Scientific Notation

Extremely Large and Small Numbers Large and small numbers are prevalent in our world: In 2011 the Use National Debt was: $15,143,572,733,215 The mass of a dust particle is approximately: 0.000000000753 kg Standard Form or Decimal Form Scientists have created an easier way to express these extreme numbers.

Introduction to Scientific Notation: Powers of Ten Decimal form Powers of 10 Name 1,000,000,000 109 One billion 1,000,000 106 One million 1,000 103 One thousand 10 101 Ten 1 100 One 0.1 10-1 One tenth 0.001 10-3 One thousandth 0.000001 10-6 One millionth Scientific Notation is based on Powers of Ten.

Definition of Scientific Notation Numbers in Scientific Notation are written as: c x 10n where c is a decimal number greater than or equal to 1 and less than 10 (Typically c is rounded to 2 or 3 decimal places) and n is an integer. Below are two examples: Example: Which expression is written in correct Scientific Notation? 3.9 is the only number greater than or equal to 1 and less than 10 AND the exponent is an integer

Converting Standard Form to Scientific Notation By converting the power of 10 to Standard Form Notation, the true value of the Scientific Notation number can be obtained. Example: Convert the following from Standard Form to Scientific Notation . Then try to find a shortcut. 1.2 0 0 0 0. The positive 5 makes the number bigger, by moving the decimal 5 places to the right. 1 2 3 4 5 0 .0 0 0 3.5 The negative 4 makes the number smaller, by moving the decimal 4 places to the left. 4 3 2 1

Examples Rewrite each Scientific Notation in Standard Form without a calculator. The + 8 moves the decimal 8 places to the right (bigger). 1.3 4 5 0 0 0 0 0. 1 2 3 4 5 6 7 8 The – 9 moves the decimal 9 places to the left (smaller). 0 .0 0 0 0 0 0 0 0 4.103 9 8 7 6 5 4 3 2 1 The – 5 moves the decimal 5 places to the left (smaller). 0 .0 0 0 0 7.8 5 4 3 2 1 The + 6 moves the decimal 6 places to the right (bigger). 3.6 7 0 0 0 0. 1 2 3 4 5 6

Converting Scientific Notation to Standard Form To convert Standard Form to Scientific Notation, find the largest power of ten such that when it divides the number, the result is between 1 and 10. Example: Convert the following from Scientific Notation to Standard Form. Then try to find a shortcut. 1013 Round to 3 decimal places 15143572733215. . The number is made smaller by moving the decimal 13 places (power of ten needs is positive) 13 12 11 10 9 8 7 6 5 4 3 2 1 10-10 0.000000000753 . The number is made larger by moving the decimal 10 places (power of ten needs is negative) 1 2 3 4 5 6 7 8 9 10

Examples Rewrite each Standard Form in Scientific Notation without a calculator. Move the decimal + 5 places to the left (positive since the number is made smaller). 2 4 5 0 0 0. . 5 4 3 2 1 Already in Scientific Notation, it is a number greater than or equal to 1 and less than 10 Move the decimal - 4 places to the right (negative since the number is made bigger). 0 .0 0 0 7 2 2 . 1 2 3 4 Move the decimal + 7 places to the left (positive since the number is made smaller). 1 7 4 3 9 5 0 2. . 7 6 5 4 3 2 1

Arithmetic with Scientific Notation In order to multiply or divide numbers in Scientific Notation, use the properties of exponents. Examples: Simplify the following expressions Scientific Notation Regroup Simplify Scientific Notation Regroup Simplify

Scientific Notation and the Calculator Most scientific calculator write in large and small numbers in Scientific Notation. In TI calculators the letter E stands for x10, for example 1.5 E12 = 1.5x1012 Make sure to rewrite what the calculator displays into correct scientific notation. Examples: Use the calculator to write the following expressions in Scientific Notation. Calculator: Calculator: