Unit Conversion

t
kg
hg
dag
g
dg
cg
mg

II. Powers of 10 and Scientific Notation

1. Powers of 10 for Beginners

For the physicist, powers of 10 simplify calculations and writing.
  • 100 = 1; 101 = 10; 102 = 100; 103 = 1000; 104 = 10000;....
  • The value of the power corresponds to the number of zeros
You need to shift the decimal point of "5" to the right since 23.5 x 105 = 23.5 x 100000.
So, 23.5 x 105 = 2350000,0 = 2350000

Regarding negative powers of 10:
  • 10-1 = 0.1; 10-2 = 0.01; 10-3 = 0.001; 10-4 = 0.0001;....
  • 321 x 10-6 can also be written (by shifting the decimal point "6" places to the left) as 0,000321

2. Important Properties of Powers of 10

We often use the following properties:
10A x 10B = 10A+B, examples:
  • 103 x 102 = 103+2 = 105
  • 103 x 10-2 = 103+(-2) = 101

1/10A = 10-A, examples:
  • 1/102 = 10-2
  • 1/10-3 = 10-(-3) = 103

10A/10B = 10A-B, examples:
  • 103/105 = 103-5 = 10-2
  • 102/10-4 = 102-(-4) = 106

3. Scientific Notation

a. What is it?🤓

Scientists agree on a way to write a numerical value: a x 10n:
  • a is a number between 1 and 9.999999....
  • n is an integer, positive or negative (a relative integer), for example, n=2 or n=-3

A small test to check if you've understood, click on the numbers that are written correctly, then click send📬.

  • 0.2x102 is written incorrectly because 0.2 is not between 1 and 9.9...
  • 22.0 and 25 x 10-3 are incorrect for the same reason
  • 5 is correctly written, the goal is to simplify the writing so writing 5 x 100 instead of 5 is not ideal
  • 5.3250 x 10-8 is correctly written, mathematically 5.3250 x 10-8 = 5.325 x 10-8, but in physics, 5.3250 makes sense,
    this means the precision is 4 digits after the decimal point

b. How to write a number in scientific notation?

Let's start with 2356978, the first thing is to determine the value of a if I want to use the notation a x 10n.
Simply move the decimal point until you get a number between 1 and 9.99...
  • 2356978 → 2.356978

    • Of course, 2356978 ≠ 2.356978, you need to determine n so that 2356978 = 2.356978 x 10n, as seen above, you must choose n > 0 to shift the decimal point to the right
    and since it needs to be "shifted by 6", n = 6:
  • 2356978 = 2.356978 x 106

    • When you want to use scientific notation for values less than 1 (for example, 0.002356), it's the same principle but the power of 10 will be negative:
  • 0.002356 → a = 2.356 → shift the decimal point 3 places to the left to maintain equality → n = -3 → 0.002356 = 2.356 x 10-3
    • In the next activity, we want to write values in scientific notation a x 10n, drag and drop the correct value of 10n.
    Score: 0/5
    The numbers are random, you can .
    •   =   x
    •  =  x
    •  =   x
    •   =   x
    •   =   x
    ◀️
    101
    102
    103
    104
    105
    106
    107
    108
    10-1
    10-2
    10-3
    10-4
    10-5
    10-6
    10-7
    10-8

    4. The Power of Powers of 10

    Can you calculate  0.00004 x 600 ____________3000 x 0.0008 without a calculator?


    If you’ve been following along so far, the answer is yes, of course, and easily! It just takes a little method:
    1. Use scientific notation:

    0.00004 x 600 ____________3000 x 0.0008= 4 x 10-5 x 6 x 102 ______________3 x 103 x 8 x 10-4

    2. Combine the powers of 10 and use the property: 10A x 10B = 10A+B

    4 x 10-5 x 6 x 102 ______________3 x 103 x 8 x 10-4=     (4 x 6) x (10-5 x 102) __________________(3 x 8) x (103 x 10-4)= 24 x 10-3 ______24 x 10-1

    3. Use the property 10A/10B = 10A-B

    24 x 10-3 ______24 x 10-1= 1 x 10 -3 -(-1) = 1 x 10-2

    Of course, the calculations can be more complicated, but in any case, it is better to simplify the powers of 10 "manually" following this method.

    III. Powers of 10 and Conversion

    1. Multiples, Submultiples, and Powers of 10

    Multiples and submultiples are widely used in everyday life: for example, 40 km instead of 40000 m.
    Here k is the symbol for the multiple kilo, which equals 1000, or 103.
    In fact, we can replace a multiple/submultiple with its value as a power of 10:
    Multiple
    Submultiple    		 femto pico nano micro milli centi deci deca hecto kilo mega giga tera peta
    Symbol f p n μ m c d da h k M G T P
    Power 10-15 10-12 10-9 10-6 10-3 10-2 10-1 101 102 103 106 109 1012 1015

    2. Using Powers of 10 for Conversions

    I would like to draw your attention to an important point:

    To convert 50 mg to g, simply replace milli(m) with its value as a power of 10, which is 10-3:
    50 mg = 50 x 10-3 g = 5 x 10-2 g
    But if I now want to convert from g to mg, I need to perform the inverse operation, which is dividing by 10-3 or multiplying by 103.

    • conversion g to mg → x10-3 || conversion g to μg → x10-6 || conversion g to ng → x10-9...
    • conversion mg to g → x103  || conversion μg to g → x106  || conversion ng to g → x109...

    Just play with the powers of 10, you already know everything!

    3. The Final Challenge

    We start with a random value, and your goal is to use the multiples or submultiples available to you
    to express this value with the smallest power of 10 (in absolute value, meaning we prefer 10-1 over 10-2 because 1<2>)

    • The values are written in scientific notation
    • They correspond to physical quantities with units such as meter(m), ampere(A), volt(V), Joule(J)...

    Your turn, goal: get a score of 5/5!

    Score: 0/5
    The numbers are random, you can .
    • x 10   =   x
    • x 10  =  x
    • x 10  =   x
    • x 10   =   x
    • x 10   =   x
    ◀️
    10-1
    10-2
    101
    102
    p
    n
    μ
    m
    k
    M
    G
    T
    P