Welcome to our Free Sig Fig Calculator with supporting tutorial which explains significant figures (significant digits), how rounding and zeroes affect significant digits, how to work with trailing zeroes and decimals when calculating significant figures and more. The detailed sig fig tutorial allows you to learn and understand the rules of significant figures (sig figures) and how to calculate sig figures manually using pen and paper. The supporting sig figures calculator allows you to check your answers. This sig fig calculator is very simple to use and designed for basic sig figures calculations and to establish a good understanding of the topic. For more advanced calculations, we provide a more detailed significant figures calculator with specific Physics formula for the computation of geometric shapes, velocity and other metrics.

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## What are Significant Figures used for?

Significant Figures eliminate error by allowing precision in measurement, in a 'human friendly' version, of a large number for the computation of precise formulas for Math, Physics, Chemistry, Engineering and other core disciplines which use mathematics. The use of sig figures is particularly useful when dealing with high number ratio calculations.

## Everything You Need To Know About Significant Figures

Believe it or not, scientific numbers are unusual and quite diverse compared to frequent mathematical numbers. If this sounds ambiguous to you right now, its nothing to worry about and will be clear by the time you have read this article and we have explained sig figures in full detail. Let's try to make this a little clearer. In general Maths, there is no difference between 4, 4.0 (1 zero after the decimal point), or even 4.00000 (5 zeroes after the decimal point). When we consider real-life situations, it makes a huge difference. Currency is a simple example. Ever wondered why is that so?

Well, apparently, when we deal with practical applications like Physics or advanced math equations, we must be precise. When using numbers in real-life, engineers and scientists need to be accurate. However, there are certain situations where it seems impossible to define the exact analysis. We need to convert massive inconsequential numbers to achieve a certain level of certainty. This is the situation when significant figures come into the play. For simplicity, sig figures are often shortened and referred to as "sig fig".

We often ignore the Sig Figs thinking that they make no sense, but we cannot ignore the fact that Sig Figs are extremely crucial to determine the accurate number of digits. This is the reason why the iCalculatorâ„¢ team decided to make an online Sig Fig calculator available for efficient calculations that helps users get significant figures for any number they want in just a few seconds. We also wanted to help clear some of the confusion surrounding significant figures and get "sig fig explained" in a clear way that will help your understanding of significant figures.

Let us start with the term "Significant Figures" itself. Read the full article to gain clarity on significant figures and most importantly, access instructions on how to use the sig fig calculator.

## What do we mean by Significant Figures?

All the numbers in a result that have some degree of accuracy (reliability) are said to be significant figures. In simple words, you can use sig figs to display how accurate a number is.

**For example:** the number 45.5 has 3 significant numbers, while 45.50 has 4 significant numbers as the last zero counts is a significant digit, the fourth significant number in this example (4 being the first significant number, 5 the second significant number, then the decimal, the 5 after the decimal is the third significant number and as mentioned, the trailing zero is the fourth significant number)..

## 5 Rules For Significant Figures

You can use our sig fig calculator to get significant figures in a number. However, before using a sig fig calculator, it is important to understand the mathematical rules that affect sig fig calculation. Let's learn the five rules of identifying whether a number is significant or not.

### 1. Non-zero numbers

All digits in a number that are non-zero are considered as significant.

**For example:** if we talk about 563.45, it has 5 digits, and all of them are non-zeros. Therefore, all the digits of this number are significant, and so, it has a total of 5 significant figures.

### 2. Zeros within a number

All zeros that appear between two non-zero digits are said to be significant.

**For example:** 200045 has a total of three non-zero digits and 3 zeros. But there are 3 zeros between 2 and 4, i.e., two non-zero digits. Therefore, these zeros will also be considered to be significant. And thus, this number has 6 significant figures (from rule 1 and rule 2). Similarly, 56.009 has 5 significant figures, including all the zeros between 6 and 9 as well.

### 3. Leading Zeros

No matter how many zeros appear before the first non-zero digit in a number, all those zeros are insignificant. Such zeros are generally used to represent the location of the decimal.

This can be a little misleading for some, particularly when considering decimal numbers and sig fig computations, let's explain this sig fig rule in more detail with an example. The number 0.004 has three zeros before its first non-zero digits, i.e., 4. Hence, according to Rule 3, this number has only one significant digit, and all the other digits are insignificant.

Similarly, 0.00074 has only two significant figures.

### 4. Zeros after the Decimal

If we talk about the significance of zeros in a number, then all those zeros that appear after the decimal are significant, the only condition is that those zeros must appear after the first non-zero digit.

**For example:** 0.6000 has 4 significant figures out of which the three zeros after the first non-zero digits are significant as well. However, 0.0056 has only two significant digits, and both of them are non-zeros. The reason is that in the first number, the zeros appear after the first non-zero digit and thus, they are considered significant. In the second number however, the zeros appear before the first non-zero digit, and hence, they are considered to be insignificant.

### 5. If zeros appear at the end of a number, but they are not after any decimal, then it is not necessary that they must be significant.

**For example:** 60,200 may have 3, 4, or 5 significant figures. 60,200 can also be written as:

- 6.02 x 104, in this case, it has 3 significant figures.
- 6.020 x 104, in this case, it has 4 significant figures.
- 6.0200 x 104, in this case, it has 5 significant figures.

Today, you will notice that most of the textbooks contain the standard rules for finding significant figures. Did you know that these rules did not exist 30 years back? Would you like to know what happened in the wonderful world of math that suddenly every textbook started mentioning significant figures? Of course I do we hear you scream "I love math and I love learning significant figures", okay, perhaps we are simply imagining that, the chances are you need to learn sig figs for your math homework and you may not be that thrilled about it. The good news is that we have a history of sig fig which should help your understanding and also help you impress your math teacher with your incredible newfound knowledge about significant figures. The history of significant figures spans a number of centuries and the majority of the great mathematicians, great thinkers and inspirational characters in the field of mathematics have explored significant figures within their work.

## The Evolution of Significant figures

If we take a look at Math and its progression over the last 300 years, from Newton to Millikan, there is a wealth of information from virtually every famous author in the field, they have all aimed to throw light on the importance of significant figures.

Some believe that sig fig's story is related to the discovery of calculators. However, the story of sig fig started much earlier that this in the 1700s. Prior to this time, any digit that was between 1 and 9 was considered to be significant. Trailing and leading zeros were not a part of significant figures at that time. People used them only as place holders so that they could spot the decimal. Sir Isaac Newton, a well-known physicist, used the concept of significant figures for the very first time to demonstrate some impressive features of multiplication.

Later in the 1800s, mathematics received a new explanation of significant figures. Silas Whitcomb Holman provided and explained a more precise definition of significant figures in his 1882 essay. He also mentioned referred to significant figures later in his textbook, "Discussion of the Precision of Measurements".

The focus on significant figures continued and numerous other scientists, mathematicians, and physicists have contributed towards the concept of sig figs beyond these two prior key moment in the history of significant figures.

## Practical use and application of Significant Figures

Now that you know a little more about the history of significant figures, there are few terminologies and factors that will help you handle significant figures in the most precise manner.

### a) What is an 'exact number' and how many significant figures are there in it?

Numbers that have "complete certainty" are said to be "exact numbers". There is no possibility of uncertainty in exact numbers.

**For example:** the number of students in a classroom will always be 12 or 34 or 50 for example. The number of students cannot be 12.23 or 50.16 as we simply cannot have parts of this unit, it is always 1. This example can be applied to a number of situations, there will always be 12 eggs in one dozen, there will always be 1000 metres in 1 kilometre and so on, these types of numbers that are absolute are said to be "exact numbers".

**Important!**

When the question arises how many significant figures are there in an exact number, the answer will always be "infinite". Yes, an exact number has infinite significant figures. We can write 4 as 4.0 or 4.00 or even 4.0000. Every time we add a zero after the decimal, the number of significant figures in 4 increases. This proves that 4 (four), being an exact number, has infinite significant figures.

### b) How many significant figures does 100 have?

By now, you most likely understood that 100 is an exact number and that the concept of significant figures cannot be applied to exact numbers. Thus, we can say that 100 has an infinite number of significant figures.

If we go into further depth, it all depends on you how you would like to use it. If you want to use only 3 sig figs in 100, write it as 100 only. You can even write it as 100.0 to measure 4 significant figures, or 100.0000000 to measure 10 significant figures. As we mentioned earlier, it all depends on you how would you like to handle the number!

### c) How to multiply significant figures?

When multiplying two significant figures, the rule for rounding the result is similar to that of addition or subtraction. Just the operations are different.

**For example:** if we multiply 4.0 (2 significant figures) and 14.20 (4 significant figures), the result will be 56.800. Now, we have to round this result in such a way that the number of significant figures in the result would be equal to that of having the least sig figs i.e, 4.0. Hence, the rounded off result will be 57 (2 significant figures).

Now that we have gathered enough information on significant figures, let us throw some light on the calculation of sig figs.

### d) How do you calculate significant figures?

Here are some of the rules that will help you to determine the number of significant figures in a given number.

#### Rules for the numbers without a decimal.

- First of all, find out the first non-zero digit in the number and start counting from that digit only.
- Then you have to stop counting for significant figures on the last digit that is a non-zero number.
- Always remember that a non-zero digit is always considered to be as significant.
- All the zeros that occur between two non-zero digits are always considered as significant. However, zeros appearing at the other places are insignificant.

#### Rules for the decimal numbers.

- Once again, you need to find out the first non-zero digit and start counting from that digit only. It does not matter whether the first non-zero digit is appearing after the decimal or before the decimal. Everything before the first non-zero digit is considered to be insignificant.
- Stop counting at the very last digit of the number for significant figures. It does not matter whether the last digit after the decimal is a zero or a non-zero; it is always considered as significant.
- Any non-zero digit in a decimal number is always considered to be significant.

### e) How do you round to 3 significant figures?

Before rounding any number, it is important to first understand the concept of rounding significant figures. While rounding any number, we generally drop a specific number of digits from the end of the actual number.

**For example:** if we want to round a 5 digit number to 3 significant figures, then we will drop the last 2 digits and round off the last digit of the remaining number.

For a better understanding, let's look at an example of how we can round off a 4 digit number to 3 significant figures.

If we want to round off 234.7 to 3 significant figures, we have to follow the above steps:

- We want to round the number up to 3 significant figures, so we have to drop the remaining digits. Luckily, here we have only one digit left after the first three digits, which is 7.
- Now, the rule says that if the digit to be dropped is greater than 5 then we increase the last remaining digit by 1. Since 7 is greater than 5 so, we will be dropping it and increasing the last remaining digit, i.e., 4 (in 234.7) by 1.
- Thus, the number is rounded off to 3 significant figures, and the result obtained is 235.

The following are few other rules that are useful when handling and calculating significant figures.

### Rules for adding and subtracting significant figures

While performing different types of calculations related to the significant figures, it is important to ensure that the results generated after the calculation is accurate and reflect the number and units appropriately.

When calculating addition and subtraction of significant figures, we have to round off the result obtained. Generally, the last digits of the result are rounded off (the numbers that occur at the farthest position to the right).

**For example:** if we add 200 (3 significant figures) and 34.765 (5 significant figures), the result will be 234.765. This number would be rounded to 235, which will have 3 significant figures.

**Important!**

When two numbers of different significant figures are added or subtracted, the result must have the number of significant figures similar to the one having the least decimal places.

## Why do we study significant figures?

It goes without saying that anything you study has some practical application in your life or is at least supposed to have a significance in the real world. Sig figs are studied mostly due to their importance in determining the accuracy of an answer. In the field of science and technology, no device can function effectively without a 100% accurate answer. How do scientists define how precise a result is? The answer is hopefully obvious now after reading this sig fig guide, they use significant figures.

Now if you are wondering where exactly do we use significant figures in science or any other technology field, the answer is: sig figs are the basics of any real-life calculation. Moreover, from Maths to Chemistry, and from Physics to other higher branches, sig figs are used everywhere.

Let's look at an example so that you understand where sig figs are used in a specific branch of science, say Physics.

In physics, we generally deal with concepts including force, motion, acceleration, etc. If someone gives you an object and asks you to find out the force on that object due to gravity, what would you do? First, you need to find out the mass of that object. Then you would multiply the obtained mass with the value of "g," i.e., 9.81. The final answer would be the force acting on that body.

Let us consider that you completed some measurements and found out that the mass of an object is 25 kg. Then you multiplied the mass with 9.81, and the resulting force came out to be 245.25 N. Did you consider if there was a slight mistake while taking the measurements? It is possible that the actual mass of the object was 25.5 kg or there an error occurred while calculating the mass. So, now, the resulting force changes to 250.155 N. This difference could create a lot of misunderstanding or lead to further calculations being distorted as they would be based on an incorrect calculation. This is the time when you will need the concept of significant figures the most. Using sig figs while performing similar calculations can eliminate the chances of these types of mathematical error. The importance of precision in math calculations, cannot be overstated, you may also find the Exponents Calculator a useful tool for consideration in significant figure calculations.

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