Scientific Notation

Motivation

There are various reasons that a standardized form for expressing numbers may be desired.

One reason is to differentiate between known digits and digits that are used to determine the size of the number. For example, if we read that the population of Puerto Rico is 3,700,000 people. Our intuition would indicate that this is probably an approximation to the nearest hundred thousand however a form of expression with which we could know this would be helpful.

Another reason is to quickly determine exactly how large or how small a number is. For example, given the numbers 525345452453413256 and 525345236345253436, it would require careful inspection to determine that the second number is in fact about ten times larger than the first. A form of expression which would allow us to immediately determine the size of a number without carefully counting digits would be helpful.

Definition of Scientific Notation

A number is in Scientific Notation if it has been expressed in the form a × 10b where 1 <= a < 10 and b is an integer. The following table presents examples of numbers and how they may be expressed in scientific notation. It is worth noting that 2.34 E4 is a shorthand notation for 2.34 104. In particular, this format is common with calculators.

Number

Scientific Notation

EE form of Scientific Notation

123

1.23 × 102

1.23 EE 2

0.0234

2.34 × 10-2

2.34 EE -2

1230000

1.23 × 106

1.23 EE 6

0.000321

3.21 × 10-4

3.21 EE -4


The Effect of Powers of 10

The goal for this tutorial is to take numbers in scientific notation and present them as simple numbers with neither products nor exponents and to take simple numbers and present them in scientific notation. The key to doing this is understanding the effect that multiplying a number by a power of ten will have. The easiest way to understand this is to associate multipliation by powers of ten with movement of a decimal point.

The following tables show the effect of multiplying the number 1.23 by various powers of 10. It is worth noting that the number was expressed as 1.23000 so that the movement of the decimal point would be clearer.

Number

Power of 10

Result

Movement of Decimal Point

1.23000

× 100

1.23000

0 units right

1.23000

× 101

12.3000

1 unit right

1.23000

× 102

123.000

2 units right

1.23000

× 103

1230.00

3 units right

1.23000

× 104

12300.0

4 units right

1.23000

× 105

123000.

5 units right



 

Number

Power of 10

Result

Movement of Decimal Point

1.23

× 100

1.23000

0 units left

1.23

× 10-1

0.123

1 unit left

1.23

× 10-2

0.0123

2 units left

1.23

× 10-3

0.00123

3 units left

1.23

× 10-4

0.000123

4 units left

1.23

× 10-5

0.0000123

5 units left



 

We can reach the following conclusions from these tables.

  • If a ≥ 0 then the effect of multiplying a number by 10a is to move the decimal point a units to the right.
  • If a ≥ 0 then the effect of multiplying a number by 10-a is to move the decimal point a units to the left.

Converting Expressions in Scientific Notation to Simple Numbers

Example: Convert the number 2.34 x 105 to a simple numerical expression by exressing the same number without exponents or products.

Solution: We can convert the above number expressed in scientific notation to a simple numerical expression without exponents or products with the following steps.

  • Place the number 2.34 by itself without its associated power of ten.

  • As 5 ≥ 0, count off 5 digits to the right. Adding zeroes as necesary



  • Move the decimal point 5 units to the right. The result is 234000

Example: Eliminate products and powers from the expression 5.581 × 10-7

Solution: We can convert the above expression in scientific notation to a simplenumber without exponents or products with the following steps.

 

  • Place the number 5.581 by itself without its associated power of 10.

  • As -7 ≤ 0, count off 7 digits to the right. Adding zeroes as necesary



  • Move the decimal point 7 units to the left. The result is .000000581


Examples:

Eliminate products and powers from the number 7.43 × 103
1. Place the number by itself 7.43
2. As 3 ≥ 0, count off 3 digits to the right. Adding zeroes as necesary.
3. Move the decimal point 3 units to the right. 7430


Eliminate products and powers from the number 1.97 × 10-9
1. Place the number by itself 1.97
2. As -9 ≥ 0, count off 9 digits to the right. Adding zeroes as necesary.
3. Move the decimal point 9 units to the left. .00000000197

To practice, enter an expression in the text box below, select an accompanying power of ten in the pull down menu to see the resulting expression. (Note, unless a power in the pull down menu is actively selected, the result will not be displayed.


Number to Scientific Notation

Example: Place 4730000 in scientific notation.

Solution: We can convert the above number to scientific notation with the following steps.

  • Remove any decimal points from the number (in this case there are none) 473000
  • Place a decimal point in the digits so that the number is between one and ten. We will henceforth refer to this number as a. a = 473000.
  • Determine the number of units and the direction that the decimal point must be moved to convert a to the initial number.



    4.73000 to 4730000 means 6 units to the right.
  • We will consider the number b to have magnitude equal to the number of units that the decimal point must be moved and its sign is positive if the decimal point moves to the right and negative if the decimal point moves to the left. In this case b = +6
  • The number in scientific notation is a × 10bor in this case 4.73000 × 106. Depending on the situation, it is generally necessary to remove trailing zeroes giving the final result
    4.73 × 106.

Example: Place -0.0000426 in scientific notation.

Solution: We can convert the above number to scientific notation with the following steps.

  • Remove any decimal points from the number -00000426.
  • Place a decimal point in the digits so that the number is between one and ten. We will henceforth refer to this number as a. a = -000004.26 = -4.26a .
  • Determine the number of units and the direction that the decimal point must be moved to convert a to the initial number.



    -4.26 to -0.0000426 means 5 units to the left. It is worth noting that we must add superfluous zeros to the left to determine this.
  • We will consider the number b to have magnitude equal to the number of units that the decimal point must be moved and its sign is positive if the decimal point moves to the left and negative if the decimal point moves to the left. In this case b = -5

  • The number in scientific notation is a × 10b or in this case -4.26 × 10-5. In this case, there are no trailing zeroes hence the final result -4.26 × 10-5.


Examples:

Place 82600000 in scientific notation.
1. Remove any decimal points from the number (in this case there are none) 8260000
2. Place a decimal point in the digits so that the number is between one and ten. We will henceforth refer to this number as a. a = 8260000.
3. Determine the number of units and the direction that the decimal point must be moved to convert a to the initial number.

8.260000 to 82600000 means 7 units to the right
4. We will consider the number b to have magnitude equal to the number of units that the decimal point must be moved and its sign is positive if the decimal point moves to the right and negative if the decimal point moves to the left. b = +7
The number in scientific notation is a × 10b. Depending on the situation, it is generally necessary to remove trailing zeroes giving the final result . 8.26000 × 107 = 8.26 × 107


Place -0.00936 in scientific notation
1. Remove any decimal points from the number. -000936
2. Place a decimal point in the digits so that the number is between one and ten. We will henceforth refer to this number as a. a = -0009.36 = -9.36a
3. Determine the number of units and the direction that the decimal point must be moved to convert a to the initial number.

-9.36 to -0.00936 means 3 units to the left.

4. We will consider the number b to have magnitude equal to the number of units that the decimal point must be moved and its sign is positive if the decimal point moves to the right and negative if the decimal point moves to the left. b = -3
5. The number in scientific notation is a × 10b . -9.36 × 10-3


To see a number expressed in scientific notation, type any number in the text box labeled expression and click submit.



Click below to practice problems associated with scientific notation.