Scientific Notation Scientific Notation - A way of succinctly writing very large or very small numbers Always has one non-zero digit in front of the decimal point Requires moving decimal points and using powers of ten Exponent rules make calculations easier To Put a Number into Scientific Notation Move the decimal point until it is behind exactly one non-zero digit o When moving the decimal to the left ( ), the exponent will be positive Moving the decimal to the left makes the number smaller. We’ll need to multiply by a large number to maintain equality o When moving the decimal to the right ( ), the exponent will be negative Moving the decimal to the right makes the number bigger. We’ll need to multiply by a small number to maintain equality Only write non-zero digits, followed by x10 raised to the power of however many places you moved the decimal point, with the sign as indicated by the direction of movement 9,203,400,000,000 = 9.2034 x 1012 To Take a Number out of Scientific Notation Undo the steps above, working backwards Look at the exponent on the 10 o If the exponent is positive, it means the decimal was moved to the left. To undo this, move it to the right however many places are indicated by the number itself o If the exponent is negative, it means the decimal was moved to the right. To undo this, move it to the left however many places are indicated by the number itself Fill in zeros as necessary to maintain the correct placement of the decimal point 8.125 x 10-10 = 0.000 000 000 812 5 Powers of ten 1 10 100 1 , 000 10 , 000 100 , 000 1 , 000 , 000 1 , 000 , 000 , 000 1 , 000 , 000 , 000 , 000 1 , 000 , 000 , 000 , 000 , 000 - - = = = = = = = = = = 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 9 10 12 10 15 10 0 = 10 -1 = 10 -2 = 10 -3 = 10 -4 = 10 -5 = 10 -6 = 10 -9 = 10 -12 = 10 -15 = 1 0. 0. 0. 0. 0. 0. 0. 0. 0. 1 01 001 000 1 000 01 000 001 000 000 001 000 000 000 001 000 000 000 000 001 Positive powers of ten represent how many zeros are behind the 1 o 105 means 1 with 5 zeros behind it, or 100,000 Negative powers of ten, because the decimal also needs to be moved behind the 1 itself, contain one less zero than indicated by the power o 10-5 means 1 with a decimal and four zeros in front of it, or 0.00001 Always verify by counting the number of places the decimal has moved, not simply zeros 1 Scientific Notation Practice Put 7,000 into scientific notation - If there is no decimal point written, it’s behind the ones digit - Count the number of places the decimal point must be moved so there is exactly one nonzero digit in front of it 7000. - Three places to the left means x 103 7 x 103 Put 0.00000524 into scientific notation - Count the number of places the decimal point must be moved so there is exactly one nonzero digit in front of it 0.00000524 - Six places to the right means x 10-6 5.24 x 10-6 Take 3.01 x 104 out of scientific notation - Look at the exponent on the 10 Positive 4 means the decimal had been moved four places to the left. Undo this by moving it four places to the right 3.01 - Fill in zeros as needed to maintain the decimal in this location 30100 Take 8.74 x 10-2 out of scientific notation - Look at the exponent on the 10 Negative 2 means the decimal had been moved two places to the right. Undo this by moving it two places to the left 8.74 - Fill in zeros as needed to maintain the decimal in this location 0.0874 Error analysis Put 98.1 x 103 into proper scientific notation - There can only be one digit in front of the decimal point - Because we’re moving the decimal point to the left, we increase the exponent 9.81 x 104 Put 0.035 x 10-6 into proper scientific notation - There needs to be one non-zero digit in front of the decimal point - Because we’re moving the decimal point to the right, we decrease the exponent 3.5 x 10-8 2 Scientific Notation Multiplying and Dividing in Scientific Notation Multiply (or divide) the digits Multiply (or divide) the powers of ten using the rules of exponents o Add the exponents when multiplying o Subtract the exponents when dividing Combine the expressions, recombining the powers of ten o Recombine with multiplication, regardless of what operation was used in the problem Practice Multiply 7 x 103 by 5.24 x 10-6 Multiply the digits - 5.24 x 7 = 36.68 - Move the decimal point so there is one digit in front of the decimal point - 3.668 x 101 Multiply the powers of ten using the rules of exponents (add the exponents) - 103 x 10-6 = 103 + -6 = 10-3 Combine the expressions, multiplying the powers of ten - 3.668 x 101 x 10-3 - 3.668 x 10-2 o This can be done all at once as follows (7 x 103) x (5.24 x 10-6) (7 x 5.24) x (103 x 10-6) 36.68 x 10-3 3.668 x 10-2 Divide 6.48 x 10-4 by 8 x 106 Divide the digits - 6.48 ÷ 8 = 0.81 - Move the decimal point so there is one digit in front of the decimal point - 8.1 x 10-1 Divide the powers of ten using the rules of exponents (subtract the exponents) - 10-4 ÷ 106 = 10-4 - 6 = 10-10 - Combine the expressions, multiplying the powers of ten - 8.1 x 10-1 x 10-10 - 8.1 x 10-11 o This can be done all at once as follows (6.48 ÷ 10-4) x (8 x 106) (6.48 ÷ 8) x (10-4 ÷ 106) 0.81 x 10-10 8.1 x 10-11 We can verify these longhand - 7000 x 0.00000524 = 0.03668 = 3.668 x 10-2 0.000648 ÷ 8,000,000 = 0.000 000 000 081 = 8.1 x 10-11 3