Problem 112 - Chemistry Matter and Change

Question

Round 20.56120 \(\mathrm{g}\) to three significant figures. (Chapter 2\()\)

Step-by-Step Solution

Answer

The first three non-zero digits of 20.56120 g are 20.5. The digit after the third significant figure is 6, so we round the third digit up, making the number 20.6 g, which has three significant figures.

Step 1: Identify the first three non-zero digits

The first three non-zero digits in the given decimal number (20.56120 g) are 20.5.

Step 2: Check the following digit

The digit following the third significant figure (5) is 6.

Step 3: Round the third digit if necessary

Since the following digit (6) is greater than 5, we round the third digit up. Thus, 5 becomes 6.

Step 4: Write the final rounded number

After rounding, the number becomes 20.6 g, which has three significant figures.

Key Concepts

Rounding

Rounding is an important mathematical technique that simplifies numbers while retaining their value within a certain level of accuracy. When you round a number, you're looking at the digits that make up the number and deciding where to "cut off" less significant digits. This is often determined by the purpose or context in which the number is used.

For instance, in the exercise given, the task is to round the number 20.56120 to three significant figures. To achieve this, you begin by identifying the first three significant digits, which are 20.5. The next digit, 6, determines whether we leave the third digit as it is or round it up. Because 6 is greater than 5, we round the number up from 20.5 to 20.6. Rounding helps manage data more efficiently and keeps calculations simple without losing essential information.

Decimal Numbers

Decimal numbers are numbers that have a decimal point, which separates the whole number part from the fractional part. They are a convenient way to express numbers that are not whole and use a base-10 number system. The decimal point is crucial as it helps to accurately represent values that are less than one or in-between whole numbers.

In the given exercise, the number 20.56120 is a decimal. The digits to the right of the decimal point (56120) show the fractional part of the number. When dealing with decimal numbers, especially in rounding, each digit’s position relative to the decimal point holds significant importance. This hierarchical significance ensures precision and accuracy in mathematics and sciences.

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. This method uses powers of ten to simplify and standardize numbers, making it easier to read, compare, and compute.

It typically involves writing the number as the product of a number between 1 and 10, and a power of ten. For example, the number 20.56120 can be written in scientific notation as 2.056120 × 10². By rounding this to three significant figures based on our original exercise, it becomes 2.06 × 10². Scientific notation is especially useful in scientific calculations and when dealing with extremely large or small values. It ensures numbers are not only easily manageable but also precise.