Scientific notation is a way of writing numbers using a power of 10. It is used to

reduce the writing of numbers that have many digits.

Very small or very large numbers are often found in the sciences in general and

writing in scientific notation makes it easier to make comparisons and calculations.

A number in scientific notation has the following format: Examples

a) 6 590 000 000 000 000 = 6.59. 10 15

b) 0.000000000016 = 1.6. 10 – 11

Convert a number to scientific notation

Here's how to convert numbers into scientific notation in a practical way:

1st Step: Write the number in decimal form, with only one digit other than 0 in front of the

comma.

2nd Step: Put in the exponent of the power of 10 the number of decimal places that we had to "walk"

with the comma. If walking with the comma the value of the number has decreased, the exponent will be positive, if

increased the exponent will become negative.

Step 3: Write the product of the number by the power of 10 Examples

1) Transform the number 32 000 into scientific notation.

 First "walk" with the comma, placing it between the 3 and the 2, because this way we will only have

the numeral 3 before the comma;

 To place the comma in this position, we verified that we had to "go" 4 decimal places, since

in whole numbers the comma is at the end of the number. In this case 4 will be the exponent of the

power of 10.

 Writing in scientific notation: 3.2. 104

2) The mass of an electron is approximately 0.000000000000000000000000000911 g.

Convert this value to scientific notation.

 First "walk" with the comma, placing it between 9 and 1, because this way we will only have

the digit 9 (which is the first digit other than 0) before the comma;

 To place the comma in this position we "walk" 28 decimal places. It is necessary to remember that when

put the comma after the 9, the number has a higher value, so in order not to change its

value the exponent will be negative;

 Writing the electron mass in scientific notation: 9.11. 10 – 28 g Operations with scientific notation

To perform operations between numbers written in scientific notation, it is important to review operations with

potentiation.

Multiplication

Multiplication of numbers in scientific notation form is done by multiplying the numbers, repeating the

base 10 and adding the exponents. Examples

a) 1.4. 10 3

x 3.1. 10 2

= (1.4 x 3.1). 10 (3 + 2)

= 4.34. 10 5

b) 2.5. 10 – 8

x 2.3. 10 6

= (2.5 x 2.3). 10 (-8 + 6) = 5.75. 10 – 2

Division

To divide numbers in scientific notation form we must divide the numbers, repeat base 10 and

subtract the exponents.

Examples

a) 9.42. 10 5

: 1.2 . 10 2

= (9.42: 1.2). 10 (5 – 2) = 7.85. 10 3

b) 8.64. 10 – 3

: 3.2 . 10 6

= (8.64: 3.2). 10(-3-6) = 2.7. 10 – 9 Addition and Subtraction

To add or subtract numbers in scientific notation, add or subtract the numbers.

numbers and repeat the power of 10. Therefore, to do these operations, it is necessary that the powers of

10 have the same exponent.

Examples

a)3,3. 10 8

+ 4.8 . 10 8

= (3.3 + 4.8). 10 8

= 8.1 . 10 8

b) 6.4. 10 3

– 8.3 . 10 3

= (6.4 – 8.3). 10 3

= – 1.9 . 10 3