ENMATH11E-2024-2025-1st-Sem-HANDOUTS-1-NUMBERS-AND-OPERATIONS-Copy.pptx

operations on numbers

1. LECTURE ONE: NUMBERS AND OPERATION ENMATH 11 E

2. TOPICS 1.) Numbers 2.) Field Axioms of Real Numbers 3.) Operations of Integers 4.) Fractions 5.) Exponents 6.) Radicals 7.) Order of Operations (GEMA/GEMS) Learning Outcomes/Objectives: By the end of this lecture, students must be able to operate and simplify numerical expressions

3. NUMBERS A number is an arithmetic value used for representing the quantity and used in making calculations. HOW DO WE CLASSIFY NUMBERS?

4. NUMBERS REAL NUMBERS natural numbers (or "whole" numbers), negative numbers, integers, fractions, decimals, square roots and special numbers like π. RATIONAL NUMBERS – can be expressed as includes terminal decimal or repeating decimal IRRATIONAL NUMBERS – cannot be expressed as   IMAGINARY NUMBERS Defined as the square root of a negative number Written in a form of real numbers multiplied by the unit called “ i “ Example: 5i, 10i

5. NUMBERS NATURAL NUMBERS - “counting numbers” Includes all the positive integers from 1 till infinity Does not include “zero” WHOLE NUMBERS Includes all natural numbers and “zero” “All natural numbers are whole number but not all whole numbers are natural numbers”

6. NUMBERS NEGATIVE NUMBERS Any number whose value is less than “zero” Example:   INTEGERS Defined as: The negative numbers 0; and The positive numbers

7. FIELD AXIOMS OF REAL NUMBERS 1. CLOSURE AXIOMS For each pair of real numbers a and b , there is a unique sum a + b , and a unique product, a  b both of which are also real numbers For every a, b  R Addition : Multiplication :   EXAMPLE: Let a = 2 ; b = 3 Addition : ; where 5 is a real number Multiplication : ; where 6 is a real number  

8. FIELD AXIOMS OF REAL NUMBERS 2. COMMUTATIVE RULE The sum or the product of any two real numbers a and b is not affected by the order in which these numbers are added or multiplied For every a, b  R Addition : Multiplication :   EXAMPLE: Let a = 2 ; b = 3 Addition : Multiplication :  

9. FIELD AXIOMS OF REAL NUMBERS 3. ASSOCIATIVE RULE The sum or the product of any triple real numbers a , b and c is not affected by the manner in which these numbers are group for addition or multiplication For every a, b, c  R Addition : Multiplication : (   EXAMPLE: Let a = 2 ; b = 3 ; c = 4 Addition : Multiplication :  

10. FIELD AXIOMS OF REAL NUMBERS 4. DISTRIBUTIVE RULE Multiplication is distributive over addition. This axiom changes the product of two factors into a sum of two terms For every a, b, c  R   EXAMPLE: Let a = 2 ; b = 3 ; c = 4  

11. FIELD AXIOMS OF REAL NUMBERS 5. IDENTITY ELEMENT When a binary operation is done on any pair of real numbers taken in the order a first, b second, and the operation produces the first element a , then the second element, b , is called identity element with respect to the operation done. For every a  R Addition : ; What is the value of b? Multiplication: ; What is the value of b?          

12. FIELD AXIOMS OF REAL NUMBERS 6. INVERSE ELEMENT If a binary operation is applied on any pair of real numbers taken in the order a first, b second, and the operation produces the identity element, then second element, b , is called the inverse of the first number, a, with respect to the operation done Addition: For each real number a , there exists a unique real number denoted by the symbol such that ; What is the value of b?      

13. FIELD AXIOMS OF REAL NUMBERS 6. INVERSE ELEMENT If a binary operation is applied on any pair of real numbers taken in the order a first, b second, and the operation produces the identity element then second element, b , is called the inverse of the first number, a, with respect to the operation done Multiplication: For each real number a, except zero, there exist a unique real number denoted by the symbol b such that ; What is the value of b?      

14. OPERATIONS OF INTEGERS ADDITION SAME SIGN Add the numbers Copy the sign DIFFERENT SIGNS Subtract the numbers Copy the sign of the larger number SUBTRACTION Change the sign of the subtrahend Use the addition rule for integers MULTIPLICATION SAME SIGN Product is positive DIFFERENT SIGNS 1. Product is negative DIVISION SAME SIGN Quotient is positive DIFFERENT SIGNS 1. Quotient is negative

15. OPERATIONS OF INTEGERS ADDITION SUBTRACTION MULTIPLICATION DIVISION Note: Sometimes, we also get a remainder. When the dividend is not completely divided by the divisor, the leftover value is called “remainder.”

16. FRACTION used to represent the portion/part of the whole thing fractions represent the division of a quantity; usually referred to as a ratio of two numbers, or sometimes a quotient.   TYPE OF FRACTIONS UNIT FRACTION – In a fraction, the numerator with 1 is called a unit fraction. For example, ½, ¼ PROPER FRACTION – If a numerator value is less than the denominator value, it is called a proper fraction. Example: 7/9, 8/10 IMPROPER FRACTION – If a numerator value is greater than the denominator value, then it is called an improper fraction. Example: 6/5, 11/10 MIXED FRACTION – If a fraction consists of a whole number with a proper fraction, it is called a mixed fraction. Example 5 ¾, 10 ½

17. FRACTION used to represent the portion/part of the whole thing fractions represent the division of a quantity; usually referred to as a ratio of two numbers, or sometimes a quotient. Additional Cases of Fractions If the numerator is zero: A fraction with a numerator of 0 (zero) will be equal to zero. Example: If the denominator is zero: A fraction is said to be undefined (or have no meaning) when the denominator = 0. Example: If both the numerator and denominator is zero: A fraction with is called an indeterminate form . The term “indeterminate” means an unknown value. (To be discussed further in ENDCAL13E)  

18. OPERATIONS OF FRACTION ADDING AND SUBRACTING FRACTIONS 1. FRACTIONS WITH LIKE DENOMINATORS A. add or subtract the numerators B. write the result over the same denominator 2. FRACTIONS WITH UNLIKE DENOMINATORS A. simplify them by finding the LCM (least common multiple) EXAMPLES:  

19. EXAMPLE: 1.)   2.)   3.)  

20. OPERATIONS OF FRACTION MULTIPLICATION OF FRACTIONS 1. MULTIPLICATION OF FRACTIONS (two or more) Multiply the numerator Multiply the denominator Simplify MULTIPLICATION OF FRACTIONS WITH MIXED NUMBER Rewrite the mixed fraction as improper fraction Multiply the numerator Multiply the denominator Simplify EXAMPLES:  

21. EXAMPLE: 1.)   2.)  

22. OPERATIONS OF FRACTION DIVISION OF FRACTIONS Solve for the reciprocal of the divisor Multiply the dividend by the reciprocal of the divisor Simplify EXAMPLES:  

23. REVIEW: RECIPROCALS 1.)   2.)   3.)   4.)  

24. EXAMPLE: 1.)   2.)   Hint: Try to reduce everything to its lowest form first before performing the operations

25. EXPONENTS The exponent of a number says how many times to use the number in a multiplication Sometimes refer to as index or power   BASE EXPONENT/INDEX /POWER Read as “3 to the power of 2” or “3 squared” or “3 to the second power” Means “multiply 3 by itself 2 times” Example:   REMEMBER: Any number a, (except 0) raise to the power 1 is a Any number a, (except 0) raise to the power 0 is 1 Any number a, (except 0) raise to the power -1 is  

26. EXPONENTS Example: 1.) = = 9 2.) = = -9 3.) = 1  

27. LAW OF EXPONENTS Example:   LAW OF EXPONENTS LAW OF EXPONENTS any base (positive or negative) raised to an even exponent = positive (+) while negative base raised to odd exponent = negative (-)  

28. LAW OF EXPONENTS Example: 1.) = = = = 216   2.) = = = =  

29. RADICALS Also called “fractional exponent” Involves the use of radical sign , sometimes these are called “SURDS”     Read as “square root” of 4     Read as “cube root” of 4 Read as “fourth root” of 4 GENERAL RULE     index radicand radical sign radical

30. LAW OF RADICALS Example:   LAW OF RADICALS LAW OF RADICALS

31. LAW OF RADICALS Example: 1.) = = =   3.) = = OR   4.) = = =   2.) = =  

32. ADDITION AND SUBTRACTION OF POWERS AND RADICALS Example:   RULE FOR POWERS : Combine terms having the same base and exponent If terms have different base and exponents, solve them individually before proceeding to the operations RULE FOR RADICALS: terms must have the same radicand terms must have the same index

33. ADDITION AND SUBTRACTION OF EXPONENTS AND RADICALS Example: 1.) = 2.) = = =  

34. Recall: PEMDAS PEMDAS - P arentheses , E xponents , M ultiplication and D ivision , A ddition and S ubtraction PEMDAS: Order of operation P: Solve first the expression which are present inside parentheses or brackets like parentheses ( ), brackets[ ], or braces { } . NOTE: Simplify first from the innermost to the outermost grouping symbol. E: Exponential expressions should be calculated first before the operations. Usually, they are expressed in power or roots, like MD: Then perform multiplication or division from left to right, whichever comes first in the equation. AS: At last, perform addition or subtraction whichever comes first while moving from left to right.  

35. GEMA/GEMS GEMS stands for Groupings, Exponents, Multiplication or Division, Subtraction or Addition. Groupings refers to all grouping symbols – parentheses, brackets, braces, etc. GEMS is another acronym that has been introduced to replace PEMDAS. These can be used interchangeably. NOTE : Perform the operation that comes first. (For Multiplication & Division and for Addition & Subtraction)

36. GEMA/GEMS

37. GEMA/GEMS PERFORM THE INDICATED OPERATION AND SIMPLIFY: EXAMPLES:  

38. GEMA/GEMS EXAMPLES: 1.) = = = = -30   2.) = = = = = = 2  

39. GEMA/GEMS Additional Sample Problems : PERFORM THE INDICATED OPERATION AND SIMPLIFY 1.) = = = = =   2 .) = = = = =  

40. GEMA/GEMS Additional Sample Problems : 3 .) = = = = =   4 .) = = = = =  

41. SEATWORK #1 – ½ CW YP PERFORM THE INDICATED OPERATION AND SIMPLIFY THE FF: 1.) 2.)  

42. QUESTIONS? “Education is the most powerful weapon which you can use to change the world” - NELSON MANDELA

43. THANK YOU!