How to Round Off Number for Physics Caliculation

The process of omitting the non significant digits and retaining only the desired number of significant digits, incorporating the required modifications to the last significant digit is called rounding off the number.

In Physics calculation is a vital part and during that we shall reduce the number to the required extend and this is called rounding off and thus we are ready to accept the error up to some extend.

Rules for rounding off numbers:

1. The preceding digit is raised by 1 if the immediate insignificant digit to the dropped is more than 5.

Ex: 4728 is rounded off to three significant figures as 4730.

2. The preceding digit is to be left unchanged if the immediate insignificant digit to be dropped is less than 5.

Ex: 472 is rounded off to three significant figures as 472

3. If the immediate insignificant digit to be dropped is 5 then there will be two different cases

a) If the preceding digit is even, its is to be unchanged and 5 is dropped.

Ex: 4.7258 is to be rounded off to two decimal places. The digit to be dropped here is 5 (along with 8) and the preceding digit 2 is even and hence to be retained as two only
4.7258=4.72

b) If the preceding digit is odd, it is to be raised by 1

Ex: 4.7158 is to be rounded off to two decimal places. As the preceding digit 1 is odd, it is to be raised by 1 as 2.

4.7158=4.72

Rules for Arithmetic Operations with significant Figures:

1. In multiplication or division, the final result should retain only that many significant figures as are there in the original number with the least number of significant figures.

Ex: . But the result should be limited to the least number of significant digits-that is two digits only. So Final answer is 9.9.
2. In addition or substraction the final result should retain only that many decimal places as are there in the number with the least decimal places.

Ex: 2.2+4.08+3.12+6.38=15.78. Finally we should have only one decimal place and hence 15.78 is to be rounded off as 15.8.

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Errors and Significant Figures in Physics

Physics is a subject of measurements and errors do creep in while measuring physical quantities.Here we are going to learn about significant figures and how to apply them while measuring physical quantities.

Significant Figures :

A significant figure is defined as the figure, which is considered reasonably, trust worthy in number.

Eg: = 3.141592654 (upto 10 digits)
=3.14 (with 3 figures )
=3.1416 (upto 5 digits )

The significant figures indicate the extent to which the readings are reliable.

Rules for determining the number of significant figures:

1. All the non-zero digits in a given number are significant without any regard to the location of the decimal point if any.

Ex: 184,52 has five significant digits.

1845.2 or 184.52 all have the same number of significant digits that is 5.

2.All zeros occruing between two non zero digits are significant without any regard to the location of deciman point if any.

Ex: 106008 has six significant digits.

106.008 or 1.06008 has also got six sifnificant digits.

3.If the number is less than one, all the zeros to the right of the decimal point but to the first non-zero digit are not significant.

Ex: 0.000308 has 3 significant digits.

a) All zeros to the right of a demcimal point are significant if they are not followed by a non-zero digit.

Ex: 30.00 has 4 significant digits

b) All zeros to the right of the last non-zero digit after the decimal point are significant.

Ex: 0.05600 has 4 significant digits

4. All zeros to the right of the last non-zero digit in a number having no decimal point are not significant.

Ex: 2030 has 3 significant digits

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Errors and Approximation in Physics part two

This present post is in continuation with what are the errors in physics and here we are going to discuss further types of errors.

We do measure different physical quantities like velocity,acceleration,force and work done and the list goes on in physics during our experimentation.During their measurement errors do occur and we have to minimize them.

RANDOM ERRORS :

They are due to uncontrolled disturbances which influence the physical quantity and the instrument.

Ex-:The errors due to line voltage changes and backlash error.
Backlash errors are due to screw and nut.

GROSS ERRORS :

The cause for gross errors are improper recording, neglect of the sources of the error, reading the instrument incorrectly, sheer carelessness .

Ex: In a tangent galvanometer experiment, the coil is to be placed exactly in the magnetic meridian and care should be taken to see that no other magnetic material are present in the vicinity.

No correction can be applied to these gross errors.

When the errors are minimized, the accuracy increases.

The systematic errors can be estimated and observations can be corrected.

Random errors are compensating type. A physical quantity is measured number of times and these values lie on either side of mean value-with random errors. These errors are estimated by statistical methods and accuracy is achieved.

Personal errors like parallax error can be avoided by taking proper care.

The instrumental errors are avoided by calibrating the instrument with a standard value and by applying proper corrections.

True value and Errors

In the measurement of a physical quantity the arithmetic mean of all readings which is found to be very close to the most accurate reading is to be taken as True value of the quantities.

Absolute Errors :

The magnitude of the difference between the true value of the measured physical quantity and the value of individual measurement is called absolute error.

Absolute error is equal to |True value - measured values|

The absolute error is always positive.

The arithmetic mean of all the absolute errors is considered as the mean absolute error or final absolute error of the value of the physical concerned and it is also positive.

Relative error:

The relative error of a measured physical quantity is the ratio of the mean absolute error to the mean value of the quantity measured.

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Errors and Approximation in Physics

Physics is the study of nature and measurement of the physical quantities is vital part of its study. During the measurements of physical quantities errors do occur and here we are dealing with different kind of errors possible.

Accuracy and precision :

The numerical values obtained on measuring physical quantities depend upon the measuring instruments, methods of measurement.

A unit of measurement of a physical quantity is the standard reference of the same physical quantity which is used for comparison of the given physical quantity.

Accuracy refers to how closely a measured value agrees with the true values.

Precision refers to what limit or resolution the given physical quantity can be measured.

Types of Errors:

Uncertainty in measurement of a physical quantity is called the error in measurement.

The difference between the measured value and true value as per standard method without mistakes is called the error.

Error = True value Measured value
Correction = -error

True value means, standard value free of mistakes.

Errors are broadly classified into 3 types :

i) Systematic errors
ii) Random errors
iii) Gross errors

SYSTEMATIC ERRORS :

The errors due to a definite cause and which follow a particular rule are called systematic errors. They always occur in one direction.

Constant error :

Systematic errors with a constant magnitude are called constant errors. The constant arised due to imperfect design, zero error in the instrument or any other such defects. These are also called instrumental errors.

Zero error :

The error due to improper designing and construction.
Ex: If a screw gauge has a zero error of -4 head scale divisions, then every reading will be 0.004cm less than the true value.

Environmental Error:

The error arise due to external conditions like changes in environment, changes in temperature, pressure, humidity etc.
Ex: Due to rise in temperature a scale gets expanded and this results in error in measuring length.

Imperfection in Experimental technique or Procedure:

The error due to experimental arrangement, procedure followed and experimental techniques called Imperfection error.
Ex: In colorimetric experiments, the loss of heat due to radiation, the effect on weighing due to buoyancy of air cannot be avoided.

Personal errors or observational errors:

These errors are entirely due to personal peculiarities like individual bias, lack of proper settings of the apparatus, carelessness in taking observations.
Ex: Parallax error

You may be interested in having a look at units and dimensions concept here below

Units and Dimensions part one and two
Uses and limitations of dimensional formulas
List of dimensional formulas part one and two

Vectors and their usage in Physics

Parallelogram law,Triangle law and applications

Scalar product and vector product

Relative Velocity and Motion of a boat across a river