Scientific Note

Scientific Notation (also called Standard Class in Britain) is a special way of writing numbers:

It makes it easy to utilise big and modest values.

OK, How Does it Work?

Example: 700

Why is 700 written every bit 7 × ten2 in Scientific Annotation ?

700 = 7 × 100

and so 700 = 7 × 102

Both 700 and seven × 102 accept the same value, merely shown in different means.

Example: 4,900,000,000

    1,000,000,000 = x9 ,
so 4,900,000,000 = 4.nine × 109 in Scientific Notation

The number is written in 2 parts:

  • Only the digits, with the decimal point placed after the first digit, followed by
  • × ten to a power that puts the decimal point where it should be
    (i.e. it shows how many places to move the decimal point).

scientific notation 5326.6 = 5.3266x10^3
In this example, 5326.6 is written every bit v.3266 × 103 ,
because 5326.half-dozen = 5.3266 × yard = 5.3266 × 10three

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

Other Ways of Writing Information technology

three.1 × ten^8

Nosotros can apply the ^ symbol (above the vi on a keyboard), as it is easy to type.

Example: 3 × 10^4 is the same equally 3 × ten4

  • 3 × 10^four = 3 × 10 × 10 × 10 × ten = 30,000

calculator E notation

Calculators frequently use "E" or "eastward" similar this:

Example: 6E+ 5 is the same equally vi × 105

  • 6E+v = 6 × 10 × x × 10 × 10 × 10 = 600,000

Example: 3.12E4 is the same as 3.12 × 10iv

  • 3.12E4 = 3.12 × ten × x × 10 × 10 = 31,200

How to Do it

To figure out the power of ten, think "how many places practise I move the decimal bespeak?"

left arrow When the number is 10 or greater, the decimal point has to move to the left, and the ability of 10 is positive.
right arrow

When the number is smaller than 1, the decimal bespeak has to movement to the right, so the power of 10 is negative.

Instance: 0.0055 is written v.v × 10-3


Considering 0.0055 = five.five × 0.001 = 5.5 × 10-three

Example: 3.2 is written 3.2 × x0


We didn't have to move the decimal point at all, so the power is x0

But it is at present in Scientific Annotation

Check!

After putting the number in Scientific Notation, simply check that:

  • The "digits" office is between 1 and x (it tin be one, simply never 10)
  • The "power" part shows exactly how many places to move the decimal point

Why Use Information technology?

Because it makes information technology easier when dealing with very big or very small numbers, which are mutual in Scientific and Engineering piece of work.

Instance: it is easier to write (and read) ane.iii × ten-ix than 0.0000000013

Information technology can as well brand calculations easier, as in this instance:

Example: a tiny space inside a figurer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.

What is its volume?

Let'due south first catechumen the iii lengths into scientific notation:

  • width: 0.000 002 56m = 2.56×ten-6
  • length: 0.000 000 14m = i.iv×10-7
  • pinnacle: 0.000 275m = 2.75×10-4

So multiply the digits together (ignoring the ×10s):

ii.56 × i.4 × two.75 = 9.856

Concluding, multiply the ×10s:

10-half dozen × 10-7 × 10-four = 10-17 (easier than it looks, only add together −six, −4 and −7 together)

The upshot is ix.856×10-17 thousand3

Information technology is used a lot in Scientific discipline:

Example: Suns, Moons and Planets

The Sunday has a Mass of i.988 × x30 kg.

Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg
(and that number gives a false sense of many digits of accuracy.)

It can also save space! Hither is what happens when you lot double on each square of a chess board:

Chess board doubling
Values are rounded off, so 53,6870,912 is shown as just v×ten8

That last value, shown every bit nine×x18 is actually nine,223,372,036,854,775,808

Applied science Notation

Engineering Notation is like Scientific Notation, except that we just use powers of ten that are multiples of iii (such as xthree, 10-3, 1012 etc).

Examples:

  • ii,700 is written 2.7 × 103
  • 27,000 is written 27 × 10iii
  • 270,000 is written 270 × ten3
  • 2,700,000 is written 2.7 × 106

Case: 0.00012 is written 120 × 10-half-dozen

Notice that the "digits" part can now be between 1 and 1,000 (information technology can be 1, simply never 1,000).

The reward is that nosotros can replace the ×10s with Metric Numbers. So nosotros can apply standard words (such as thou or million), prefixes (such as kilo, mega) or the symbol (k, K, etc)

Instance: xix,300 meters is written 19.3 × 103 m, or nineteen.3 km

Case: 0.00012 seconds is written 120 × 10-6 south, or 120 microseconds