Scientific Notation: 1. Writing a Decimal Number in Scientific Notation 2. Converting Scientific Notation to Decimal Form 3. Arithmetic of Numbers in Scientific Notation

**Scientific Notation**

Suppose you
are told that the diameter of the Sun is 13,92,000 km and that of the Earth is 12,740
km, it would seem to be a daunting task to compare them. In contrast, if 13,92,000 is written as 1.392 ×10^{6}
and 12,740 as 1.274×10^{4},
one will feel comfortable. This sort
of representation is known as scientific notation.

Since [1.392 ⋅ 10^{6} ] / [1.274 ⋅ 10^{4 }] ^{= }14/13
10^{2} = 108

You can imagine
108 Earths could line up across the face of the sun.

Scientific
notation is a way of representing numbers that are too large or too small, to be
conveniently written in decimal form. It allows the numbers to be easily recorded
and handled.

** **

Here are
steps to help you to represent a number in scientific form:

(i)
Move the decimal point so that there is only one non-zero digit to its left.

(ii)
Count the number of digits between the old and new decimal point. This gives ‘*n*’,
the power of 10.

(iii)
If the decimal is shifted to the left, the exponent *n* is positive. If the
decimal is shifted to the right, the exponent *n* is negative.

Expressing a number *N* in the form of *N* = *a* ×10* ^{n}* where, 1 ≤

The following table of base 10 examples may make things clearer:

**Decimal notation** : Scientific notation

100 : 1×10^{2}

1,000 : 1×10^{3}

10,000 : 1×10^{4}

1,00,000 : 1×10^{5}

10,00,000 : 1×10^{6}

1,00,00,000 : 1×10^{7}

**Decimal notation **: Scientific notation

0.01 : 1 × 10^{-2}

0.001: 1 × 10^{-3}

0.0001: 1 × 10^{-4}

0.00001: 1 × 10^{-5}

0.000001: 1 × 10^{-6}

0.0000001: 1 × 10^{-7}

Let
us look into few more examples.

**Example 2.28**

Express
in scientific notation (i) 9768854 (ii) 0.04567891 (iii) 72006865.48

*Solution*

The
decimal point is to be moved six places to the left. Therefore *n* = 6.

The
decimal point is to be moved two places to the right. Therefore *n* = −2
.

The
decimal point is to be moved seven places to the left. Therefore *n* =
7 .

** **

The
reverse process of converting a number in scientific notation to the decimal form
is easily done when the following steps are followed:

(i)
Write the decimal number.

(ii)
Move the decimal point by the number of places specified by the power of 10, to
the right if positive, or to the left if negative. Add zeros if necessary.

(iii)
Rewrite the number in decimal form.

**Example 2.29**

Write
the following numbers in decimal form: (i) 6. 34×10^{4} (ii) 2. 00367 ×10^{−}^{5}

*Solution*

=
0.0000200367

(i) If the indices in the scientific notation of two numbers are
the same, addition (or subtraction) is easily performed.

**Example 2.30**

The
mass of the Earth is 5.97×10^{24} kg and that of the Moon is 0.073 ×10^{24}
kg. What is their total mass?

*Solution*

Total mass
= 5.97×10^{24} kg + 0.073 ×10^{24} kg

= (5.97
+ 0.073) ×10^{24} kg

= 6.043
×10^{24} kg

(ii) The product or quotient of numbers in scientific notation can
be easily done if we make use of the laws of radicals appropriately.

**Example 2.31**

Write the following in scientific notation :

(i)
(50000000)^{4} (ii) (0.00000005)^{3} (iii) (300000)^{3}
× (2000)^{4} (iv) (4000000)^{3} ÷ (0.00002)^{4}

*Solution*

(i) (50000000)^{4} = (5.
0×10^{7} )^{4}

= (5.0)^{4} ×(10^{7})^{4}

= 625.0
×10^{28}

= 6.25×10^{2}×10^{28}

= 6.25×10^{30}

(ii) (
0. 00000005)^{3} = (5. 0×10^{−}^{8} )^{3}

= (5.0)^{3} ×(10^{−}8 )^{3}

= (125.0 ) ×(10)^{−}^{24}

= 1.25×10^{2} ×10^{−}^{24}

= 1.25×10^{−}^{22}

(iii) ( 300000 )^{3} ×(2000)^{4}

= (3.0×10^{5} )^{3} ×(2.0×10^{3} )^{4}

= (3.0)^{3} ×(10^{5} )^{3} ×(2.0)^{4} ×(10^{3} )^{4}

= (27. 0 ) ×(10^{15} )×(16.
0)×(10^{12} )

= (2.7×10^{1})×(10^{15} )×(1.6×10^{1})×(10^{12} )

= 2.7×1.6×10^{1} ×10^{15}
×10^{1}
×10^{12}

= 4. 32 ×10^{1}^{+}^{15} ^{+}^{1}^{+}^{12} = 4 .32 ×10^{29}

(iv) ( 4000000 )^{3} ÷ ( 0. 00002)^{4}

= (4.0×10^{6} )^{3} ÷ (2.0×10^{−}^{5} )^{4}

= (4.0)^{3} ×(106 )^{3} ÷ (2.0)^{4} ×(10^{−}^{5})^{4}

= 64.0×10^{18} / 16.0×10^{−}^{20}

= 4×10^{18}
×10^{+}^{20}

=
4.0×10^{38}

1. Write two numbers in scientific notation whose product is 2. 83104.

2. Write two numbers in scientific notation whose quotient is 2.
83104.

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