Question

Write \(19.919\) to (a) 2 d.p., (b) 1 d.p.

Short answer

Question: Round the number 19.919 to 2 decimal places and 1 decimal place. Answer: When rounded to 2 decimal places, 19.919 becomes 19.92. When rounded to 1 decimal place, 19.919 becomes 19.9.

Step-by-step solution

Identify the decimal places

In the given number \(19.919\), we can see there are three decimal places. Let's identify the digits in each position: 1st decimal place: 9 2nd decimal place: 1 3rd decimal place: 9 Now, we'll proceed to round the number to both 2 decimal places and 1 decimal place.

Round \(19.919\) to 2 decimal places

To round \(19.919\) to 2 decimal places, we'll analyze the digit in the 3rd decimal place, which is 9. Since 9 is greater than 5, we'll round up. To do this, we add 1 to the digit in the 2nd decimal place and keep the first decimal place as it is: \(19.91\boxed{9} \rightarrow 19.92\) So, \(19.919\) rounded to 2 decimal places is \(19.92\).

Round \(19.919\) to 1 decimal place

To round \(19.919\) to 1 decimal place, we'll analyze the digit in the 2nd decimal place, which is 1. Since 1 is less than 5, we'll round down, which means we'll keep the first decimal place as it is and remove the following decimal places: \(19.\boxed{91}9 \rightarrow 19.9\) So, \(19.919\) rounded to 1 decimal place is \(19.9\).

Key concepts

Understanding Decimal Places

Decimal places refer to the number of digits to the right of the decimal point in a number. These places help provide precise measurements or calculations. For example, in the number \(19.919\), the three digits after the decimal point are the decimal places: 9, 1, and 9. Let's break it down a little further:

• 1st decimal place: This is the digit immediately after the decimal point, which is 9.
• 2nd decimal place: This is the second digit from the decimal point, which is 1.
• 3rd decimal place: This is the third digit from the decimal point, which is another 9.

Identifying these positions helps when performing rounding or other mathematical operations. This is because the number of decimal places can affect the rounding decisions and the overall precision of a representation. Remember, more decimal places mean more accuracy but sometimes require simplification for practicality.

Rounding Numbers Explained

Rounding numbers is the process of adjusting the digits of a number to make it simpler or to reflect a precision level that's needed for a particular task. Often, we round numbers for ease of computation or for better presentation. The rule of thumb for rounding is:

• Look at the digit immediately after your target position. If this digit is less than 5, you round down.
• If it's 5 or greater, you round up.

Let's see how this works with our example:- **Rounding to Two Decimal Places:** Take the number \(19.919\). To round to two decimal places, look at the third decimal place, which is 9. Since 9 is greater than 5, you round up the number to \(19.92\).- **Rounding to One Decimal Place:** Using the same number \(19.919\), when rounding to one decimal place, observe the second decimal place, which is 1. Since 1 is less than 5, the number rounds down to \(19.9\).Rounding might seem mechanical, but it plays a crucial role in mathematics, especially when dealing with financial figures and scientific data, where exact values are not always necessary.

Significant Figures Demystified

Significant figures are the digits in a number that contribute to its precision. These include all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion. Grasping this concept is essential as it represents how precise a measurement is.When considering significant figures:

• All non-zero numbers are always significant. For example, in \(19.919\), all digits are significant.
• Zeros between significant figures are also significant. For instance, in \(306.05\), all five digits are significant.
• Trailing zeros in the decimal part are significant. For example, \(0.500\) has three significant digits.

Understanding how to count and manage significant figures is often utilized in science and engineering, ensuring that calculated results do not imply excessive precision. When rounding, significant figures help in determining which digits are retained, ensuring that numbers remain meaningful.