Scientific Notations

We can write very big or small numbers or remember them easily with the help of scientific notation of numbers.

Let us say, the speed of light in the air is approximately 30,00,00,000 m/sec.

To write this number as powers of 10,
  First write it as product of two numbers.
  30, 00, 00, 000 =3 × 10,00,00,000
  Count the number of zeros to write index of 10. 3 × 108 m/sec


Note:

  •  Scientific notation of numbers is written as powers of 10.
  •  From which first number is between 1 to 10 and the second is written as power of 10.

Numbers with decimal point


In order to write the given numbers with decimal point as a power of 10, Move the decimal point over to the right until the coefficient lies between 1 to 10.

Note:

  •  For each place shift the decimal in excess of the exponent will be lowered 1 power of 10.

Simple Example


 2.6544× 103

Explanation:

2654.4
= 2.6544 × 1000
= 2.6544× 103



 2.595 × 104

Explanation:

25,950
=2.595 × 10,000
= 2.595 × 104


Medium Example:


 3.23 × 106

Explanation:

3,230,000
= 3.23 × 1,000,000
= 3.23 × 106



 1.002 × 1010

Explanation:

10,020,000,000
=1.002 × 10,000,000,000
= 1.002 × 1010


Advance Example:


 5.1× 108 sq km

Explanation:

51 00,00,000 sq m
= 51× 1,00,00,000
= 51× 107
= 5.1× 10×107
= 5.1× 108 sq km


 5.1× 108 sq km

Explanation:

0. 0000000000027 m
= `27/100000000000`
= 27/ 1011
= 27× 10-11
=2.7× 10-10 m

 37500

Explanation:

Land in a village is about 3.75 × 10 4 acres
= 3.75 × 10 4
= 3.75 ×10×10×10×10
= 3.75 ×10000
= 37500


 3000 kg

Explanation:

= 0.3 × 102 ×100
= 0.3 × 104
= 0.3 × 10× 10 × 10× 10
= 3000 kg.


How to multiply Scientific notation


To multiply numbers given in Scientific notation multiply the numbers and add the exponents of 10.

Let us say,
(x×10a) × (y×10b)
  Multiply the numbers (x×y)
 Add the exponents of 10.
 So that, (x×10a) × (y×10b) = (x×y) × 10a+b

Simple Example


 (10) × 10a+b

Explanation:

(5×10a) × (2×10b)
Multiply the numbers and add the exponent of 10.
= (5×2) × 10a+b
= (10) × 10a+b


Medium Example:


 5.6× 108

Explanation:

(8×105) × (7×102)
Multiply the numbers and add the exponent of 10.
= (8×7) × 105+2
=(56) × 107
=5,60,00,0000
= 5.6× 108


How to divide Scientific notation


To divide numbers given in Scientific notation divide the numbers and subtract the exponents of 10.

Let us say,
(x×10a) ÷ (y×10b)
 Divide the numbers (x/y)
 Subtract the exponents of 10. 10a-b
 So that, (x×10a) ÷ (y×10b) = (x/y) × 10a-b

Simple Example


 2× 10a-b

Explanation:

(10×10a) ÷ (5×10b)
Divide the numbers and subtract the exponents of 10.
= `(10/5)` × 10a-b
= 2× 10a-b


Medium Example:


 15× 104

Explanation:

(75×108) ÷ (5×104)
= `(75/5)` × 108-4
= 15× 104


Complementary examples Scientific notation


Example:
 1.4× 103

Explanation:

(8×105) × (7×102) / (4×105)
= (8×7) × 105+2/ (4×105)
= (56× 107) ÷ 4×105
= 56/4 × 107-5
= 14× 102
= 1400
= 1.4× 103


Example:
 5× 102

Explanation :

Speed of light is 300,000,000 m/s. = 300,000 km/s= 3× 105
Distance between the sun and the earth is 15,00,00,000 km.= 15,00,00,000 km= 15 × 107
We have to find the time that sun's light to reach to the earth.
Now time= distance / speed
So that, if we put the values here for distance and speed we can find the time
Time= distance / speed
= 15× 107/ 3× 105
= 5× 107-5
= 5× 102






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