Question

For each of the following numbers, by how many places must the decimal point be moved to express the number in standard scientific notation? In each case, will the exponent be positive, negative, or zero?

Short answer

For 126, move the decimal point 2 places left (\(1.26 \times 10^2\), positive exponent). For 0.0185, move the decimal point 2 places right (\(1.85 \times 10^{-2}\), negative exponent). For 530000, move the decimal point 5 places left (\(5.3 \times 10^5\), positive exponent). For 6, no need to move the decimal point (\(6.0 \times 10^0\), zero exponent). For 0.000459, move the decimal point 4 places right (\(4.59 \times 10^{-4}\), negative exponent).

Step-by-step solution

Identify the number in standard scientific notation

To determine the number of places to move the decimal point and the resulting exponent, identify the correct position of the decimal point so that there is only one non-zero digit to the left of the decimal point.

Count the number of decimal places moved and determine the exponent's sign

Count the number of decimal places the point was moved. If the decimal point was moved to the left, the exponent will be positive. If the decimal point was moved to the right, the exponent will be negative. If the decimal point does not need to be moved, the exponent will be zero. Now let's apply these steps to the given numbers.

Example 1: 126

To convert 126 to standard scientific notation, we have to move the decimal point 2 places to the left. This gives us \(1.26 \times 10^2\). Therefore, the exponent is positive and has a value of 2.

Example 2: 0.0185

To convert 0.0185 to standard scientific notation, we have to move the decimal point 2 places to the right. This gives us \(1.85 \times 10^{-2}\). Therefore, the exponent is negative and has a value of -2.

Example 3: 530000

To convert 530000 to standard scientific notation, we have to move the decimal point 5 places to the left. This gives us \(5.3 \times 10^5\). Therefore, the exponent is positive and has a value of 5.

Example 4: 6

To convert 6 to standard scientific notation, we do not need to move the decimal point. This gives us \(6.0 \times 10^0\). Therefore, the exponent is zero and has a value of 0.

Example 5: 0.000459

To convert 0.000459 to standard scientific notation, we have to move the decimal point 4 places to the right. This gives us \(4.59 \times 10^{-4}\). Therefore, the exponent is negative and has a value of -4.

Key concepts

Decimal Point

The decimal point is a key concept when working with numbers that need to be written in a standardized form, such as scientific notation. It serves as a marker to separate the whole number part of a number from its fractional part. Knowing the position of the decimal point is crucial for accurately reading and calculating numbers.

When you convert a number to scientific notation, you must adjust the decimal point to ensure there is only one non-zero digit to the left. For example, in the number 126, the decimal point originally follows the last digit. To convert this to scientific notation, you move it two places to the left ending up with 1.26. The new placement of the decimal point determines the magnitude of the number and, consequently, the sign and value of the exponent in the notation system.

Understanding the importance of the decimal point placement will help simplify calculations and improve accuracy in interpreting results.

Exponent

In scientific notation, the exponent is part of the expression that indicates how many times the number should be multiplied or divided by 10. Once we shift the decimal point to create only one non-zero digit before it, the number of places the decimal is moved becomes the exponent.

The exponent's sign depends on the direction in which the decimal point is moved:

• If the decimal moves to the left, the exponent is positive. This shows that the original number is large, multiplies by a power of 10.
• If the decimal moves to the right, the exponent is negative, indicating that the original number is small, divided by a power of 10.
• If there is no movement of the decimal point needed, the exponent is zero.

For instance, to convert 0.0185 into scientific notation, the decimal is moved two places to the right, resulting in a negative exponent of -2, leading to the representation of 1.85 × 10^-2.

Standard Form

Standard form is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It arises commonly in scientific notation, where numbers are expressed as the product of a number between 1 and 10 and a power of 10.

Consider the number 530,000: to express it in standard scientific notation, you move the decimal point five places to the left, converting it into 5.3, with an exponent of 5. It then becomes 5.3 × 10^5. This format makes it easy to handle calculations involving very large or very small numbers.

Using standard form simplifies comparing numbers, performing computations, or understanding scales in scientific work or calculations. It enhances both clarity and precision, which are pivotal in mathematical operations and scientific communication.