Question

Show how converting numbers to scientific notation can help you decide which digits are significant.

Short answer

Converting numbers to scientific notation helps in determining the significant digits by presenting the number in a more simplified form, where only the significant digits are part of the coefficient and any trailing zeros serve as rounding digits. For example, 0.000345 becomes $3.45\times {10}^{-4}$, where 3 and 4 are significant digits, and 76500 becomes $7.65\times {10}^3$, with significant digits 7 and 6. This process highlights the significant digits, making it easier to identify and work with them.

Step-by-step solution

Example 1: Convert number to scientific notation

Let's take an example, 0.000345, and convert it to scientific notation. To do this, move the decimal point to the right until we have only one digit to its left. This digit will be multiplied by a power of 10, where the exponent is equal to the number of units we moved the decimal point. In this case, we need to move the decimal point 4 places to the right. So, the number becomes: $0.000345=3.45\times {10}^{-4}$ Now, it is easier to identify the significant digits: the 3 and the 4 (5 is not significant because it is primarily used for rounding).

Example 2: Convert number to scientific notation

Let's take another example, 76500. Convert it to scientific notation by moving the decimal point three places to the left. $76500=7.65\times {10}^3$ The significant digits are 7 and 6. The 5 serves as a rounding digit.

Example 3: Convert number to scientific notation

Let's take a third example, 0.084200. We move the decimal point two places to the right. $0.084200=8.4200\times {10}^{-2}$ Here, all the digits are significant, including the trailing zeros. #Conclusion:# Converting numbers to scientific notation makes it easier to identify the significant digits, as they will be the digits that are part of the coefficient (the number before the "x"). The trailing zeros serve as rounding digits but can become significant if they are explicitly included after the decimal point.

Key concepts

Scientific Notation

Scientific notation is a powerful tool used to express long numbers in a compact form, particularly in the fields of science and engineering. It simplifies numbers by reducing the number of zeros, at the same time making identification of significant figures quite straightforward. Scientific notation follows the format:

$a\times {10}^n$

where $a$ is a number greater than or equal to 1 and less than 10, often referred to as the coefficient, and $n$ is an integer that represents the number of places the decimal point is moved. In scientific notation, significant digits are contained within the coefficient $a$, which easily conveys the precision of a number without unnecessary or ambiguous zeros.

For example, consider the number 0.000345, in scientific notation, it is written as $3.45\times {10}^{-4}$, clearly showing two significant digits, 3 and 4; while the 5 is for rounding purposes and doesn't count as significant.

Significant Figures

Significant figures, or significant digits, are a way to express the precision of a number. These are the digits that carry meaning contributing to its measurement resolution. This includes all non-zero digits, any zeros between significant digits, and trailing zeros in a decimal number. Understanding which digits are significant ensures accurate and meaningful communication of measurements.

For instance, in the number 0.084200, all the digits are considered significant because the trailing zeros in the decimal signify precision. The application of scientific notation also highlights exactly which digits are significant, as in the scientific notation $8.4200\times {10}^{-2}$, where all digits in the coefficient, including the trailing zeros, are significant.

Decimal Place Movement

When converting a number into scientific notation, you move the decimal place to create a new number from 1 to less than 10. This process is crucial for identifying significant figures and for simplifying calculations. The direction in which the decimal point is moved reflects the size of the number. For smaller numbers, the decimal point is moved to the right, making the exponent negative, while for large numbers, it is moved to the left with a positive exponent.

To illustrate this with a real number, 76500 is transferred to scientific notation as $7.65\times {10}^3$, moving the decimal three places to the left. Here, the exponent is positive three, indicating the number's largeness and showing its significant digits distinctly.

Exponent

An exponent in scientific notation indicates how many times the base, which is always 10, should be multiplied by itself. It's a shorthand for representing the zeros that follow or precede a number. A positive exponent shows how many places the decimal has moved to the left, indicating a large number. Conversely, a negative exponent denotes how many places to move the decimal to the right, which signals a smaller number. Exponents help us quickly understand the scale and precision of the original number without counting numerous zeroes.

For example, the exponent in $3.45\times {10}^{-4}$ is -4, showing that the decimal point was moved four places to the right from its original position, pointing out that the number is much smaller than one. Similarly, $7.65\times {10}^3$ with an exponent of 3 denotes a much larger number due to the decimal being shifted three places to the left.