Scientific Notation


Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 109. So, how does this work?

We can think of 5.6 x 109 as the product of two numbers: 5.6 (the digit term) and 109 (the exponential term).

Here are some examples of scientific notation.

10000 = 1 x 104 24327 = 2.4327 x 104
1000 = 1 x 103 7354 = 7.354 x 103
100 = 1 x 102 482 = 4.82 x 102
10 = 1 x 101 89 = 8.9 x 101 (not usually done)
1 = 100
1/10 = 0.1 = 1 x 101 0.32 = 3.2 x 101 (not usually done)
1/100 = 0.01 = 1 x 102 0.053 = 5.3 x 102
1/1000 = 0.001 = 1 x 103 0.0078 = 7.8 x 103
1/10000 = 0.0001 = 1 x 104 0.00044 = 4.4 x 104

As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.

In scientific notation, the digit term indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example,

46600000 = 4.66 x 107 This number only has 3 significant figures. The zeros are not significant; they are only holding a place. As another example,

0.00053 = 5.3 x 104 This number has 2 significant figures. The zeros are only place holders.

Addition and Subtraction:

  • All numbers are converted to the same power of 10, and the digit terms are added or subtracted.
  • Example: (4.215 x 102) + (3.2 x 104) = (4.215 x 102) + (0.032 x 102) = 4.247 x 102
  • Example: (8.97 x 104) – (2.62 x 103) = (8.97 x 104) – (0.262 x 104) = 8.71 x 104

Multiplication:

  • The digit terms are multiplied in the normal way and the exponents are added. The end result is changed so that there is only one non-zero digit to the left of the decimal.
  • Example: (3.4 x 106)(4.2 x 103) = (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 1010
    (to 2 significant figures)
  • Example: (6.73 x 105)(2.91 x 102) = (6.73)(2.91) x 10(5+2) = 19.58 x 103 = 1.96 x 102
    (to 3 significant figures)

Division:

  • The digit terms are divided in the normal way and the exponents are subtracted. The quotient is changed (if necessary) so that there is only one non-zero digit to the left of the decimal.
  • Example: (6.4 x 106)/(8.9 x 102) = (6.4)/(8.9) x 10(6-2) = 0.719 x 104 = 7.2 x 103
    (to 2 significant figures)
  • Example: (3.2 x 103)/(5.7 x 102) = (3.2)/(5.7) x 103(2) = 0.561 x 105 = 5.6 x 104
    (to 2 significant figures)

Powers of Exponentials:

  • The digit term is raised to the indicated power and the exponent is multiplied by the number that indicates the power.
  • Example: (2.4 x 104)3 = (2.4)3 x 10(4×3) = 13.824 x 1012 = 1.4 x 1013
    (to 2 significant figures)
  • Example: (6.53 x 10-3)2 = (6.53)2 x 10(3)x2 = 42.64 x 106 = 4.26 x 105
    (to 3 significant figures)

Roots of Exponentials:

  • Change the exponent if necessary so that the number is divisible by the root. Remember that taking the square root is the same as raising the number to the one-half power.
  • Example:

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From here down is the same as above, just phrased differently. Sometimes it helps to read different accounts of the topic you are studying. So read the piece on scientific notation in the Arithmetic section of your school text book as well. Do not panic if there is anything you do not understand. Just note the bits you have difficulty with and we will work through them in the next class. If you have a question you can put clearly into words send me an email and I will try to answer it sooner. Remember it is more important to get to understand what we are trying to do than to remember any formulas. The formulas for this stuff are all given in the formulae and tables booklet page 21 and you will be given this in any State exams.

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Do you know this number, 300,000,000 m/sec.?

It’s the Speed of light !

Do you recognize this number, 0.000 000 000 753 kg. ?

This is the mass of a dust particle!

Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on powers of the base number 10.

The number 123,000,000,000 in scientific notation is written as :


The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called the base . It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 1011 the number 11 is referred to as the exponent or power of ten.


To write a number in scientific notation:

Put the decimal after the first digit and drop the zeroes.


In the number 123,000,000,000 The coefficient will be 1.23

To find the exponent count the number of places from the decimal to the end of the number.

In 123,000,000,000 there are 11 places. Therefore we write 123,000,000,000 as:


Exponents are often expressed using other notations. The number 123,000,000,000 can also be written as:

1.23E+11 or as 1.23 X 10^11



For small numbers we use a similar approach. Numbers less smaller than 1 will have a negative exponent. A millionth of a second is:

0.000001 sec. or 1.0E-6 or 1.0^-6 or

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Link to interactive practise scientific notation generator.

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Part 1: How Do we Multiply numbers in Scientific Notation?

Scientific Notation is based on powers of the base number 10.

The number 123,000,000,000 in scientific notation is written as :


The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called the base . It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 1011 the number 11 is referred to as the exponent or power of ten.

Rules for Multiplication in Scientific Notation:

1) Multiply the coefficients

2) Add the exponents (base 10 remains)

Example 1: (3 x 104)(2x 105) = 6 x 109

What happens if the coefficient is more than 10 when using scientific notation?

Example 2: (5 x 10 3) (6x 103) = 30. x 106

While the value is correct it is not correctly written in scientific notation, since the coefficient is not between 1 and 10. We then must move the decimal point over to the left until the coefficient is between 1 and 10. For each place we move the decimal over the exponent will be raised 1 power of ten.

30.x106 = 3.0 x 107in scientific notation.

Example 3:

(2.2 x 10 4)(7.1x 10 5) = 15.62 x 10 9 = 1.562 x 10 10

Example 4:

(7 x 104)(5 x 106)(3 x 102) = 105. x 10 12 –now the decimal must be moved two places over and the exponent is raised by 2. Therefore the value in scientific notation is: 1.05 x 10 14

Now Try these:

(write your answers in the form of coefficientx10^exponent) If your answer is 3.5 x 10 3 you should type 3.5×10^3 in the box then click the submit button).

(2 X 103)(4X104)=

(6 X 105 )(7 X 10 6)=

(5.5 X 107)(4.2 x 104)= What happens when the exponent(s) are negative?

We still add the exponents, but use the rules of addition of signed numbers.

Example 5: (3 x 10 -3) (3x 10-3) = 9. x 10-6

Example 6: (2 x 10 -3) (3x 108) = 6. x 105

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Part 2: How Do we Divide numbers in Scientific Notation?

Scientific Notation is based on powers of the base number 10.

The number 123,000,000,000 in scientific notation is written as :


The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called the base . It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 1011 the number 11 is referred to as the exponent or power of ten.

Rules for Division in Scientific Notation:

1) Divide the coefficients

2) Subtract the exponents (base 10 remains)

Example 1: (6 x 106) / (2 x 103) = 3 x 103

What happens if the coefficient is less than 10?

Example 2: (2 x 10 7) / (8 x 103) = 0.25 x 104

While the value is correct it is not correctly written in scientific notation since the coefficient is not between 1 and 10. We must move the decimal point over to the right until the coefficient is between 1 and 10. For each place we move the decimal over the exponent will be lowered 1 power of ten.

0.25×10 4 = 2.5 x 103in scientific notation.

What happens when the exponent(s) are negative?

We still subtract the exponents (apply the rules for subtracting signed numbers)

Example 5: (9 x 10 -6) / (3x 10-3) = 3. x 10-3

Example 6: (2 x 10 3) / (4 x 10-8) = 0.5 x 1011 = 5 x 10 10

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Scientific Notation problems involving multiplication and division.

Example 1:

(6 x 106) (2x 103) (2x 103)
_________________________ =
______4 x 10 4
Calculate the coefficient:6 x 2 x 2
___4
Find the base/exponent term:
(106) (103) (103)
___________________
____ 10 4
= 6 x 10 8

Example 2:

(4 x 106) (2x 103)
__________________ =
(8 x 10 -4 )(2x 103)
Calculate the coefficient:4 x 2
8 x 2
Find the base/exponent term:
(106) (103)
____________
(10 -4
) (103)
= 0.5x 1010 = 5 x 10 9

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