Scientific Notation
See how to write and solve a number
in scientific notation.
8 examples and their solutions.
Scientific Notation
Definition
a × 10n
1 ≤ a < 10
n: integer
Scientific notation is a way to write a number1 ≤ a < 10
n: integer
that is too big or too small.
a: shows how the number looks like.
n: shows how big the number is
1.23 = 1.23 × 100
12.3 = 1.23 × 101
123 = 1.23 × 102
1230 = 1.23 × 103
Scientific Notation → Standard Notation
Example
310200
Solution 310200● - [1] [2]
↓
3●10200◌ - [3]
5 digits
310200 = 3.102 × 105 - [4]
↓
3●10200◌ - [3]
5 digits
310200 = 3.102 × 105 - [4]
[1]
First, find the a part:
numbers next to the consecutive 0s.
→ 3102
numbers next to the consecutive 0s.
→ 3102
[2]
Next, find the decimal point.
[3]
Move the decimal point
to make 3102
between 1 ≤ a < 10.
to make 3102
between 1 ≤ a < 10.
[4]
The decimal point moved 5 digits.
So the 10n part is either 105 or 10-5.
310200 is greater than 1 (= 100).
So the 10n part is 105.
So 310200 = 3.102 × 105.
So the 10n part is either 105 or 10-5.
310200 is greater than 1 (= 100).
So the 10n part is 105.
So 310200 = 3.102 × 105.
Close
Example
0.00509
Solution 0●00509 - [1]
↓
0◌005●09 - [2]
3 digits
0.00509 = 5.09 × 10-3 - [3]
↓
0◌005●09 - [2]
3 digits
0.00509 = 5.09 × 10-3 - [3]
[1]
First, find the a part:
numbers next to the consecutive 0s.
→ 509
numbers next to the consecutive 0s.
→ 509
[2]
Move the decimal point
to make 509
between 1 ≤ a < 10.
to make 509
between 1 ≤ a < 10.
[3]
The decimal point moved 3 digits.
So the 10n part is either 103 or 10-3.
0.00509 is less than 1 (= 100).
So the 10n part is 10-3.
So 0.00509 = 5.09 × 10-3.
So the 10n part is either 103 or 10-3.
0.00509 is less than 1 (= 100).
So the 10n part is 10-3.
So 0.00509 = 5.09 × 10-3.
Close
Standard Notation → Scientific Notation
Example
7.95 × 106
Solution 7●95 - [1]
↓
7◌950000● - [2]
7.95 × 106 = 7950000
↓
7◌950000● - [2]
7.95 × 106 = 7950000
[1]
Write the a part.
[2]
7.95 × 106 is greater than 1 (= 100).
So move the decimal point
6 digits →.
(to make the number greater than 1)
Fill the 0s, 0000, in the blank digits.
So move the decimal point
6 digits →.
(to make the number greater than 1)
Fill the 0s, 0000, in the blank digits.
Close
Example
8.6 × 10-4
Solution 8●6 - [1]
↓
0●0008◌6 - [2]
8.6 × 10-4 = 0.00086
↓
0●0008◌6 - [2]
8.6 × 10-4 = 0.00086
[1]
Write the a part.
[2]
8.6 × 10-4 is less than 1 (= 100).
So move the decimal point
4 digits →.
(to make the number less than 1)
Fill the 0s, 0.000, in the blank digits.
So move the decimal point
4 digits →.
(to make the number less than 1)
Fill the 0s, 0.000, in the blank digits.
Close
Multiplying Scientific Notation
Example
(2.15 × 103)⋅(1.98 × 105)
Solution (2.15 × 103)⋅(1.98 × 105)
= (2.15⋅1.98)⋅(103⋅105)
= 4.26 × 108 - [1] [2]
= (2.15⋅1.98)⋅(103⋅105)
= 4.26 × 108 - [1] [2]
[1]
2.15⋅1.98 = 4.2570
2.15 and 1.98 have 3 significant digits.
(215, 198)
So, to make the a part 3 significant digits,
round 4.2570 to the nearest hundredth.
4.2570 → 4.26
2.15 and 1.98 have 3 significant digits.
(215, 198)
So, to make the a part 3 significant digits,
round 4.2570 to the nearest hundredth.
4.2570 → 4.26
[2]
Close
Example
(8.73 × 104)⋅(9.01 × 102)
Solution (8.73 × 104)⋅(9.01 × 102)
= (8.73⋅9.01)⋅(104⋅102)
= 78.7 × 106 - [1]
= 7.87⋅10 × 106
= 7.87 × 107 - [2]
= (8.73⋅9.01)⋅(104⋅102)
= 78.7 × 106 - [1]
= 7.87⋅10 × 106
= 7.87 × 107 - [2]
[1]
8.73⋅9.01 = 78.6573
8.73 and 9.01 have 3 significant digits.
(873, 901)
So, to make the a part 3 significant digits,
round 78.6573 to the nearest hundredth.
78.6573 → 78.7
8.73 and 9.01 have 3 significant digits.
(873, 901)
So, to make the a part 3 significant digits,
round 78.6573 to the nearest hundredth.
78.6573 → 78.7
[2]
Make 78.7 between 1 ≤ a < 10.
78.7 → 7.87⋅10
78.7 → 7.87⋅10
Close
Dividing Scientific Notation
Example
(9.02 × 107) ÷ (3.75 × 104)
Solution (9.02 × 107) ÷ (3.75 × 104)
= (9.02 ÷ 3.75)⋅(107 ÷ 104)
= 2.41 × 103 - [1] [2]
= (9.02 ÷ 3.75)⋅(107 ÷ 104)
= 2.41 × 103 - [1] [2]
[1]
9.02 ÷ 3.75 = 2.405...
9.02 and 3.75 have 3 significant digits.
(902, 375)
So, to make the a part 3 significant digits,
round 2.405... to the nearest hundredth.
2.405... → 2.41
9.02 and 3.75 have 3 significant digits.
(902, 375)
So, to make the a part 3 significant digits,
round 2.405... to the nearest hundredth.
2.405... → 2.41
[2]
Close
Example
(1.57 × 108) ÷ (8.06 × 105)
Solution (1.57 × 108) ÷ (8.06 × 105)
= (1.57 ÷ 8.06)⋅(108 ÷ 105)
= 0.174 × 103 - [1]
= 1.74⋅10-1 × 103
= 1.74 × 102 - [2]
= (1.57 ÷ 8.06)⋅(108 ÷ 105)
= 0.174 × 103 - [1]
= 1.74⋅10-1 × 103
= 1.74 × 102 - [2]
[1]
1.57 ÷ 8.06 = 0.1742...
1.57 and 8.06 have 3 significant digits.
(157, 806)
So, to make the a part 3 significant digits,
round 0.1742... to the nearest hundredth.
0.1742... → 0.174
1.57 and 8.06 have 3 significant digits.
(157, 806)
So, to make the a part 3 significant digits,
round 0.1742... to the nearest hundredth.
0.1742... → 0.174
[2]
Make 0.174 between 1 ≤ a < 10.
0.174 → 1.74⋅10-1
0.174 → 1.74⋅10-1
Close