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# Order of Magnitude

In order to compare the measured values of physical quantities, we make use of the idea of the order of magnitude.

The power, when the magnitude of a physical quantity is expressed in terms of the nearest power of ten, is called the order of magnitude. Instead of knowing the actual value of a physical quantity, in many cases, it is sufficient to know only the order of magnitude of that quantity. Order-of-magnitude helps us to write numbers that are too large or too small in a standard or convenient form and are commonly used by scientists, mathematicians, engineers etc.

## How do you calculate the order of magnitude?

The order of magnitude of a numerical value N is 𝑥, such that Large begin{aligned} 0.5 < frac{N}{10^{x}}leq 5 end{aligned}

That is, in order to find the order of magnitude of a quantity, it is first expressed as the power of 10 with the numerical part or coefficient should be greater than 0.5 and less than or equal to 5. Then the power of 10 is usually called as the order of magnitude of that quantity. It can be any integer.

The order of magnitude of all the numerical numbers can be calculated in the same way as given below.

• The number 72 can be expressed as large begin{aligned} 0.5< frac{72}{10^{2}}leq 5 end{aligned}. That is, it becomes 0.5 < 0.72 ≤ 5. Therefore, the order-of-magnitude of the numerical number 72 is 2.
• Similarly,  the number 5234 can be expressed as large begin{aligned} 0.5< frac{5234}{10^{4}}leq 5 end{aligned}. That is, it becomes 0.5 < 0.5234 ≤ 5. Therefore, the order of magnitude of the numerical number 5234 is 4.
• Consider the number 4234. It can be expressed as large begin{aligned} 0.5< frac{4234}{10^{3}}leq 5 end{aligned}. That is, it becomes 0.5 < 4.234 ≤ 5. Therefore, the order-of-magnitude of the numerical number 4234 is 3.
• Consider the number 0.35. It can be expressed as large begin{aligned}0.5< frac{0.35}{10^{-1}}leq 5 end{aligned}. That is, it becomes 0.5 < 3.5 ≤ 5. Therefore, the order-of-magnitude of the numerical number 0.35 is -1.
• The number 0.74 can be expressed as largebegin{aligned} 0.5< frac{0.74}{10^{0}}leq 5end{aligned}. That is, it becomes 0.5 < 0.74 ≤ 5. Therefore, the order of magnitude of the numerical number 0.74 is 0.
• Consider the number 0.95. It can be expressed as largebegin{aligned} 0.5< frac{0.95}{10^{0}}leq 5 end{aligned}. That is, it becomes 0.5 < 0.95 ≤ 5. Therefore, the order-of-magnitude of the numerical number 0.95 is 0.
• The number 0.035 can be expressed as largebegin{aligned} 0.5< frac{0.035}{10^{-2}}leq 5 end{aligned}. That is, it becomes 0.5 < 3.5 ≤ 5. Therefore, the order-of-magnitude of the numerical number 0.035 is -2.
• Consider the number 0.065. It can be expressed as large begin{aligned} 0.5< frac{0.065}{10^{-1}}leq 5end{aligned}. That is, it becomes 0.5 < 0.65 ≤ 5. Therefore, the order-of-magnitude of the numerical number 0.035 is -1.

Similarly, we can write the order-of-magnitude for all numbers. The order of magnitudes of some numerical numbers is given below.

 Number Expressed in nearest power of 10 Order of magnitude 1 100 zero 6 0.6 × 101 1 28 2.8  × 101 1 46 4.6  × 101 1 63 0.63 × 102 2 94 0.94 × 102 2 289 2.89 × 102 2 431 4.31 × 102 2 541 0.541 × 103 3 832 0.832 × 103 3 2348 2.348 × 103 3 6859 0.6859 × 104 4 0.03 3 × 10-2 -2 0.06 0.6 × 10-1 -1 0.00047 4.7 × 10-4 -4

Table 1: Order-of-magnitude of some numerical numbers

## Order of magnitude of length

The sizes of the objects, we come across in the universe, may vary over a wide range of the order of 10-14 m of the tiny nucleus of an atom to the size of the range of 1026 m of the extent of the observable universe. The size and order of lengths of some of these objects are given in the below table 2.

 Sl.No Size of object or distance measured Order of length (m) 1. The radius of a proton 10-15 2. Size of the atomic nucleus 10-14 3. Size of the hydrogen atom 10-10 4. Size of poliomyelitis virus 10-8 5. Wavelength of light 10-7 6. Size of red blood corpuscle 10-5 7. Thickness of paper 10-4 8. The height of a person 100 9. The height of Mount Everest above sea level 104 10. The radius of the moon 106 11. Radius of earth 107 12. Mean distance of the moon from earth 108 13. Radius of sun 109 14. Mean distance of the sun from earth 1011 15. The distance of Pluto from the sun 1013 16. The distance of our galaxy (Milky Way) 1020 17. The distance of nearest galaxy(Andromeda) to Milky Way 1022 18. Distance to the boundary of the observable universe 1026

Table 2: Range and order of lengths

## Order of magnitude of mass

Mass is a basic property of matter and its SI unit is kilogram. The mass of a body does not depend on the temperature, pressure or location of the object in space. The masses of the objects, we come across in the universe, may vary over a very wide range of the order of a tiny mass of 10-30 kg of an electron to the huge mass of about 1055 kg of the known universe. The range and order of the typical masses of various objects are given in the below table 3.

 Sl.No Object Mass (kg) 1. Electron 10-30 2. Proton 10-27 3. Uranium atom 10-25 4. Red blood cell 10-13 5. Dust particle 10-9 6. Raindrop 10-6 7. Mosquito 10-5 8. Grape 10-3 9. Human 102 10. Automobile 103 11. Moon 1023 12. Earth 1025 13. Sun 1030 14. Milky way galaxy 1041 15. Observable universe 1055

Table 3: Range and order of masses

## Order of time intervals

To measure any time interval we need a clock. We now use an atomic standard of time, which is based on the periodic vibrations produced in a caesium atom. This is the basis of the caesium clock, sometimes called an atomic clock. The time interval of events that we come across in the universe varies over a very wide range. The range and order of some typical time intervals are given in the below table 4.

 Sl.No Event Time interval (s) 1. Life – span of most unstable particle 10-24 2. The time required for light to cross a nuclear distance 10-22 3. Period of X – rays 10-19 4. Period of atomic vibrations 10-15 5. Period of the light wave 10-15 6. The lifetime of an excited state of an atom 10-8 7. Period of radio wave 10-6 8. Period of the sound wave 10-3 9. Wink of eye 10-1 10. The time between successive human heart beats 100 11. Travel time for light from the moon to the Earth 100 12. Travel time for light from the sun to the Earth 102 13. The time period of a satellite 104 14. The rotation period of the Earth 105 15. Rotation and revolution periods of the moon 106 16. Revolution period of the Earth 107 17. Travel time for light from the nearest star 108 18. Average human life – span 109 19. Age of Egyptian pyramids 1011 20. Time since dinosaurs became extinct 1015 21. Age of the universe 1017

Table 4: Range and order of time intervals

## Solved Examples

1. Write the order of magnitude of the given numbers from smallest to biggest:
0.7     0.004     0.04     1      3.54      0.1      11.5      0.4      0.008     57     0.00089      0.05

 Number Expressed in nearest power of 10 Order of magnitude 0.7 0.7 × 100 0 0.004 4 × 10-3 -3 0.04 4 × 10-2 -2 1 1 × 100 0 3.54 3.54 × 100 0 0.1 1 × 10-1 -1 11.5 1.15 × 101 1 0.4 4 × 10-1 -1 0.008 0.8 × 10-2 -2 57 0.57 × 102 2 0.00089 0.89 × 10-3 -3 0.05 5 × 10-2 -2

The order-of-magnitude of numbers can be written from smallest to biggest as

0.00089     0.004     0.008     0.04     0.05     0.1     0.4      0.7       1      3.54        11.5      57

2. What will be the order of magnitude of 105+103

Ans:

The resultant order-of-magnitude will be the highest order which is added, i.e, here the resultant order of magnitude will be 5.

Let’s verify,

`105+103 = 100000 + 1000               = 101000               = 1.01 × 105`
i.e., the order of magnitude is 5.

3. Find the order of magnitude of the sum of 586482 and 745899.

Ans:

`586482 + 745899 =  1332381                              =   1.332381 × 106`
i.e., the order of magnitude is 6.

4. Find the order of magnitude of the product of 2456 and 127.

Ans:

`24563 × 127 =  3119501                      =  3.119501 × 106`
i.e., the order of magnitude is 6.

5. Find the order of magnitude of the acceleration due to gravity.

We know, acceleration due to gravity, g = 9.8 m/s2
i.e., g can be expressed as 0.98 × 101 m/s2
Therefore, order of magnitude of acceleration due to gravity,  g = 1

6. Find the order of magnitude of the Gravitational constant.

We know, Gravitational constant, G = 6.673 x 10-11 Nm2kg-2
i.e., G can be expressed as 0.6673 x 10-10 Nm2kg-2
Therefore, the order of magnitude of gravitational constant,  G = –10

7. Find the order of magnitude of the Faraday’s constant.

We know, Faraday’s constant, F = 9.648 x 104 C/mol
i.e., F can be expressed as 0.9648 x 105 C/mol
Therefore, the order of magnitude of Faraday constant,  F = 5

8. A school is having 6314 students. What is the order of magnitude of the number of students?