Problem 51 - Prealgebra

Question

Express each number in scientific notation. $$-0.00203$$

Step-by-Step Solution

Answer

\(-0.00203\) in scientific notation is \(-2.03 \times 10^{-3}\).

Step 1: Identify the non-zero digits

Look at the provided number, \(-0.00203\). Identify the non-zero digits, which are 2, 0, and 3, as these are the significant figures in the number.

Step 2: Place the decimal point

To write the number in scientific notation, place the decimal point so only one non-zero digit appears before it. For \(-0.00203\), place the decimal point after the first non-zero digit to get \(-2.03\).

Step 3: Determine the power of 10

Count the number of decimal places the decimal has moved from the original number \(-0.00203\) to \(-2.03\). It moves 3 places to the right. Hence, the power is \(-3\) (because moving the decimal to the right represents a negative exponent).

Step 4: Write in scientific notation

Combine the results from the previous steps to express the number in scientific notation. Therefore, \(-0.00203\) in scientific notation is \(-2.03 \times 10^{-3}\).

Key Concepts

Significant Figures

In scientific notation, significant figures play a crucial role as they determine the precision and accuracy of your numbers. Significant figures are the digits in a number that contribute to its precision. In the number \(-0.00203\), the significant figures are 2, 0, and 3. This means these digits provide important information about the value's accuracy.

To identify significant figures:

• Start from the first non-zero digit and count to the last digit.
• In decimal numbers like \(-0.00203\), zeros before the first non-zero digit are not considered significant.
• Zeros between non-zero digits (like the 0 between 2 and 3) are significant.

Recognizing significant figures is essential when converting numbers into scientific notation as it affects the precision of the outcome.

Decimal Point

The position of the decimal point affects how numbers are expressed and calculated. In scientific notation, placing the decimal correctly is vital to ensure the expression is correct.

Steps to position the decimal point properly:

• Start with your given number, for example, \(-0.00203\).
• Move the decimal point to the right of the first non-zero digit. This transforms \(-0.00203\) into \(-2.03\).
• The position you've decided for the decimal ensures that there is only one non-zero digit to its left.

Correctly placing the decimal point helps in writing the number in a standard scientific format, ensuring clarity and accuracy in computations.

Exponent

An exponent in scientific notation reflects how much the decimal point has shifted to transform the number to its standard form. It is essentially the power of 10 by which the number should be multiplied or divided.

Understanding exponents:

• If the decimal point moves to the right, your exponent is negative because you are dividing by a power of 10. For instance, in moving from \(-0.00203\) to \(-2.03\), the point shifted three places right, resulting in an exponent of \-3\.
• If the decimal moves left, the exponent becomes positive, as you multiply by a power of 10.
• The exponent indicates how many places and in which direction the decimal point has travelled.

Exponents are key to simplifying the process of representing large or small numbers in scientific notation.

Powers of 10

Powers of 10 are fundamental to scientific notation, acting as the tool to scale numbers up or down based on the decimal point's position. Each movement of the decimal point represents a power of 10.

Key aspects of powers of 10:

• Each increase in the power of 10 corresponds to multiplying the number by 10, while decreasing means dividing by 10.
• In the expression \(-2.03 \times 10^{-3}\), \(-3\) indicates that the number must be divided by \10^3\, which correctly scales it to \-0.00203\.
• Using powers of 10 provides a convenient way to express very large or small values without losing precision.

Powers of 10 are crucial for converting numbers into scientific notation efficiently, making large and tiny numbers more manageable to work with.