Question

Round each number to three significant digits. $$29.555$$

Short answer

29.555 rounded to three significant digits is 29.6.

Step-by-step solution

Identify the Significant Digits

Initially, identify the first three digits of the number that are significant, starting from the first non-zero digit. For the number 29.555, the first three significant digits are 29.5

Determine the Rounding Digit

The third significant digit is 5. Look at the digit to the right of it to determine how to round. The digit to the right is a 5.

Apply Rounding Rules

Since the digit to the right (5) is equal to or greater than 5, round the third significant figure up. This changes 29.5 to 29.6.

Key concepts

Significant Figures

Significant figures, often known as 'significant digits,' play a vital role in scientific and mathematical calculations. They represent the precision of a number, reflecting the meaningful digits that contribute to its accuracy.

When working with significant figures, the first step is to determine which digits in the number are significant. All non-zero digits are considered significant. Zeros between non-zero digits are also significant, while leading zeros are not. For example, in the number 29.555, all the digits are significant because they occur between non-zero digits, or are themselves non-zero.

However, the concept extends beyond just identifying digits; it's also about understanding how many of those digits are allowed to remain in a final answer. This is commonly dictated by the requirements of the exercise or the precision of the measurement instruments used to obtain the data.

Rounding Rules

Rounding is a technique used to reduce the number of digits in a figure while keeping it as close as possible to the original number. When rounding to a specific number of significant figures, we apply certain rules to determine which digits to keep and which to lose.

The most common rounding rule involves looking at the digit immediately to the right of the last digit we want to keep. If this digit is 5 or greater, we round up the last significant figure. If it is less than 5, we leave the last significant figure as it is. In our exercise example, since the digit to the right of the last significant figure (5) is 5, we round up, changing 29.5 to 29.6.

Another important aspect to consider is that when we round up the last significant figure and it is a 9, it could cause a 'chain reaction' of rounding across preceding digits, sometimes increasing the order of magnitude of the number.

Step-by-Step Solution

Providing a step-by-step solution not only helps in arriving at the correct answer but also enhances comprehension and enables problem-solving skills. Each step is crucial and should be presented clearly.

In the exercise provided, the first step involves identifying the first three significant digits. This educates students on evaluating which numbers have an impact on precision. Subsequently, the second step requires students to examine the number that will dictate whether to round up or stay the same. Here, understanding the rounding digit's significance is crucial. The final step is applying the appropriate rounding rule, rounding up if necessary and demonstrating the immediate application of the identified rule.

Such structured guidance reinforces the learning objective and ultimately leads to better retention and understanding. Tailoring explanations to the critical points in each step helps students focus on the learning process, not just the final answer.