## Introduction:

In this unit, we will be introducing and discussing rounding and approximating numbers. Rounding and approximating numbers can prove to be very useful in mathematics when we are dealing with complex calculations. These approximation tactics can be used to simplify difficult math problems and give us accurate estimations to the math problems we are presented with.

## Place Values:

When rounding numbers it is important to have an understanding of place values. **The place value is the position of a digit in a given number**. Each place value has a name to determine its location in a number. The names of the place values are as follows:

**Ones: **This place is the first digit from the right**Tens: **This place is the second digit from the right**Hundreds: **This place is the third digit from the right

When a number has a decimal, the place values to the right of the decimal have their own names:

**Tenths: **This place is the first digit to the right of the decimal point**Hundredths:** This place is the second digit to the right of the decimal point**Thousandths:** This place is the third digit to the right of the decimal point

When we are rounding, it is important to remember the names of the place values as they will tell us which number we are rounding. In the next few examples we will be focusing on rounding to places to the left of the decimal point and to the right of the decimal point. Let’s start to the left of the decimal point.

**Rounding To The Nearest Ten:** When we are rounding to the nearest ten, we will want to look at the ones place. If the ones place is less than 5, then we will round down such that the ones place is truncated to 0. If the ones place is equal to or greater than 5, then we will round up such that the tens place is increased by 1 and the ones place is truncated to 0.

**Rounding To The Nearest Hundred: **When we are rounding to the nearest hundred, we will want to look at the tens place. If the tens place is less than 5, then we will round down such that the tens and ones place are truncated to 0. If the tens place is equal to or greater than 5, then we will round up such that the hundreds place is increased by 1 and the tens and ones place are truncated to 0.

Now let’s look at rounding to the places to the right of the decimal place.

**Rounding To The Nearest Whole Number: **When we are rounding to the nearest whole number, we will want to look at the tenths place. If the tenths place is less than 5, then we will round down such that the entire decimal place is truncated and the whole number is kept the same. If the tenths place is equal to or greater than 5, then we will round up such that the ones place is increased by 1 and the entire decimal is truncated.

**Rounding To The Nearest Tenth: **When we are rounding to the nearest tenth, we will want to look at the hundredths place. If the hundredths place is less than 5, then we will round down such that the hundredths place and every place to the right of it is truncated. If the hundredths place is equal to or greater than 5, then we will round up such that the tenths place is increased by 1 and the hundredths place and every place to the right of it is truncated.

**Rounding To The Nearest Hundredth: **When we are rounding to the nearest hundredth, we will want to look at the thousandths place. If the thousandths place is less than 5, then we will round down such that the thousandths place and every place to the right of it is truncated. If the thousandths place is equal to or greater than 5, then we will round up such that the hundredths place is increased by 1 and the thousandths place and every place to the right of it is truncated.

## Significant Figures:

Another way we can round numbers to approximate calculations is by using significant figures. **A significant figure is a number that holds significance in the accuracy of the rounded number. **We determine significant numbers by counting the digits in a given number. This method is used in order to maintain a level of accuracy when rounding numbers.

When identifying significant numbers, we will need to follow a few rules:

- Every non-zero number is significant.
- Every zero in between non-zero numbers is significant.
- Zeros at the end of a whole number are not significant. These zeros are called
**trailing zeros.** - If a decimal begins with zeros to the left or right of the decimal point, those zeros are not significant.
- Trailing zeros to the right of the decimal point are significant.

**Rounding to One Significant Figure:** In the event that a problem requires a number to be rounded to one significant figure, we follow the same steps as listed above. However if the first significant number is in the tens place or greater, the numbers to the right of the first significant figure will be turned into zeros instead of being truncated. This will result in a whole number. For example, if the number 395.094 is rounded to one significant figure, the result will be 400.

If the first significant number is located to the right of the decimal place, follow the above steps as listed. For example, if the number 0.00674 is rounded to one significant figure, the result will be 0.007.

## Using Rounding in Multiplication and Division:

So far, we have introduced two different methods of rounding. It’s now time to use these rounding methods in practice by conducting some calculations.

**Multiplication: **Consider the following problem: Estimate 72.34 x 0.52.

**Rounding: **Round 72.34 to the nearest whole number and 0.52 to the nearest tenth to estimate the product.

When we round 72.34 to the nearest whole number, we get 72. And when we round 0.52 to the nearest tenth, we get 0.5.

Then 72 x 0.5 = 36.

**Significant Figures: **Now consider the same problem as above, but this time estimate the product to one significant figure.

When we round 72.34 to one significant figure, we get 70. And when we round 0.52 to one significant figure, we get 0.5.

Then 70 x 0.5 = 35.

Now try the same problem, but this time estimate the product to two significant figures.

When we round 72.34 to two significant figures, we get 72. And when we round 0.52 to two significant figures, we get 0.52.

Then 72 x 0.53 = 38. (Remember: The answer cannot have more than two significant figures.) The exact answer for the problem is 37.62 Which method was closest to the exact answer?

**Division: **Consider the following problem: Estimate 54.5÷5.2.

**Rounding: **Round to the nearest whole number to estimate the quotient.

When we round 54.5 to the nearest whole number, we get 55. And when we round 5.2 to the nearest whole number, we get 5.

Then 55 ÷ 5 = 11.

**Significant Figures: **Now try the same problem, but this time estimate the product to two significant figures.

When we round 54.5 to two significant figures, we get 55. And when we round 5.2 to two significant figures, we get 5.2.

Then 55 ÷ 5.2 = 10.57

Remember: The answer cannot have more than two significant figures, so we must round the answer to two significant figures. Then,10.57 will round to 11.

The exact answer for the problem is 10.48. Which method was closest to the exact answer?