Scientific Notation

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.

100 = 1

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

100 = 1 101 = 10

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

100 = 1 101 = 10 102 = 100

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

100 = 1 101 = 10 102 = 100 103 = 1000

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

Note:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

Note:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

Note:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific NotationAn important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

Note:

The number of “0’s” in the front matches the exponent.

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific Notation

Scientific Notation

An important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

Note:

The number of “0’s” in the front matches the exponent.

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific Notation

Scientific NotationAny number can be written in the form A x 10N

where 1 < A < 10.

An important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

Note:

The number of “0’s” in the front matches the exponent.

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific Notation

Scientific NotationAny number can be written in the form A x 10N

where 1 < A < 10. This form is called the scientific notation of the number.

An important application for exponents is the use of the powers of 10 in calculation of very large or very small numbers.Powers of 10:

The number of “0’s” is the exponent.

The number of “0’s” in the front matches the exponent.

Note:

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N,

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Example A. Write the following numbers in scientific notation.a. 12300.

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Move left 4 places.

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 .

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Move left 4 places.

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative,

Move left 4 places.

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

b. 0.00123

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Move right 3 places

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

b. 0.00123 = 0. 001 23

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Move right 3 places

Example A. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

b. 0.00123 = 0. 001 23 = 1. 23 x 10 –3

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right,

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.

Move right 4 places,

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left,

Move right 4 places,

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

b. 1. 23 x 10 –3

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Move left 3 places

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Move left 3 places

Example B. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123

Scientific notation simplifies multiplication and division of very large and very small numbers.

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12)

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

= 6.32.1

x 10 – 2 – ( – 10)

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

= 6.32.1

x 10 – 2 – ( – 10)

= 3 x 108

Scientific Notation

Example C. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

= 6.32.1

x 10 – 2 – ( – 10)

= 3 x 108 = 300,000,000

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025 0.00015

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4 = 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.51.5

x 10 8 + (–6) – ( – 4)

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.51.5

x 10 8 + (–6) – ( – 4)

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

= 4 x 108 – 6 + 4

Scientific Notation

Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.51.5

x 10 8 + (–6) – ( – 4)

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

= 4 x 108 – 6 + 4

= 4 x 106 = 4,000,000

Scientific Notation