# Scientific notation and significant figures

Scientific notation and significant figures are two important terms in physics. In scientific notation, numbers are expressed by some power of ten multiplied by a number between 1 and 10, while significant figures are accurately known digits and the first doubtful digit in any measurement.

## Scientific notation definition

In scientific notation, a number is expressed as some power of ten multiplied by a number between 1 and 10.

A simple and scientific method to write small or large numbers is to express them in some power of 10. The distance of the moon from Earth is 384000000 meters. This is a large number, it can also be expressed as 3.84×10 ^{8} m. This way of expressing the number is in scientific notation.

### Why do we use scientific notation?

We use it as it saves writing down large numbers of zeroes.

### Scientific notation examples

- The number 15000000 km can be written as 1.5 ×10
^{11}m. - 0.00000548 S can be written as 5.48 ×10
^{-6}s. - Mass of Earth is written as 6×10
^{24 kg.} - The radius of the earth is written as 6.4 ×10
^{4}m.

### Decimal to scientific notation

- 0.2

Solution: 0.2 = 2 × 10¹

- 0.006

Solution: 0.006=6 × 10^{3}

- 0.00063

Solution: 6 × 10 ^{4}

- 0.00000678

Solution: 0.00000678 = 6.78 ×10 ^{8}

### The scientific notation to decimal

- 3 ×10
^{-1}

Solution:3 ×10^{-1} =0.3

- 6 × 10-
^{4}

Solution: 0.0006

### Scientific notation to standard notation

In standard notation, there is only one non-zero number on the left side of the zero. Below examples has been given:

- 1168 ×10
^{-27}

Solution: 1168 ×10 ^{-27} =1.168 ×10 ^{-24}

- 725 ×10
^{-5}

Solution: 725 ×10 ^{-5 }

=7.25 ×10 ^{-3}

## What is a significant figure?

A significant figure is one, which is known to be reasonable and reliable.

or

All the accurately known digits and the first doubtful digit in an expression are called significant figures.

Physics is based on measurements. But unfortunately when a physical quantity is measured, then there is inevitably some uncertainty about its found value. This uncertainty may be due to a number of reasons.

One reason is the type of instrument being used. We know that every measuring instrument is calibrated to a certain smallest division and this fact puts a limit to the degree of accuracy which may be achieved while measuring with it.

Suppose we want to measure the length of a straight line with the help of a rod calibrated in millimeters. Let the endpoint of the line lies between 10.3 and 10.4 cm marks. By convention, if the end of the line does not touch or cross the midpoint of the smallest division, the reading is confined to the previous division.

In case the end of the line seems to be touching or has crossed the midpoint, the reading is extended to the next division.

By applying the above rule the position of the edge of a line recorded as 12.7 cm with the help of a meter rod calibrated in millimeters may lie between 12.65 cm and 12.75 cm. Thus in this example, the maximum uncertainty is ±0.05 cm. It is, in fact, equivalent to an uncertainty of 0.1 cm equal to the least count of the instrument divided into two parts, half above and half below the recorded reading.

The uncertainty or accuracy in the value of a measured quantity can be indicated conveniently by using significant figures. The recorded value of the length of the straight line I.e., 12.7 cm contains three digits (1,2,7) out of which two digits (1 and 2) are accurately known while the third digit (7) is a doubtful one. As a rule:

“In any measurement, the accurately known digits and the first doubtful digit are called significant figures.”

In other words, a significant figure is the one that is known to be reasonably reliable. If the above-mentioned measurement is taken by a better measuring instrument which is exactly up to a hundredth of a centimeter, it would have been recorded as 12.70 cm. In this case, the number of significant figures is four.

Thus, we can say that as we improve the quality of our measuring instrument and techniques, we extend the measured result to more and more significant figures and correspondingly improve the experimental accuracy of the result. While calculating a result from the measurements, it is important to give due attention to significant figures and we must know the following rules in deciding how many significant figures are to be retained in the final result.

### Significant figures rules

- Digits other than zero are always significant.
- Zeroes between significant digits are also significant.
- Zero on the left of the significant figures is not significant.
- Zero on the right of the significant figure is not significant.
- Zero on the right of a fractional number is significant

### Significant figures examples

Significant figures examples according to the rules which are mentioned above are:

- In 15.2 significant figures are 3.
- In 203 significant figures are 3.
- In 2100 significant figures are 2
- In 21.00 significant figures are 4.
- In 0.002 significant figures are 1.
- In 10.40 significant figures are 4.

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