Scientific NotationScientific Notation

Mrs Vass Mrs Vass BJHBJH

Scientific NotationScientific Notation

Definition: Scientific notation is a symbol that expresses Definition: Scientific notation is a symbol that expresses any number as a power of ten multiplied by a number any number as a power of ten multiplied by a number between 1 and 10 (including 1).between 1 and 10 (including 1).

Scientific notation allows you to work with very large Scientific notation allows you to work with very large numbers and very small numbers.numbers and very small numbers.

A number like 5 000 000 000 would be 5.0 x 10 A number like 5 000 000 000 would be 5.0 x 10 9 9

5 > 1 and < 10 and there are 9 places past where the 5 > 1 and < 10 and there are 9 places past where the new decimal place would be.new decimal place would be.

Write the scientific notation for 67 000 000 000Write the scientific notation for 67 000 000 000

which is 6.7 x 10which is 6.7 x 101010

Scientific NotationScientific Notation

Which of the following Which of the following numbers are in scientific numbers are in scientific notation? If it is notation? If it is notnot, , explain why. explain why. 

a.    3 x 10a.    3 x 1066  

b.   4.6 x 10b.   4.6 x 10-4-4  

c.    0.2 x 10c.    0.2 x 1088

d.   2.8 x 100d.   2.8 x 100

AnswersAnswers

A) A) is correctis correct scientific notation as the scientific notation as the decimal number is less than 10 and equal decimal number is less than 10 and equal to or greater than 1. It is also written in a to or greater than 1. It is also written in a power of 10power of 10

B)B) is correctis correct for the same reasons as A for the same reasons as A

C)C) is not correctis not correct as the decimal number as the decimal number is less than 1is less than 1

D)D) is not correctis not correct as the decimal is as the decimal is correct but it is not written in a power of 10 correct but it is not written in a power of 10 . If it was changed to 10. If it was changed to 102 2 it would be it would be correctcorrect

Scientific NotationScientific Notation How do we represent very small numbers in scientific How do we represent very small numbers in scientific

notation?notation? Recall that 0.005 = 5 x Recall that 0.005 = 5 x 1 1 or 5 x or 5 x 11 1000 101000 1033

Therefore, armed with the above knowledge, our previous Therefore, armed with the above knowledge, our previous knowledge of negative exponents, and the knowledge of negative exponents, and the

definition for scientific notation,definition for scientific notation, 0.005 is represented in0.005 is represented in scientific scientific

notation notation as as 5. 0 x 10 5. 0 x 10 -3 .-3 .

Scientific NotationScientific Notation The calculator only holds 7 digits.The calculator only holds 7 digits. So if you have a number like 760 000 000 it So if you have a number like 760 000 000 it

would show it as 7.62 x 10 would show it as 7.62 x 10 88 . . If you have a positive exponent for 10, then the If you have a positive exponent for 10, then the

decimal place will move to the right and make decimal place will move to the right and make the number bigger such as the number bigger such as

3.45 x 103.45 x 105 5 = 345 000= 345 000 If you have a negative exponent , the decimal If you have a negative exponent , the decimal

point will move to the left and make the number point will move to the left and make the number smaller such as 3.45 x 10 smaller such as 3.45 x 10 –5 –5 = . 0000345= . 0000345

Scientific Notation TriviaScientific Notation Trivia 101066 million million 101099 billion billion 10101212 trillion trillion 101015 15 quadrillionquadrillion 101018 18 quintillionquintillion 10102121 sextillion sextillion 1010100100 googolgoogol

Scientific Notation TriviaScientific Notation Trivia The word The word googol googol was created in 1938 was created in 1938

by the 11 year old nephew of the by the 11 year old nephew of the American mathematician Edward American mathematician Edward Kasner.Kasner.

Operations with Scientific Operations with Scientific NotationNotation

Numbers can be written in standard form ( as a Numbers can be written in standard form ( as a number) and scientific notation.number) and scientific notation.

You can do operations such as add, subtract , You can do operations such as add, subtract , multiply and divide with scientific notation. multiply and divide with scientific notation.

Scientific notation allows you to solve more easily Scientific notation allows you to solve more easily with very large or very small numbers. with very large or very small numbers.

Remember that scientific notation is written in Remember that scientific notation is written in POWERS OF 10POWERS OF 10

Adding numbers with Adding numbers with scientific notation:scientific notation:

1.4 x 101.4 x 10-3-3 + 2.3 x 10 + 2.3 x 10-3-3 If the power of 10 is the same then you If the power of 10 is the same then you

can take out the power of 10 and then can take out the power of 10 and then ADD the other two factors. ADD the other two factors.

For example:For example: (1.4 + 2.3 ) x 10(1.4 + 2.3 ) x 10-3-3

3.7 x 103.7 x 10-3-3 written in standard form it written in standard form it would be 0.0037would be 0.0037

Addition with Scientific Addition with Scientific NotationNotation

If the numbers are not in the same power of 10, If the numbers are not in the same power of 10, then you might be able to rewrite the numbers then you might be able to rewrite the numbers so that they are in the same power of 10.so that they are in the same power of 10.

5.3 x 105.3 x 1044 + 6.2 x 10 + 6.2 x 10 55 could be rewritten as could be rewritten as 0.53 x 10 0.53 x 10 55 + 6.2 x 10 + 6.2 x 10 55 ( Now you can solve it as shown in previous slide)( Now you can solve it as shown in previous slide) (0.53 + 6.2) x 10 (0.53 + 6.2) x 10 55

6.73 x 10 6.73 x 10 55 written in standard form iswritten in standard form is 674000 674000

Subtraction with Subtraction with Scientific NotationScientific Notation

Subtraction is done the same way as Subtraction is done the same way as addition. Take out the power of 10 and addition. Take out the power of 10 and then subtract the other factors. then subtract the other factors.

7.3 x 107.3 x 1033 - 6.2 x 10 - 6.2 x 10 33 ( 7.3 – 6.2) x 10 ( 7.3 – 6.2) x 10 33 1.1 x 10 1.1 x 10 33 written in standard form it is written in standard form it is 1100 1100

Multiplying in Scientific Multiplying in Scientific NotationNotation

When you multiply or divide with scientific notation , When you multiply or divide with scientific notation , you will use your exponent laws. Remember , when we you will use your exponent laws. Remember , when we multiply powers with the same base , we add the multiply powers with the same base , we add the exponents:exponents:

(2.2 x 10(2.2 x 1044 ) x ( 1.2 x 10 ) x ( 1.2 x 10 77) ) Remember that the Remember that the Commutative PropertyCommutative Property allows us to rearrange the allows us to rearrange the FACTORSFACTORS withoutwithout

affecting the answer.affecting the answer. SO : 2.2 x 1.2 x 10SO : 2.2 x 1.2 x 1044 x 10 x 10 77

2.64 x 10 2.64 x 10 (4 + 7)(4 + 7) 2.64 x 10 2.64 x 10 1111 or 264 000 000 000 or 264 000 000 000

Dividing with Scientific Dividing with Scientific Notation Notation

12.4 x 10 12.4 x 10 1010 ÷ 3.2 x 10 ÷ 3.2 x 10 66

and and

3.875 x 10 3.875 x 10 (10 – 6)(10 – 6)

= 3.875 x 10= 3.875 x 1044 OR 38750 OR 38750 You need to divide the first factors and then You need to divide the first factors and then

apply the exponent laws to the powers of 10.apply the exponent laws to the powers of 10.

2.34.12

6

10

1010