a number to scientific notation, the power of 10 will be positive. This is because you are making the original number smaller by dividing it by a power of 10, resulting in a positive exponent.

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in standard decimal notation. It is written in the form: \(a \times 10^b\), where \(a\) is a number between 1 and 10 (1 ≤ a < 10), and \(b\) is an integer (positive, negative, or zero).

Moving Decimal Place to the Left

When moving the decimal place to the left to convert a number to scientific notation, we are making the original number smaller. In other words, we are dividing the original number by a power of 10. When dividing by a power of 10, the exponent becomes negative. For example, consider the following number: \(53,000\) To convert it to scientific notation, we move the decimal place 4 places to the left: \(5.3\) Thus, the number in scientific notation is \(5.3 \times 10^4\).

Determining the Power of 10: Positive or Negative?

As we saw in the example above, when we move the decimal place to the left, the power of 10 is positive. This is because we are dividing the original number by a power of 10 to make it smaller, and division with a power of 10 results in a negative exponent. So, if you move the decimal place to the left to convert a number to scientific notation, the power of 10 will be positive.

Key concepts

Expressing Large Numbers

Understanding how to express large numbers succinctly is a fundamental skill in various scientific and mathematical fields. Scientific notation serves as a convenient method for handling such extensive figures. To visualize, imagine writing down the distance from the Earth to the Sun in standard decimal notation, which is approximately 150,000,000 kilometers. In scientific notation, this becomes a more manageable expression: \(1.5 \times 10^8\) kilometers.

Scientific notation format consists of two parts: a coefficient and a power of ten. The coefficient is a decimal number between 1 and 10, while the exponent indicates how many places the decimal point has been moved. This system allows for clear representation of both incredibly large and extremely small numbers, enhancing comprehension and reducing the likelihood of errors when reading, writing, or calculating.

Decimal Notation

Decimal notation is the standard method for writing numbers that most people are familiar with. It uses a base-10 system with each digit having a place value ten times that of the digit to its right. For instance, in the number 304, the 3 represents 300, or \(3 \times 10^2\), the 0 represents 0, and the 4 represents 4, or \(4 \times 10^0\).

However, decimal notation can become unwieldy when dealing with very large or very small numbers because it requires writing out many zeros. This is where scientific notation comes in handy by providing a compact and precise form. By shifting the decimal point to the right of the first non-zero digit and noting the number of moves as the power of 10, we achieve an equivalent expression of the original number.

Negative Exponent

In scientific notation, the concept of a negative exponent often causes confusion. A negative exponent indicates division by the base raised to the absolute value of the exponent. For example, \(10^{-3}\) is the same as \(\frac{1}{10^3}\) or \(\frac{1}{1000}\).

When converting a very small number, less than 1, to scientific notation, the decimal must be moved to the right to position it after the first non-zero digit, resulting in a negative exponent. It is critical to understand that the negative exponent is not indicating a negative number but a very small positive number. As an exercise, try converting 0.0005 to scientific notation, which becomes \(5 \times 10^{-4}\), indicating that the decimal moved four places to the right. Conversely, when dealing with large numbers, as in the textbook exercise, moving the decimal place to the left will yield a positive exponent, simplifying our notation for extensive values.