Lab Skills and Chemical Analysis S3 UNIT 1 Lab Skills and Chemical Analysis Scientific Notation

Scientific Notation Context for learning: Scientific notation allows you to write and work with very large or very small numbers without having to write loads of zeros. Learning Intentions (WALT): We are learning to: Write really big and really small numbers in scientific notation. Complete calculations with numbers written in scientific notation Adding and subtracting Dividing and multiplying Extension Understand the zero power rule Success Criteria(WILF): What I’m looking for: Be able to convert easily from a number to scientific notation and vice versa. Be able to carry out calculations using scientific notation for real-life examples. Extension Be able to explain using mathematical examples why any number raised to the power of 0 is 1.

New words Meaning GLOSSARY Index (powers) (exponent) The index of a number says how many times to use the number in a multiplication. It is written as a small number to the right and above the base number. In this example: 82 = 8 × 8 = 64. The plural of index is indices. (Other names for index are exponent or power.) Scientific notation (standard form) Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 10-9. Coefficient A number used to multiply a variable. Decimal System So, our Decimal System lets us write numbers as large or as small as we want, using the decimal point. Digits can be placed to the left or right of a decimal point, to show values greater than one or less than one. The decimal point is the most important part of a Decimal Number.

Indices Indices are also called Powers or Exponents, they are used as a means of writing repeated calculations in a shorter way.

Scientific Notation Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 10-9.

Very large number Very small number Scientific Notation Very large number Very small number Mass of the sun is: 2000000000000000000000000000000 kg Mass of a proton: 0.000000000000000000000000001673kg

The mass of one gold atom is .000 000 000 000 000 000 000 327 grams. Scientific Notation is used to express the very large and the very small numbers so that problem solving will be made easier. Examples: The mass of one gold atom is .000 000 000 000 000 000 000 327 grams. One gram of hydrogen contains 602 000 000 000 000 000 000 000 hydrogen atoms. Scientists can work with very large and very small numbers more easily if the numbers are written in scientific notation.

How to Use Scientific Notation In scientific notation, a number is written as the product of two numbers….. …..a coefficient and 10 raised to a power.

For example: 4.5 x 103 The coefficient is _________. 4.5 The number 4,500 is written in scientific notation as ________________. The coefficient is _________. 4.5 The coefficient must be a number greater than or equal to 1 and smaller than 10. The power of 10 or exponent in this example is ______. 3 The exponent indicates how many times the coefficient must be multiplied by 10 to equal the original number of 4,500.

Very large numbers! BOOM! 92,000,000 miles How far?

Very large numbers! SPLAT! How many seconds in 70 years? Happy 70th Birthday! How many seconds in 70 years? SPLAT! 70 years = 2 200 000 000 seconds!

Dinosaurs roamed the earth 228 million years ago Very large numbers! Dinosaurs roamed the earth 228 million years ago

Short hand! 10 100 = 10 x 10 1 000 = 10 x 10 x 10 10 000 = 10 x 10 x 10 x 10 100 000 = 10 x 10 x 10 x 10 x 10 1 000 000 = 10 x 10 x 10 x 10 x 10 x 10

Scientific Notation This is also known as Standard form. 200 = 2 x 100

Exercise 1 2 000 (2) 20 000 (3) 500 (4) 800 000 (5) 9 000 000 = 2 x 1000 = 2 x 10 000 = 5 x 100 = 8 x 100 000 = 9 x 1 000 000

A short cut 8 000 000 . . Move the point to get a number between 1 and 10 The point moved 6 places 8 000 000 . . Add the decimal point Between 1 and 10

A short cut 92 000 000 . . Move the point to get a number between 1 and 10 The point moved 7 places 92 000 000 . . Add the decimal point Between 1 and 10

Happy Birthday: Seconds old! A short cut Move the point to get a number between 1 and 10 The point moved 9 places 2 200 000 000 . . Add the decimal point Between 1 and 10 Happy Birthday: Seconds old! SPLAT!

A short cut 228 000 000 . . Move the point to get a number between 1 and 10 The point moved 8 places 228 000 000 . . Add the decimal point Between 1 and 10

Exercise 2 (1) 30 000 (2) 700 000 (3) 5 300 (4) 470 000 (5) 9 500 000 (6) 18 300 000 (7) 329 000 (8) 2 560 000 (9) 12 000 000 000 (10) 9 990 000

Add 7 zeros, although you probably won’t need them all. Changing back The point moves 7 places 8.6 0000000 . =86 000 000 Hint: Add 7 zeros, although you probably won’t need them all. Zeros after the point aren’t needed.

Add 5 zeros, although you probably won’t need them all. Changing back The point moves 5 places 3.46 00000 . =346 000 Zeros after the point aren’t needed. Hint: Add 5 zeros, although you probably won’t need them all.

Exercise 3 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) = 600 000 = 8 000 = 650 000 = 120 000 000 = 3 710 000 = 33 000 = 7 910 000 = 55 500 000 = 10 500 = 3 033 000 000

Name that number! 1 000 What’s this called? one thousand 1 000 000 What’s this called? one million 1 000 000 000 and this one? one billion 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 one googol !!! What’s this called?

Very small numbers! How wide is an atom? 0.000 000 000 1 metres wide!

Standard Form for small numbers Move the point to get a number between 1 and 10 0.000 000 000 1 . The point moved 10 places. Negative sign for small numbers.

Standard Form for small numbers Move the point to get a number between 1 and 10 0.000 000 76 . The point moved 7 places. Negative sign for small numbers.

Standard Form for small numbers Move the point to get a number between 1 and 10 0.000 001 93 . The point moved 6 places. Negative sign for small numbers.

Exercise 4 (1) 0.0003 (2) 0.00007 (3) 0.00045 (4) 0.0034 (5) 0. 000724 (6) 0.000000 494 (7) 0.000095 (8) 0.000000 098 (9) 0.000103 (10) 0.00000000066

Changing back small numbers Add 3 zeros to the left of the number. -3 so remember to move point left for small numbers. Hint: Add 3 zeros to the left of the number. 000 . 2 . = 0.002

Changing back small numbers Add 5 zeros to the left of the number. -3 so remember to move point left for small numbers. Hint: Add 5 zeros to the left of the number. 00000 . 8.6 = 0.000 086

Changing back small numbers Add 6 zeros to the left of the number. -3 so remember to move point left for small numbers. Hint: Add 6 zeros to the left of the number. 000000 . 5.16 = 0.000 00516

Exercise 5 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) = 0.0005 = 0.00009 = 0.0058 = 0.00000062 = 0.00000645 = 0.00000053 = 0.00000917 = 0.00212 = 0.0000203 = 0.000000006032

Rules to Remember! If a number is greater than 10, the exponent will be _____________ and is equal to the number of places the decimal must be moved to the ________ to write the number in scientific notation. positive left If a number is less than 10, the exponent will be _____________ and is equal to the number of places the decimal must be moved to the ________ to write the number in scientific notation. negative right

A number will have an exponent of zero if: ….the number is equal to or greater than 1, but less than 10.

1. Move the decimal to the right of the first non-zero number. To write a number in scientific notation: 1. Move the decimal to the right of the first non-zero number. 2. Count how many places the decimal had to be moved. 3. If the decimal had to be moved to the right, the exponent is negative. 4. If the decimal had to be moved to the left, the exponent is positive.

Practice Problems ANSWERS PROBLEMS 1.2 x 10-4 1 x 103 1 x 10-2 Put these numbers in scientific notation. PROBLEMS 1.2 x 10-4 1 x 103 1 x 10-2 1.2 x 101 9.87 x 10-1 5.96 x 102 7.0 x 10-7 1.0 x 106 1.26 x 10-3 9.88 x 1011 8 x 100 ANSWERS .00012 1000 0.01 12 .987 596 .000 000 7 1,000,000 .001257 987,653,000,000 8

EXPRESS THE FOLLOWING AS WHOLE NUMBERS OR AS DECIMALS PROBLEMS ANSWERS 4.9 X 102 3.75 X 10-2 5.95 X 10-4 9.46 X 103 3.87 X 101 7.10 X 100 8.2 X 10-5 490 .0375 .000595 9460 38.7 7.10 .000082

In the United States, 15,000,000 households use private wells for their water supply. Write this number in scientific notation. 1.5 X 107

The U.S. has a total of 1.2916 X 107 acres of land reserved for state parks. Write this in standard form. 12,916,000 acres

.000007 meters The nucleus of a human cell is about 7 X 10-6 meters in diameter. What is the length in Scientific notation? .000007 meters

A ribosome, another part of a cell, is about 0 A ribosome, another part of a cell, is about 0.000000003 of a meter in diameter. Write the length in scientific notation. 3 X 10-9

Using Scientific Notation in Multiplication, Division, Addition and Subtraction Scientists must be able to use very large and very small numbers in mathematical calculations. As a student in this class, you will have to be able to multiply, divide, add and subtract numbers that are written in scientific notation. Here are the rules.

Sample Problem: Multiply (3.2 x 10-3) (2.1 x 105) Multiplication When multiplying numbers written in scientific notation…..multiply the coefficients and add the exponents. Sample Problem: Multiply (3.2 x 10-3) (2.1 x 105) Solution: Multiply 3.2 x 2.1. Add the exponents -3 + 5 Answer: 6.7 x 102

Sample Problem: Divide (6.4 x 106) by (1.7 x 102) Division Divide the numerator by the denominator. Subtract the exponent in the denominator from the exponent in the numerator. Sample Problem: Divide (6.4 x 106) by (1.7 x 102) Solution: Divide 6.4 by 1.7. Subtract the exponents 6 - 2 Answer: 3.8 x 104

Multiplication and Division

Addition and Subtraction To add or subtract numbers written in scientific notation, you must….express them with the same power of ten. Sample Problem: Add (5.8 x 103) and (2.16 x 104) Solution: Since the two numbers are not expressed as the same power of ten, one of the numbers will have to be rewritten in the same power of ten as the other. 5.8 x 103 = .58 x 104 so .58 x 104 + 2.16 x 104 =? Answer: 2.74 x 104

Addition and Subtraction

Scientific notation and your calculator

Kahoot Scientific Notation