Answer

3.66

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## Table of Contents

- Significant figures
- Directions for use
- Rounding significant figures
- Rounding numbers algorithm
- Rounding decimals

- Calculation example

This calculator rounds the given number to the necessary quantity of significant figures, replacing the “leftover numbers” with zeros. For example, rounding 11 to one significant figure will give 10 as an answer.

## Significant figures

Significant figures in a numerical value represent the digits that carry meaning contributing to its precision. This includes all non-zero digits, any zeroes between non-zero digits, and trailing zeros in a decimal number. For example, in 103.00, all five digits are significant: the '1' and '3' as non-zero digits, the '0's as they are between non-zero digits, and the last '0' because it's a trailing zero in a decimal number. Leading zeros, like those in 0.0025, are not significant as they only indicate the position of the decimal point.

The concept of significant figures is crucial in scientific, engineering, and mathematical calculations as it reflects the accuracy of measurements and computations. When performing calculations, maintaining the correct number of significant figures ensures that the precision of the results is neither artificially increased nor reduced. This principle is vital for expressing the reliability of data and for making meaningful comparisons between different measurements.

## Directions for use

To use this significant figures rounder, enter the given number and the necessary number of significant figures, then press “Calculate.”The given number can consist of up to 30 symbols. You can use number notation, scientific notation, or e-notation as input. You can also use commas to separate thousands, but it’s unnecessary. Some examples of accepted inputs:

- 150987
- 3,000,000
- 2.456e7
- -7.5 x 10^3

The number of significant figures should be less than 16, i.e., 15 is the largest number of significant figures this calculator can round to.

## Rounding significant figures

Let’s first define “rounding”. Rounding is the process of rewriting the number in a simpler form, while keeping its value close to the original value. For example, 1001 can be rounded to 1000. And 6.999999 can be rounded to 7. The resulting number is (slightly) less accurate than the original, but it’s much easier to pronounce and write it down.

Now, to significant figures. The number of significant figures is basically the number of figures you keep in a number. All other figures are turned into zeros.

## Rounding numbers algorithm

The process of rounding a number basically means finding a number with fewer digits whose value is close to the value of the original number. For example, it is intuitively clear that 6.1 will round down to 6, since it’s “closer” to 6 than to 7. Similarly, 6.2, 6.3, and 6.4 will all round down to 6. While 6.9 will round up to 7, since it’s closer to 7 than to 6. Same with 6.8, 6.7, and 6.6. But what do we do with 6.5? It’s exactly in the middle between 6 and 7. Several different rounding rules exist. Here we will discuss the most common method. In the most common rounding method, 5 is rounded “up,” so 6.5 is rounded up to 7. The algorithm for rounding numbers, in that case, consists of the following steps:

- Identify the number of significant figures you want to keep.
- Look at the last digit you are keeping. If the NEXT digit is smaller than 5, keep the last digit the same; if the next digit is greater than or equal to 5, increase the last significant digit by 1.

For example, round each number to two significant figures: 1015 and 876. Let’s start with 1015:

- We want to round to 2 significant figures, so the last figure we are keeping (and not turning to 0) is zero: 1015 – here, we keep the bold digits and turn the other ones to zero.
- Let’s look at the digit following the zero – it’s one. 1 is less than 5. Therefore, the last significant digit is kept the same. The number becomes \$1\bar{0}00\$. The horizontal line above the second digit indicates that this number is rounded to the second significant figure.

Now let’s look at 876:

- The last digit we keep is 7, and the second digit of the number is 876 – again, we keep the bold digits and turn the rest into zeros.
- The next digit after 7 is 6. 6 is greater than 5. Therefore, we have to add 1 to the last kept digit: 7 + 1 = 8. The final number will be \$8\bar{8}0\$. Also, here, the horizontal bar is added above the second digit to demonstrate that the number was rounded to the second significant figure.

### Rounding decimals

The algorithm for rounding decimals is the same as for rounding whole numbers. It is important to note that leading zeros are not significant numbers. Therefore, they are disregarded when choosing the last preserved digit. For example, round each number to three significant figures: 9.05675, 0.01234.

Starting with 9.05675, we get:

- We want to round to three significant figures, so the last digit we keep is 5: 9.05675, where we only keep the bold digits.
- Looking at the digit after 5, we see that it’s a 6. 6 is greater than 5. Therefore, the last significant digit has to be increased by 1: 5 + 1 = 6. The final number is 9.06000. Unlike in the case of whole numbers, the trailing zeros do not change the value of the final answer. Therefore, they can be deleted. The final answer is 9.06.

Now let’s look at 0.01234:

- We want to round to 3 significant figures. Therefore, the last digit we keep is 3. Note that the first zeros are not significant figures: 0.01234, where we only keep the bold digits.
- The digit after 3 is 4. 4 is smaller than 5. Therefore, the last digit does not change; the final number is 0.01230, or 0.0123.

## Calculation example

Imagine buying a dress in a store, which costs $15 + income tax. The income tax is 6.25%. Now you, of course, want to calculate the final price of the dress. To do that, you will first calculate the value of 6.25% as follows:

*6.25% of 15 = (15/100) × 6.25 = 0.15 × 6.25 = 0.9375*

Then you will calculate the final price of the dress:

*Final price = 15 + 0.9375 = 15.9375*

Since a hundredth of a dollar is the smallest unit we can use, we round up the resulting number to two digits after the decimal point.

In this case, rounding to hundredths is the same as rounding to 4 significant figures. (Note that you might need a different number of significant figures to round a different number to hundredths. For example, to round **5.63**25 to hundredths, you would use 3 significant figures, while to round **132.12**5 to hundredths, you would use 5 significant figures).

Rounding **15.93**75 to 4 significant figures, we get:

- The last digit we keep is 3: 15.9375.
- The digit after 3 is 7. 7 is greater than 5. Therefore, the last digit should increase by 1: 3 + 1 = 4. The rounded number will be 15.94.

This means that if you pay for the dress with 20 dollars, you will get $(20 - 15.94) = $4.06 as change.

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