Scientific Notation

1. Scientific Notation

2. Scientific Notation • Do you know this number, 300,000,000 m/sec.? • It's the Speed of light ! • Do you recognize this number, 0.000 000 000 753 kg. • This is the mass of a dust particle!

3. Scientists have developed a shorter method to express very large numbers or very small numbers. • This method is called scientific notation. • The number 123,000,000,000 in scientific notation is written as :

4. Scientific Notation • The first number 1.23 is called the base. It must be greater than or equal to 1 and less than 10. • The second number is written in exponent form or 10 to some power. • The exponent is the number of decimal places needed to arrive at the bass number.

5. To write a number in scientific notation: • Put the decimal after the first digit and drop the zeroes. • This gives you the base number. • In the number 123,000,000,000 The base number will be 1.23 • To find the exponent count the number of places from the decimal to the end of the number. • In 123,000,000,000 there are 11 places

6. Multiplying Scientific Notated Numbers • Multiply the base numbers • Add the exponents of the Tens • Adjust the base number to have one digit before the decimal point by raising or lowering the exponent of the Ten + 3.25 X 10 3 X 2.50 x10 5 = 3.25 X 2.50 3 + 5= 8 8.125 X 10 8

7. Dividing Scientific Notation Numbers • Divide the base numbers • Subtract the exponents of the Tens • Adjust the base number to have one digit before the decimal point by raising or lowering the exponent of the Ten

8. Dividing • Divide 3.5 x 108 by 6.6 x 104 • You may rewrite the problem as: • 3.5 x 108 6.6 x 104 • Now divide the two base numbers • Subtract the two powers of 10 • Adjust base number to have one number before the decimal

9. 3.5 x 108 6.6 x 104 4 is now subtracted from 8 3.5 is now divided by 6.6 in this order on the calculator. 0.530303 x 104 Change to correct scientific notation to get: 5.3 x 103Note - We subtract one from the exponent because we moved the decimal one place to the right.

10. Scientific Notation - Addition and Subtraction • All exponents MUST BE THE SAME before you can add and subtract numbers in scientific notation. The actual addition or subtraction will take place with the numerical portion, NOTthe exponent. • You must change the base number on one of the digits by moving the decimal. • Always make the powers of ten the same as the largest. • Move the decimal on the smallest number until its power of ten matches that of the largest exponent.

11. Ex. 1  Add 3.76 x 104 and 5.5 x 102 • move the decimal to change 5.5 x 102 to 0.055 x 104 • add the base numbers and leave the exponent the same:  3.76 + 0.055 = 3.815 x 104 • following the rules for rounding, our final answer is 3.815 x 104 • Subtraction is done exactly in the same manor.

12. Dimensional Analysis • Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is a useful technique. • Unit factors may be made from any two terms that describe the same or equivalent "amounts" of what we are interested in. • For example, we know that • 1 inch = 2.54 centimeters

13. Unit Factors • We can make two unit factors from this information:

14. When converting any unit to another there is a pattern which can be used. • Begin with what you are given and always multiply it in the following manner. • Given units X =Want units • You will always be able to find a relationship between your two units. • Fill in the numbers for each unit in the relationship. • Do your math from left to right, top to bottom. Want units Given units

15. Given units X = Want units Want Units Given Units • (1) How many centimeters are in 6.00 inches?

16. Metric System of Measurement System International Or International System of Measurement Based on units of ten

17. Basic Units of Measure • Length – Distance : meter (metre) m • Time – second s • Mass – grams g or kilograms kg • Volume – liter (litre) l 1cc=1cm3=1ml 1dm3=1liter (l) • Temperature – Celsius C or • Kelvin K = C + 273

18. Metric Prefixes

19. Conversion in the Metric System • If you can remember something silly,  • ("King Henry Died Monday Drinking Chocolate Milk"),  the metric conversions are so easy. • King Henry Died Monday Drinking Chocolate Milk • (km) (hm) (dam) (m/unit) (dm) (cm) (mm) • Remember the 1st letter is the symbol for the prefix and the second is the unit you are measuring in. • Just sketch the  chart above (K, H, D, M, D, C, M) and place the number you wish to convert under the proper slot. • Move the decimal point left or right the correct number of places to make the conversion.

20. Example: convert 43.1 cm to km. • King Henry  Died  Monday Drinking Chocolate Milk • (km) (hm) (dam) (m/unit) (dm) (cm) (mm) • 43.1 • This is a move of 5 places to the left filling spaces with zeros and you get • .000431 km Example:convert 43.1 dm to mm. • King Henry  Died  Monday Drinking Chocolate Milk • (km) (hm) (dam) (m/unit) (dm) (cm) (mm) • 43.1 • This is a move of 2 places to the right filling spaces with • zeros and you get • 4310 mm

21. Significant Digits or Figures • Significant digits, which are also called significant figures. Each recorded measurement has a certain number of significant digits. • The significance of a digit has to due with whether it represents a true measurement or not. • Any digit that is actually measured or estimated will be considered significant

22. Rules For Significant Digits • Digits from 1-9 are always significant. • Zeros between two other significant digits or counting numbers are always significant. • One or more zeros to the right of both the decimal place and another significant digit are significant.

23. Significant Digit Examples • All counting numbers: 1,238 there are 4 significant digits in this number. • Zero’s between counting number: 123,005 in this number there are 6 significant digits • Zero’s to the right of the decimal and the right or end of the number count as significant digits: 123.340 in this number there are 6 significant digits

24. Significant Digit Rules For Multiplication and Division Answers • Your final answer will have the same number of digits as that with the least number of significant digits in the problem. • Ex: 2.43 X 35 = 85.05 since in the problem the least number of significant digits is two then your answer will be 85 • Do not forget to round up or leave where needed.

25. Significant Digit Rules for Subtraction and Addition • The correct number of digits in the final answer will be the same as the least number of decimal places in the problem. • Ex: 123.45 12.456 +1045.4 1181.306 since the least number of decimal places is one then the final answer is 1181.3 3. Remember to check for round off or not.