1. Slide 1 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 1 Introductory Mathematics & Statistics for Business 4th Edition John S. Croucher

2. Slide 2 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 2 Basic mathematics n Learning Objectives Carry out calculations involving whole numbers Carry out calculations involving fractions Carry out calculations involving decimals Carry out calculations involving exponents Use and understand scientific notation Use and understand logarithms Chapter M1

3. Slide 3 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 3 Whole numbers n The decimal system –Numerals symbols i.e. 0, 1, 2, 3  are numerals represent natural numbers or whole numbers used to count whole objects or fractions of them –Integer is another name for a whole number –Digits numerals consist of one or more digits

4. Slide 4 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 4 Mathematical operations n Four basic mathematical operations performed on numbers –multiplication represented by: x –division represented by: –addition represented by: + –subtraction represented by: -

5. Slide 5 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 5 Rules for mathematical operations n Order of operations: Multiplication and division BEFORE Addition and subtraction

6. Slide 6 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 6 Rules for mathematical operations n Multiplication and division –same signs give positive result –different signs give negative result –perform calculations in brackets first

7. Slide 7 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 7 Rules for mathematical operations n Addition –like signs—use the sign and add –unlike signs—use sign of greater and subtract n Subtraction Two signs next to each other –minus a minus is a plus  -(-3)=3 –minus a plus is a minus  -(+3)=-3

8. Slide 8 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 8 Fractions n A fraction appears as: –Proper fraction –Proper fraction—numerator less than denominator –Improper fraction –Improper fraction—numerator greater than denominator

9. Slide 9 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 9 Addition & subtraction of fractions n Different denominators denominatorslowest common multiple –change denominators to lowest common multiple –LCM –LCM is the smallest number into which all denominators will divide

10. Slide 10 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 10 Multiplication & division of fractions numerators –Multiply numerators to get new numerator denominators –Multiply denominators to get new denominator –Cancel common factors of nominators and numerators by multiplying

11. Slide 11 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 11 Decimals n Any fractions can be expressed as a decimal by dividing the numerator by the denominator. n A decimal consists of three components: an integer a decimal point another integer.

12. Slide 12 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 12 Rules for decimals n Addition and subtraction –Align the numbers so that the decimal points are directly underneath each other.

13. Slide 13 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 13 Rules for decimals n Multiplication and division 1.Count the number of digits to the right of each decimal point for each number. 2.Add the number of digits in Step 1 to obtain a number, say x. 3.Multiply the two original decimals, ignoring decimal points. 4.Mark the decimal point in the answer to Step 3 so that there are x digits to the right of the decimal point.

14. Slide 14 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 14 Exponents n An exponent or power of a number is written as a superscript to a number called the base. n The base number is said to be in exponential form. n Exponential form—a n »where a is the base »where n is the exponent or power

15. Slide 15 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 15 Rules for exponents n Positive exponents Two numbers with same base—a n & a m The product will have the same base; the exponent will be the sum of the two original exponents—a n x a m = a n+m The quotient of the two numbers will have the same base; the exponent will be the difference between the original exponents—a n a m = a n-m

16. Slide 16 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 16 Rules for exponents n Positive exponents –A number in exponential form is raised to another exponent. The result is the original base raised to the product of the exponents. (a n ) m = a nm n Negative exponents –A number expressed with a negative exponent is equal to the reciprocal of the same number with the negative sign removed.

17. Slide 17 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 17 Rules for exponents n Fractional exponents –Exponents can be expressed as a fraction where k is an integer and is said to be the k th root of a when k =2 it is the square root; k =3 is the cube root

18. Slide 18 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 18 Rules for exponents n Scientific notation –Scientific notation is a shorthand way of writing very large and very small numbers. –Scientific notation expresses the number as a numeral (less than 10) multiplied by the base number 10 raised to an exponent. –The reference position for the decimal point in a number is immediately to the right of the first non-zero digit.

19. Slide 19 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 19 Logarithms n Logarithms are closely connected to the theory of exponents. n Calculations using logarithms have been replaced by calculators since the 1970s. n An understanding of logarithms can be useful in statistics, physics, engineering etc. The logarithm of a number N to a base b is the power to which b must be raised to obtain N. log b N That is, if x = log b N, then N = b x