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Rounding

- Calculus
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When dealing with numbers we can sometimes end up with numbers that are very precise and contain a lot of digits. Some values such as pi even have infinitely many digits! High levels of precision are not always necessary when representing the numbers in written form and we may wish to write the number in an approximation that is shorter and takes less physical space. **Rounding** is a technique that allows us to do this.

Rounding is the process of replacing a number with an approximated version of it that is shorter.

For example, 24.7 can be rounded to 25. When a number is rounded, some accuracy is lost as a rounded value of a number is not the same as the true value of the number.

Numbers can be rounded to the nearest digit, 10, 100 and so on. To round a number to the nearest 10:

Locate the digit that represents the number of 10's in the number. For example, in the number 468 the number 6 gives the number of 10's.

Look at the digit to the right of this digit. If the digit is 5 or above, add 1 to the 10's digit. If the digit is 4 or below, keep the 10's digit the same. If adding 1 to the 10's digit causes it to become 10, change it to 0 and add 1 to the digit before the 10's digit.

Change all digits to the right of the 10's digit to 0. If the number has a decimal point, remove the decimal point and any digits after it.

This process can be repeated with any digit. For example, to round to the nearest 100, simply follow the above steps but with the digit that represents the number of 100's in this number.

**Round 6647 to the nearest 10.**

**Solution:**

We begin by locating the digit that represents the number of 10's. In the case of 6647, this number is 4. Next, we look at the digit to the right of the 4, which is 7. The number 7 is greater than or equal to 5 so we add 1 to the 10's digit. This means that the 10's digit now becomes 1 more than 4 which is 5. Lastly, we change every digit to the right of the 10's column to 0, giving us our answer:

6650

**Round 7593 to the nearest 100.**

**Solution:**

We begin by locating the digit that represents the number of 100's. In the case of 7593, this number is 5. Next, we look at the digit to the right of the 5, which is 9. The number 9 is greater than or equal to 5 so we add 1 to the 100's digit. This means that the 100's digit now becomes 1 more than 5 which is 6. Lastly, we change every digit to the right of the 100's column to 0, giving us our answer:

7600

**Round 43491 to the nearest 1000.**

**Solution:**

We begin by locating the digit that represents the number of 1000's. In the case of 43491, this number is 3. Next, we look at the digit to the right of the 3, which is 4. The number 4 is less than 5 so we keep the 1000's digit the same. Lastly, we change every digit to the right of the 1000's column to 0, giving us our answer:

43000

Significant figures are the digits that are significant in indicating a numerical value.

47891 rounded to 3 significant figures is 47900. As we can see, there are three digits that indicate the value of the number, and the extra precision has been lost due to rounding.

You may be asked to round a number to x significant figures. How can this be done?

To round a number to x significant figures:

Starting from the first non-zero digit in the number, count x digits to the right. For example, if we had the number 1234 and we wanted to round the number to 2 significant figures, the first digit (1) would be our first significant figure, and the second digit (2) would be the second significant figure. We have counted 2 significant figures so we stop at the 2

^{nd}non-zero digit.Look at the digit to the right of the digit that you stopped at. If this digit is 5 or more, add 1 to the digit you stopped at. If it is 4 or less, keep the digit the same. If adding 1 to the last significant digit causes it to go over 9 (no longer a digit), change it to 0 and add 1 to the digit before the last significant digit.

Change all digits to the right of the last significant digit to 0. If there is a decimal point to the right of the last significant digit, remove it along with every digit after it.

Another way of rounding a number is to round it to a certain amount of decimal places. Decimal places are digits that come after the decimal point of a number. For example, 76.443 rounded to 2 decimal places would be 76.44 as there are 2 digits that come after the decimal point.

To round a number to x decimal places:

Count x digits to the right of the decimal point in the number. For example, if we were rounding 12.67 to 1 decimal place we would count 1 digit to the right of the decimal point, so we would stop at the 6.

Look at the digit to the right of the digit that you stopped at. If this digit is 5 or more, add 1 to the digit you stopped at. If it is 4 or less, keep the digit the same. If adding 1 to the last digit causes it to go over 9, change it to 0 and add 1 to the digit before.

Remove all digits to the right of the digit you stopped at.

**Round the number 4789.4 to 3 significant digits.**

**Solution:**

First, we count the first 3 digit from the left of the number. This takes us to the 3^{rd} digit, 8. Next, we look at the number to the right of this digit, 9. 9 is greater than or equal to 5, so we add 1 to the 3^{rd} digit. Adding 1 to 8 gives us 9. Lastly, we change all digits to the right of the 3^{rd} digit to 0 and remove the decimal point and every digit after it. This gives us our answer:

4790.

**Round the number 76.4894 to 3 d.p.**

**Solution:**

The abbreviation d.p refers to decimal places. Counting 3 digits from the decimal points, we end up at the digit with the value 9. The digit to the right of this digit is 4. This is less than 5 so we keep the 3^{rd} digit after the decimal point the same. Finally, we remove every digit to the right of the 3^{rd} digit after the decimal point, giving us our answer:

76.489

- Rounding is the process of replacing a number with an approximated version of it that is shorter.
- To round a number to the nearest 10:
- Locate the digit that represents the number of 10's in the number.
- Look at the digit to the right of this digit. If the digit is 5 or above, add 1 to the 10's digit. If the digit is 4 or below, keep the 10's digit the same. If adding 1 to the 10's digit causes it to go over 9, change it to 0 and add 1 to the digit before the 10's digit.
- Change all digits to the right of the 10's digit to 0. If the number has a decimal point, remove the decimal point and any digits after it.

- To round a number to x significant figures:
- Starting from the first non-zero digit in the number, count x digits to the right.
- Look at the digit to the right of the digit that you stopped at. If this digit is 5 or more, add 1 to the digit you stopped at. If it is 4 or less, keep the digit the same. If adding 1 to the last significant digit causes it to go over 9, change it to 0 and add 1 to the digit before the last significant digit.
- Change all digits to the right of the last significant digit to 0. If there is a decimal point to the right of the last significant digit, remove it along with every digit after it.

- To round a number to x decimal places:
- Count x digits to the right of the decimal point in the number.
- Look at the digit to the right of the digit that you stopped at. If this digit is 5 or more, add 1 to the digit you stopped at. If it is 4 or less, keep the digit the same. If adding 1 to the last digit causes it to go over 9, change it to 0 and add 1 to the digit before.
- Remove all digits to the right of the digit you stopped at.

To round a number to x significant figures:

- Starting from the first non-zero digit in the number, count x digits to the right. For example, if we had the number 1234 and we wanted to round the number to 2 significant figures, the first digit (1) would be our first significant figure, and the second digit(2) would be the second significant figure. We have counted 2 significant figures so we stop at the 2nd non-zero digit.
- Look at the digit to the right of the digit that you stopped at. If this digit is 5 or more, add 1 to the digit you stopped at. If it is 4 or less, keep the digit the same. If adding 1 to the last significant digit causes it to become 10, change it to 0 and add 1 to the digit before the last significant digit.
- Change all digits to the right of the last significant digit to 0. If there is a decimal point to the right of the last significant digit, remove it along with every digit after it.

Rounding is the process of replacing a number with an approximated version of it that is shorter.

To round a number to x decimal places:

- Count x digits to the right of the decimal point in the number. For example, if we were rounding 12.67 to 1 decimal place we would count 1 digit to the right of the decimal point, so we would stop at the 6.
- Look at the digit to the right of the digit that you stopped at. If this digit is 5 or more, add 1 to the digit you stopped at. If it is 4 or less, keep the digit the same. If adding 1 to the last digit causes it to become 10, change it to 0 and add 1 to the digit before.
- Remove all digits to the right of the digit you stopped at.

Numbers can be rounded to the nearest digit, 10, 100 and so on. To round a number to the nearest 10:

- Locate the digit that represents the number of 10's in the number. For example, in the number 468 the number 6 gives the number of 10's.
- Look at the digit to the right of this digit. If the digit is 5 or above, add 1 to the 10's digit. If the digit is 4 or below, keep the 10's digit the same. If adding 1 to the 10's digit causes it to become 10, change it to 0 and add 1 to the digit before the 10's digit.
- Change all digits to the right of the 10's digit to 0. If the number has a decimal point, remove the decimal point and any digits after it.

More about Rounding

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