Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ 25.86134 $$

Short answer

Rounded: 25.861, Truncated: 25.861

Step-by-step solution

- Understand Rounding

To round a number to three decimal places, look at the fourth decimal place. If it is 5 or more, round the third decimal place up by one. If it is less than 5, leave the third decimal place unchanged.

- Round the Given Number

The given number is 25.86134. The fourth decimal place is 3, which is less than 5. Therefore, the number rounded to three decimal places is 25.861.

- Understand Truncating

To truncate a number to three decimal places, simply cut off the number after the third decimal place without rounding up.

- Truncate the Given Number

The given number is 25.86134. Truncating this number to three decimal places results in 25.861.

Key concepts

rounding numbers

Rounding is a mathematical procedure used to approximate numbers. It helps simplify numbers, making them easier to work with. To round a number to a specific decimal place:

1. Identify the digit at the place you are rounding to. For three decimal places, this is the third digit to the right of the decimal point.
2. Look at the digit immediately following the identified place. This is the crucial digit that determines whether you round up or stay the same.
3. If the crucial digit is 5 or higher, increase the identified digit by one. If it is less than 5, leave the identified digit unchanged.

Let’s apply this to the number 25.86134. To round it to three decimal places, observe the fourth digit (1). Since 1 is less than 5, the third digit (6) remains unchanged. Therefore, 25.86134 rounded to three decimal places is 25.861.

truncating numbers

Truncating is another way to approximate numbers but without rounding. Instead of adjusting digits based on their value, truncation simply cuts the number after the specified decimal place.

Here’s how truncating works:
1. Identify up to the digit at the place you want to truncate. For three decimal places, you consider up to the third digit after the decimal point.
2. Remove all digits following this position.

Using our example, 25.86134, if we truncate to three decimal places, we only keep the first three digits after the decimal point: 861. Therefore, truncating 25.86134 to three decimal places results in 25.861.

three decimal places

Working with numerical values to three decimal places is common in many fields, including science and finance. It means you consider three digits to the right of the decimal point.
Examples include:

• 0.123
• 45.678
• 123.456

When rounding to three decimal places, you look at the fourth digit to determine whether the third digit changes. In truncation, you simply discard any digits beyond the third place, regardless of their value.

mathematical procedures

Mathematical procedures like rounding and truncating help manage numbers in calculations. They ensure results are practical and understandable.

The chosen method can influence the outcome. Rounding can slightly adjust final values, potentially affecting precision. Truncation, however, cuts off values without alteration, which might be ideal in specific contexts where no rounding is preferred.

For example:

• If rounding impacts financial estimations, the small changes could affect profit and loss statements.
• In scientific measurements, accuracy to several decimal places might be crucial, and one method may be chosen over the other.

Understanding these procedures is key to properly applying them in various scenarios.