Scientific Notation (PreAP)

Scientific Notation

How to take really BIG numbers (like the number of grains of sand on the beach) or really SMALL numbers (like the size of an atom) and make them more manageable (easier to use and calculate).

The Formula

• What is it? Scientific Notation is a formula that is equal to the really LONG number you want to express. It's like a mathematical abbreviation, like writing Mr. instead of Mister.

• Why use it? The number might be too long to fit on the calculator screen or to write over and over.

• Long numbers can be small decimals like 0.000000245 or big whole numbers like 602,000,000 & they can be positive or negative.

Parts of the Formula

• C stands for Coefficient. Any number between 1 and 10, but not including 10. It can have decimals and it can be positive or negative.

• Base 10 (x10) allows us to move the decimal point without changing the numbers. Example: 3.15 x 10 = 31.5

• n is an exponent and has to be a whole number, positive or negative. It shows how many times to move the decimal, and what direction to move it.

Multiple Choice

Which number could be a Coefficient?

• 0.3

• 5.6

• 13.1

Multiple Choice

Which number could not be an exponent?

• 5

• -3

• 23

• 1.6

Converting LARGE #s

• Step 1) Find your coefficient! Put the decimal to the right of the first non-zero & drop zeros.

• Step 2) Write in the base 10!

• Step 3) For the exponent count how many places the decimal moved.

• Step 4) Determine if your exponent is positive or negative by the direction it moves. Big #s move left so it's positive

Converting SMALL #s

• Step 1) Find your coefficient! It's 4.3 because 4 is the first # between 1 & 10. The decimal goes to the right of the 1st nonzero and the 3 goes after the decimal.

• Step 2) Write in the base 10!

• Step 3) Count the places the decimal moved for the exponent

• Step 4) Determine if your exponent is positive or negative by the direction it moves. Small #s move right so it's negative

Multiple Choice

What would be the coefficient for 67,230,000,000 if expressed in scientific notation?

• 6

• 6.723

• 67.23

• 0.6723

Multiple Choice

Which of these numbers would have a negative exponent when expressed in Scientific notation?

• 123,46,789

• 0.000583

Multiple Choice

Which option is correct way to express 45,000,000 in scientific notation?

• 45 x10 ⁶

• 4.5 x10⁷

• 0.45 x10 ⁸

• 4.5 x10^-7

Multiple Choice

Which option is correct way to express 0.00000045 in scientific notation?

• 45 x10 ⁶

• 4.5 x10⁷

• 0.45 x10 ⁸

• 4.5 x10^-7

You have scientific notation already, now what?

• If you need the normal/regular number and not a formula then you can convert the scientific notation back to standard form.

• Positive exponents are BIG numbers - move your decimal to the right. Move it the same number as the exponent and fill in the 0's and commas.

Multiple Choice

Is 6.3 x10¹² a large whole number or a small decimal in standard form?

• LARGE

• _small

You have scientific notation already, now what?

• If you need the normal/regular number and not a formula then you can convert the scientific notation back to standard form.

• Negative exponents are _tiny numbers that look long, but are less than 0. Move your decimal to the left. Move it the same number as the exponent and fill in the 0's.

Multiple Choice

To convert 2.54 x10^-6 back to standard form, how does the decimal move?

• 10 to the left

• 6 to the left

• 6 to the right

• 2 and 1/2 times

Multiple Choice

Which answer correctly converts 6.02 x10²³ into standard notation? (count carefully!)

• 602,000,000,000,

  000,000,000,000

• 6,020,000,000,

  000,000,000

• 0.0000000000

  000000000000602

• 0.0000000000

  00000602000

Multiple Choice

How do you write 1.28 x 10^-4 in standard form?

• 0.128

• 0.0128

• 0.0000128

• 0.000128

Multiple Select

RECAP: Select the true statements

• The coefficient must be bigger than/equal to 1 and smaller than 10

• Scientific notation is for LONG numbers, big or small

• Exponents can only be positive

• Exponents must be whole numbers

• Base 10 is optional, use any base you like

Multiple Choice

MORE PRACTICE: Put the following number in scientific notation:

4,281

• 4.281 x 10³

• 4.281 x 10^-3

• 0.4281 x 10⁴

• 42.81 x 10²

Multiple Choice

MORE PRACTICE: Put the following in scientific notation:

23,948

• 23.948 x 10³

• 2.3948 x 10⁴

• 2.3948 x 10^-4

• 0.23948 x 10⁵

Multiple Choice

MORE PRACTICE: How would you write -5.6 x 10^-3 in standard form?

• 0.0056

• 0.00056

• -5,600

• -0.0056

Multiple Choice

MORE PRACTICE: How would you write 4.3756 x 10⁴ in standard form?

• 437,560,000

• 0.00043756

• 43,756

• 43,756.000

Doing math w/ Scientific Notation

Scientific notation is just another way of representing numbers, so it can be used in formulas in place of typing the really long numbers into the calculators.
We will review multiplying and dividing using scientific notation.
This allows you to check the calculator and make sure you didn't make any typos or other mistakes.

Step 1: Divide the coefficients
Step 2: Move the decimal to be after the first non-zero (if needed)
Step 3: Fill in the base 10
Step 4: Subtract the exponents and then add or subtract the adjustment you made.

Dividing

Step 1: Multiply the coefficients
Step 2: Move the decimal to be after the first non-zero (if needed)
Step 3: Fill in the base 10
Step 4: Add the exponents and then add or subtract the adjustment you made.

Multiplying

The Steps - Compare & Contrast

3×10⁸ × 6×10²³

Step 1: 3x6 = 18
Step 2: 18 --> 1.8 = +1 to the exponent
Step 3: 1.8×10
Step 4: 8+23 = 31 + 1 from moving the decimal in step 2.
= 1.8×10³²

Example

Step 1: Multiply the coefficients
Step 2: Move the decimal to be after the first non-zero (if needed)
Step 3: Fill in the base 10
Step 4: Add the exponents and then add or subtract the adjustment you made. If there is no exponent, treat it as 0.

The Steps

Multiplying Scientific Notation

Multiple Choice

3x10⁸ x 2x10²²

• 6x10³⁰

• 5x10³⁰

• 6x10^-14

• Err

Multiple Choice

4x10¹⁴ x 5x10^-4

• 2x10¹¹

• 20x10¹⁰

• 2x10¹⁸

• 2x10^-10

Multiple Choice

7x10¹⁵ x 5x10⁴

• 35x10¹⁹

• 3.5x10¹⁹

• 3.5x10²⁰

• 12x10¹⁹

Multiple Choice

-2x10³ x 2x10³

• -4x10⁶

• 0x10⁶

• 4x10⁶

• -4x10⁹

Multiple Choice

6 x 4x10⁵

• 24x10⁵

• 2.4x10⁶

• 2.4x10⁵

• 10x10⁵

2×10¹⁸ / 4×10¹⁰

Step 1: 2/4 = 0.5
Step 2: 0.5 --> 5 = -1 to the exponent
Step 3: 5×10
Step 4: 18-10 = 8 - 1 from moving the decimal in step 2.
= 5×10⁷

Example

Step 1: Divide the coefficients
Step 2: Move the decimal to be after the first non-zero (if needed)
Step 3: Fill in the base 10
Step 4: Subtract the exponents and then add or subtract the adjustment you made.

The Steps

Dividing Scientific Notation

Multiple Choice

6x10¹² / 3x10⁶

• 2x10⁶

• 3x10⁶

• 2x10²

• 3x10²

Multiple Choice

1x10⁴ / 4x10⁸

• 2.5x10^-5

• -3x10⁴

• 0.25x10^-4

• 2.5x10^-4

Multiple Choice

4x10² / 2x10^-4

• 2x10⁶

• 2x10^-2

• 2x10^-6

• 6x10^-4

Multiple Choice

6 x 10⁴ / 12

• 5x10³

• 0.5x10⁴

• 5x10⁴

• -6x10^-4

Multiple Choice

5x10⁴ / -1x10²

• -5x10²

• 5x10²

• -5x10^-2

• 4x10²