## Scientific Notation

Subject: Compulsory Maths

#### Overview

Scientific notation is, essentially, a method for writing really big or really small numbers. It is called scientific notation because these huge numbers are often found in scientific work. For example, you might have the number 6000000000000. That’s really big, right? Unfortunately, it isn’t easy to tell exactly how big at first glance with all those zeroes stuck on the end. Instead, the number could be written as 6 * 1000000000000. Then you can change it.

##### Scientific Notation

Scientific Notation is used to handle very large or very small numbers. Scientists have developed a shorter method to express very large numbers. It is written as the product of a number (integer and decimal) and a power of 10.

Here are some examples of scientific notation.

 10,000 = 1$\times$104 34,567 = 3.4567$\times$ 104 1000 = 1$\times$103 495 = 4.95 $\times$102 10 = 1$\times$101 98 = 9.8 $\times$101(not usually done) $\frac{1}{10}$=1$\times$10-1 0.23 = 2.3 $\times$10-1(not usually done) $\frac{1}{100}$= 1$\times$10-2 0.026 = 2.6$\times$10-2 $\frac{1}{1000}$ = 1$\times$10-3 0.0064 =6.4$\times$10-3 $\frac{1}{10,000}$ = 1$\times$10-4 0.00088 = 8.8$\times$10-4

Example:

0.000457

= $\frac{0.000457\times 1000000}{1000000}$

=$\frac{457}{10^6}$

=457$\times$ 10-6

=4.57$\times$102$\times$10-6

= 4.57$\times$10-4

##### Things to remember
• Scientific notation is the way that scientists easily handle very large numbers or very small numbers.
• Scientific notation can be used to turn 0.0000053 into 5.3 x 10-6
• A power of ten with  a positive exponent, such as 105, means the decimal was moved to the left.
• A power of ten with a negative exponent, such as 10-5, means decimal was moved to the right.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

324000000

= 324 × 106

= 3.24 × 102× 106

= 3.24 × 102+6

= 3.24 × 108

Solution:

45000000

= 45×106

= 4.5×101×106

= 4.5×101+6

= 4.5×107

Solution:

0.00671

= $\frac{6.71}{100000}$

= $\frac{6.71×100}{10^5}$

= 6.71×102-5

=6.71×10-3

Solution:

0.0000435

= $\frac{0.0000435×10000000}{10000000}$

= $\frac{435}{10^7}$

Solution:

0.0000435

= $\frac{0.0000435×10000000}{10000000}$

= $\frac{435}{10^7}$

Solution:

5.7× 101×-4-1

= 57× 10-4-1

= 57× 10-5

= $\frac{57}{10^5}$

= $\frac{57}{100000}$

= 0.00057

Solution:

2.6×1000 + 3.4×100

=2600 +340

= 2940

= 294×101

= 2.94×102×101

= 2.94× 103

Solution:

8.3×100 - 2.4×100

= 830-240

= 590

= 59×101

= 5.9×102

Solution:

72,000,000,000,000,000,000,000 kg.

= 72 × 1018kg

= 7.2×101×1018 kg

= 7.2×1019 kg

Solution:

3.2 × 104 + 2.3 × 104

= ( 3.2 + 2.3 )× 104

= 5.5× 104

Solution:

9.70 × 106 + 8.3× 105

= 9.70× 106 + 0.83× 106

= (9.70 + 0.83)× 106

= 10.53×106

= $\frac{10.53}{10}$× 10× 106

= 1.053×101+6

= 1.053×107

Solution:

1.2× 105

= $\frac{1.2}{10}$ × 10× 105

= 0.12× 106

Now,

( 1.2× 105) + (5.35× 106 )

= ( 0.12× 106) + (5.35× 106)

= (0.12 + 5.35)× 106

= 5.47× 106

Solution:

7.00× 10-6

= $\frac{7.00}{10}$× 101×10-6

= 0.70× 101-6

= 0.70× 10-5

Now,

8.41× 10-5 - 0.70× 10-5

= 8.41× 10-5 - 0.70× 10-5

= (8.41 - 0.70)× 10-5

= 7.71× 10-5

Solution:

$\frac{1.20× 10^{-8}}{3.0×10^{-3}}$

= $\frac{1.20}{3}$$\times$10-8 + 3

= 0.40$\times$10-5

= $\frac{40}{100}$$\times$ 10-5

= $\frac {4.0 \times 10^1 \times 10^{-5}}{10^2}$

=$\frac{4.0\times10^{1-5}}{10^2}$

= 4.0 $\times$10-4-2

= 4.0$\times$10-6

Solution:

2.4× 10-3

= $\frac{2.4}{10}$× 10× 10-3 = 0.24× 10-2

Now,

6.91× 10-2 + 0.24× 10-3

= 6.91× 10-2 + 0.24× 10-2

=(6.91 + 0.24 )× 10-2

= 7.15× 10-2

Solution:

4.9 $\times$104

= $\frac{4.9}{10}$$\times$101$\times$104

= 0.49 $\times$101+4

= 0.49$\times$105

Now,

1.33$\times$105 - 4.9 $\times$104

= 1.33 $\times$105 - 0.49$\times$105

= (1.33 - 0.49) $\times$105

= 0.84 $\times$105

= $\frac{8.4}{10}$$\times$105

= 8.4$\times$104