Scientific Notation

Multiplying and Dividing Scientific Notation

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Multiplying and Dividing Scientific Notation

As you already know, scientific notation (a form of exponential notation) is a concise way to express very small or very large numbers.
Consider the speed of light, 300,000,000 m/sec. When writing this value it is very easy to “lose” one, or more, of the zeros. It is much faster and easier to write 3.0 × 108 m/sec or 3.0 E+8 m/sec.

Remember that correctly written scientific notation has two components:

  1. a number between 1 and 10, such that
    1 ≤ n < 10
    multiplied by….  n × 10x
  2. a power of 10.

One of the advantages of scientific notation is its ease of use when performing computations.
Watch the laws of exponents at work!

To multiply

To multiply two numbers expressed in scientific notation, simply multiply the numbers out front and add the exponents.
Generically speaking, this process is expressed as:
(n × 10a) • (m × 10b) = (n • m) × 10a+b
Multiply the numbers out front and add the exponents.

Example 1: (5.1 × 104) • (2.5 × 103) = 12.75 × 107      Oops!!
This new answer is no longer in proper scientific notation.
Proper scientific notation is 1.275 × 108

Example 2: (3.4 × 103) • (5.6 × 105) 
(3.4 × 103) • (5.6 × 105) = 19.04 × 108  = 1.904 × 109 
Notice that the first multiplication did not give the answer in proper scientific notation.

NOTE: In real life situations, answers obtained from the multiplication (or division) of values expressed in scientific notation may result in answers with “more decimal accuracy” than the original values.
Regarding ACCURACY: If values are stated to the greatest accuracy that they are known, then the result of multiplication (or division) with these values cannot be determined to any better accuracy than to the number of digits in the least accurate number. Regarding accuracy, the answer to Example 1 would be 1.3 × 108.
On this site, we will be finding the mathematical results to the multiplication, or division, of scientific notation, WITHOUT a determination of accuracy.

To divide

To divide two numbers expressed in scientific notation, simply divide the numbers out front and subtract the exponents.
Generically speaking, this process is expressed as:
Multiplying and Dividing Scientific Notation 1
Divide the numbers out front and subtract the exponents.

Example 2:
Multiplying and Dividing Scientific Notation 2
Example 3:
Multiplying and Dividing Scientific Notation 3

Ever wonder how these numbers are Added or Subtracted?

To add (or subtract) two numbers expressed in scientific notation, be sure that the exponents in each number are the SAME. Generically speaking:
To ADD or SUBTRACT two numbers in scientific notation, the exponents on the power of 10 must be the same. You may need to “adjust” the numbers, moving them out of scientific notation, so the exponents are alike.
Multiplying and Dividing Scientific Notation 4
If the exponents are NOT the same, the decimal of one of the numbers has to be repositioned so that its exponent is the same as the other number being added or subtracted. Think of it as lining up the decimals for addition or subtraction.

Example 1: (3.2 × 105) + (5.1 × 104)
(3.2 × 105) + (5.1 × 104) = (3.2 × 105) + (0.51 × 105)
= 3.71 × 105

Example 1: (6.3 × 106) + (4.7 × 104)
(6.3 × 106) + (4.7 × 104) = (6.3 × 106) + (0.047 × 106)
= 6.347 × 106
The decimal point in the second number was moved two places to the left so that the base of 10 could be raised to a power of 6.

Base Quantities and Derived Quantities Definition, Units Examples

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Base Quantities and Derived Quantities Definition, Units Examples

  1. Physical quantities are quantities that can be measured.
  2. Usually, a specific scientific instrument is used to measure a particular physical quantity.
    Base Quantities and Derived Quantities Definition, Units Examples 1
  3. To describe a physical quantity we first define the unit in which the measurement is made. There are many systems of units but the most common system of units used by scientists is based on the metric system.
  4. The modernised version of the metric system is called International System of Units, officially abbreviated as SI.
  5. We can represent a physical quantity by the symbol of the quantity, the numerical value of the magnitude of the quantity and the unit of measurement of the quantity. For example, Figure shows a footballer scoring a goal. The ball was kicked a distance of 8 m.
    Base Quantities and Derived Quantities Definition, Units Examples 2
  6. There are two types of physical quantities, that is, base quantities and derived quantities.
  7. Base quantities are physical quantities that cannot be defined in terms of other quantities. Table shows five base quantities and their respective SI units.
    Base quantitySymbolSI unitSymbol of SI unit
    Lengthlmeterm
    Massmkilogramkg
    TimetsecondS
    TemperatureTkelvinK
    Electric currentIampereA

    Derived quantities are physical quantities derived from combinations of base quantities through multiplication.

  8. Table shows some derived quantities and their respective derived units.
  9. Base Quantities and Derived Quantities Definition, Units Examples 3

Example 1
It was already noon when Lela woke up. The temperature was 38°C and she was sweating all over. As it was already late, she was given only 10 minutes to pack her things. She wondered how she would pack a 1.5 kg tin of milk powder, 850 cm3 of lake water, 980 g of a rare rock, a 1.2 m long stem of a special plant and finally not to forget 6.5 m2 of tent material into her bag.

Base Quantities and Derived Quantities Definition, Units Examples 4From the above description, identify the physical quantities and then classify them into base quantities and derived quantities.
Solution:
The physical quantities are temperature (38°C), time (10 minutes), mass (1.5 kg and 980 g), length (1.2 m), volume (850 cm3) and area (6.5 m2).
Classification:
Base quantity: Mass, length, temperature, time
Derived quantity: Area, volume v

Prefixes

  1. Prefixes are used to simplify the description of physical quantities that are either very big or very small in SI units.
  2. Table lists some commonly used SI prefixes and their multiplication factors.
    PrefixSymbolValue
    picoP10-12
    nanon10-9
    microp10-6
    millim10-3
    centic10-2
    decid10-1
    kilok103
    megaM106
    gigaG109
    teraT1012

Base Quantities and Derived Quantities Definition, Units Examples 5

Example 2
It is difficult for Hawa to figure out the mass of a piece of paper which is 0.0042 kg and the mass of a cat which is 5800 g. To simplify the description of these physical quantities, prefixes are used.
Base Quantities and Derived Quantities Definition, Units Examples 6Please help Hawa to express
(a) 0.0042 kg in g,
(b) 5800 g in kg.
Solution:
Base Quantities and Derived Quantities Definition, Units Examples 7

Example 3
Convert
(a) 0.000 006 Mm to cm,
(b) 570 000 cm to km.
Base Quantities and Derived Quantities Definition, Units Examples 8
Solution:
Base Quantities and Derived Quantities Definition, Units Examples 9

Scientific Notation

  1. The distance of Pluto from the Earth is about 6 000 000 000 000 m and the radius of a hydrogen atom is about 0.000 000 000 05 m. These quantities are either too large or too small and a simpler way of expressing them is by using standard form of representation or scientific notation.
  2. In a standard form or scientific notation, a numerical magnitude can be written as:
    A × 10n, where 1 ≤ A < 10 and n is an integer
    Hence, the distance of Pluto from the Earth can be written as 6 × 1012 m and the radius of a hydrogen atom as 5 × 10-11 m.
    Base Quantities and Derived Quantities Definition, Units Examples 12

Example 4
For each of the following, express the magnitude using scientific notation.
(a) The length of a virus = 0.000 000 08 m
(b) The mass of a ship = 75 000 000 kg
Solution:
Base Quantities and Derived Quantities Definition, Units Examples 10

Conversion of Units Involving Derived Quantities

When converting the units of a derived quantity, each of its base units involved must be converted. The following Example illustrates the conversion of derived units.
Base Quantities and Derived Quantities Definition, Units Examples 13Example 5
Convert each of the following from one particular unit to another and represent the quantity in standard form.
(a) Convert the area of a button from 1.2 cm2 into m2.
(b) Convert the volume of a water tank from 2.5 m3 into cm3.
(c) Convert the density of mercury from 13.6 g cm-3 into kg m-3.
Solution:
Base Quantities and Derived Quantities Definition, Units Examples 11

Converting To and From Scientific Notation

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Converting To and From Scientific Notation

Converting To Scientific Notation

To Change from Standard Form to Scientific Notation:

  1. Place the decimal point such that there is one non-zero digit to the left of the decimal point.
  2. Count the number of decimal places the decimal has “moved” from the original number. This will be the exponent of the 10.
  3. If the original number was less than 1, the exponent is negative; if the original number was greater than 1, the exponent is positive.

Converting To and From Scientific Notation 1
Examples:

  1. Given: 4,750,000
    4.75 (moved decimal point 6 decimal places)
    Answer: 4.75 × 106
    The original number was greater than 1 so the exponent is positive.
  2. Given: 0.000789
    7.89 (moved decimal point 4 decimal places)
    Answer: 7.89 × 10-4
    The original number was less than 1 so the exponent is negative.

Converting From Scientific Notation

To Change from Scientific Notation to Standard Form:

  1. Move the decimal point to the right for positive exponents of 10. The exponent tells you how many places to move.
  2. Move the decimal point to the left for negative exponents of 10. Again, the exponent tells you how many places to move.

Converting To and From Scientific Notation 2
Examples:

  1. Given: 1.015 × 10-8
    Answer: 0.00000001015 (moved decimal 8 places left)
    Negative exponent moves decimal to the left.
  2. Given: 5.024 × 103
    Answer: 5,024 (move decimal 3 places right)
    Positive exponent moves decimal to the right.

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Maths

Correct Scientific Notation

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Correct Scientific Notation

Scientific Notation

  1. Scientific Notation is a way to express very small or very large numbers.
  2. Scientific Notation is most often used in “scientific” calculations where the analysis must be very precise.
  3. Scientific Notation consists of two parts*:
    1. a number between 1 and 10, such that
      1≤n<10
      and
    2. a power of 10.

* A large or small number may be written as any power of 10; however, CORRECT scientific notation must satisfy the above criteria.
3.2 × 1013       is correct scientific notation
23.6 × 10-8     is not correct scientific notation
Remember that the first number MUST BE greater than or equal to one and less than 10.