WARM UP Calculate: 3.1  10 = 6. 5.4  10 = = 31 0.54 1,000,000 0.0001

SCIENTIFIC NOTATION

OBJECTIVES To convert between scientific and standard notation To use scientific notation in multiplication and division To use scientific notation in approximating and estimating To work problems involving large numbers easily

DEFINITION Scientific notation for a number is notation of the form a x 10 where 1 ≤ a < 10 and n is an integer.

INTRODUCING THE CONCEPT Scientific notation is useful for calculating with very large or very small numbers. It is also helpful for estimating Scientific notation for 2,800,000,000 miles, the distance from the planet Neptune to the sun is . For 0.000000022 cm. the diameter of a helium atom, it is For 100 it is .

CONVERSIONS We can convert to scientific notation by multiplying by 1, choosing for the number 1, where b is the number of decimal places to be moved. Example: Write scientific notation for 20,000,000 degrees Kelvin, the temperature near the sun’s center. We need to move the decimal point 7 places to the left, so that it is between the digits 2 and 0, so we choose for 1 and then multiply. Multiplying by 1 Using the associative property

MORE EXAMPLES The wavelength of a certain red light is 0.000066 cm. Write scientific notation for this number. We need to move the decimal point 5 places to the right. We choose for 1, and then multiply. Multiplying by 1 Using the associative property

MORE EXAMPLES Light travels about 9,460,000,000,000 km. in one year. Write scientific notation for this number. Multiplying by 1 moved the decimal point 12 places to the left

TRY THIS…. 1. Convert 460,000,000,000 to scientific notation. 3. The mass of a hydrogen atom is 0.0000000000000000000000017 grams. Write scientific notation for this number. 4. The distance from Earth to the sun is about 150,000,000 km. Write scientific notation for this number.

EXAMPLES Convert to standard notation: Moving the decimal point 5 places to the right. Moving the decimal point 8 places to the left.

MULTIPLYING To multiply and divide using scientific notation, we use the commutative and associative properties and then use the properties of exponents to simplify the powers of ten. Multiply and write the scientific notation for the answer: We apply the commutative and associate properties: We convert 13.95 to scientific notation and then simplify:

DIVIDING Divide and write the scientific notation for the answer: Factoring Dividing Converting 0.9 to scientific notation

TRY THIS…. Multiply or divide write the scientific notation for the answer: 1. 2. 3.

ESTIMATING AND APPROXIMATING Scientific notation is helpful in estimating or approximating. Such estimating is an important skill in problem solving. Estimate 780,000,000 x Converting to scientific notation & multiplying Regrouping Rounding Converting to scientific notation Rounding

TRY THIS…. Estimate. Answer may vary depending on how you round:

COMPUTING WITH SCIENTIFIC NOTATION Many calculators switch their displays automatically to scientific notation when results are greater than the capacity of the display. If we multiply 20,820,000 by 5000 on a calculator, the display may show 1.041 11. this means . Calculators with keys EXP or EE allow you to enter numbers in scientific notation. To enter 0.0035698 in scientific notation, press: 3.5698 EXP 3 +/ – Calculate 34,700,000 × 5000 using scientific notation: 3.47 EXP 7 × 5 EXP 3 = 1.735 11