Chapter 1- Learning Outcome 1_Mathematics for Technologists

Chapter 1- Learning Outcome 1_Mathematics for Technologists

1. 21
st
Century Mathematics for
the Industrial Technology
Learners: Automotive
Technology
BABY 'RLENE I. BAYOC

2. The Numeral System, and
the Real Number
Chapter I
21
st
Century Mathematics for the Industrial Technology Learners: Automotive Technology

3. CHAPTER
 I: The Numeral System and the Real Number
Learning Outcome
1:
Defined numeral system and the
real number and applied the
properties of reals in solving
problems;

4. THE NUMERAL
SYSTEM
A number is a count or
measurement that is really an idea
in our minds while a numeral is a
symbol or name that stands for a
number. A digit is a single symbol
used to make numerals.
CHAPTER
 I: The Numeral System and the Real, Numbers

5. CHAPTER
 I: The Numeral System and the Real, Numbers

6. Properties of real numbers
Property Examples
1. Closure
If a and b are elements of the real
number, then a + b and a x b are both
elements of the real number. Therefore,
addition and multiplication are closed.
Definition/Condition/s
 
3 + 5 = 8
3 x 5 = 15
CHAPTER
 I: The Numeral System and the Real Number

7. Properties of real numbers
Property Examples
2. Commutative
If a and b are elements of the
real number, then a + b = b + a and
a x b = b x a.
Definition/Condition/s

2 + 3 = 3 + 2
    
 
                   
 
        
2 x 3 = 3 x 2
 
CHAPTER
 I: The Numeral System and the Real Number

8. Properties of real numbers
Property Examples

3. Associative
If a, b, and c are real numbers
then, (a + b) + c = a + (b + c) and
(a x b) x c = a x (b x c).
Definition/Condition/s
 
(1+2)+3=1+(2+3)
    
 
     
(1x2)x3 = 1x(2x3)
  
CHAPTER
 I: The Numeral System and the Real Number

9. Properties of real numbers
Property Examples
4. Distributive
If a, b, and c are real numbers
then,
 
 

c (a ± b) = ca ± cb ↔ ac± bc = (a ± b)
c
 
Definition/Condition/s
 
3(4+5) = 3x4+3x5
 
  
(4+5)3 = 4x3+5x3
CHAPTER
 I: The Numeral System and the Real Number

10. Properties of real numbers
Property Examples
5A. Existence
of Additive
Identity
If zero (0) is the additive identity,
then
 
           
a + 0 = a = 0 + a
Definition/Condition/s
 
±5 + 0 = ±5
  = 0 + (
±5) 
 
 
               
CHAPTER
 I: The Numeral System and the Real Number

11. Properties of real numbers
Property Examples
5B. Existence
of Multiplicative
Identity
If one (1) is the
  multiplicative
identity, then
 
                 
a x 1 = a = 1 x a.
Definition/Condition/s
 
± 3 x 1 = ±3
  =  1 x (
± 3)
 
               
CHAPTER
 I: The Numeral System and the Real Number

12. Properties of real numbers
Property Example
6A. Existence
of Multiplicative
Inverse
The additive inverse of a is
negative a and vice versa such that
a + (-a) = 0 = (-a) + a.
Definition/Condition/s
 
10 + (-10) = 0
 = -10 + 10
 
               
CHAPTER
 I: The Numeral System and the Real Number

13. Properties of real numbers
Property Example
6B. Existence
of Multiplicative
Inverse
The multiplicative inverse of
a
  is 1/a and vice versa such that
a x 1/a = 1 =
  1/a x a.
Definition/Condition/s
 
7(1/7) = 1 =
 (1/7)7
 
               
CHAPTER
 I: The Numeral System and the Real Number

14. Properties of real numbers
Property Example
7. Reflexive
Property of
Equality
If a is an element of the real number,
then, a = a.
Definition/Condition/s
 
21 = 21
 
               
CHAPTER
 I: The Numeral System and the Real Number

15. Properties of real numbers
Property Examples
8. Symmetric
Property of
Equality
If a and b are elements of the real
number then, if a = b, then b = a.
Definition/Condition/s
 
If 2 + 3 = 4 + 1,
then 4 + 1 = 2 + 3
 
               
CHAPTER
 I: The Numeral System and the Real Number

16. Properties of real numbers
Property Examples
9. Transitive
Property of
Equality
If a and b are elements of the real
number then, if a = b, and b = c, then a
= c.
Definition/Condition/s
 
If 4 – 3 = 5 – 4,
and 5 – 4 = 3 – 2,
then 4 – 3 = 3 – 2.
CHAPTER
 I: The Numeral System and the Real Number

17. Properties of real numbers
Property Examples
10. Trichotomy
Axiom
For any two real numbers a and be,
exactly one of the following must be
true. Either a < b, a = b, or a > b.
Definition/Condition/s
 
3 > 1; 3 ≠ 1; 3 1
 
               
CHAPTER
 I: The Numeral System and the Real Number

18. Properties of real numbers
Property Examples
11. Zero
Product
If a and b are elements of the real
number then, if a × b = 0 then a = 0 or
b = 0 (or both a = 0 and b = 0).
Definition/Condition/s
 
±78 x 0 = 0
 
    
CHAPTER
 I: The Numeral System and the Real Number

19. Properties of real numbers
Property Examples
12. Addition
Property of
Equality
(APE)
If a and b are elements of the
real number then, if a = b, then

a + c = b + c.
Definition/Condition/s
 

If 2 + 3 = 4 + 1, then
(2+3) + 5 = (4+1) + 5.
CHAPTER
 I: The Numeral System and the Real Number

20. Properties of real numbers
Property Example
13. Multiplication
Property of
Equality (MPE):
If a and b are elements of the
real number then, if a = b,
then a • c = b • c.
Definition/Condition/s
 
If 2 + 3 = 4 + 1,
then (2+3) x 5 = (4+1) x 5.
CHAPTER
 I: The Numeral System and the Real Number

21. Properties of real numbers
Property Examples
14 Substitution
Property
If a and b are elements of the real
number then, if a = b, then a may be
substituted for b, or conversely.
Definition/Condition/s
 

If a = b,
then a + b = a + a = 2a,
or a + b = b + b = 2b
in effect a + b = 2a = 2b.
CHAPTER
 I: The Numeral System and the Real Number

22. FUNDAMENTAL THEOREM OF ARITHMETIC
Every integer greater than 1 is either a prime number
or a composite number that can be expressed in the form
of primes.
Examples:
Prime numbers: 2, 3, 5, 7, 11, 13, …
Composite number: 88 = 2x2x2x11 or 2
3
x 11.
 
CHAPTER
 I: The Numeral System and the Real Number

23. DIVISIBILITY RULES
Divisible
by
Conditions/Rule/Solution Examples
2
A number is divisible by 2 if It is an
even number or the last digit is 0, 2,
4, 6, or 8.
810, 52, 104,
306, 678
CHAPTER
 I: The Numeral System and the Real Number

24. Divisible
by
Conditions Example/s
3
8673 -> 8+6+7+3=24
-> 24/3=7
(8673/3 = 2891)
The sum of all the digits of the
number is divisible by 3
DIVISIBILITY RULES
CHAPTER
 I: The Numeral System and the Real Number

25. Divisible
by
Conditions Examples
DIVISIBILITY RULES
4
The last two digits of the number is 2
zeros or divisible by 4.
2100->2100/ 4 = 525


1332 -> 32/4 = 6
(1332/4=333)
CHAPTER
 I: The Numeral System and the Real Number

26. DIVISIBILITY RULES
Divisible
by
Conditions Examples
5
The last digit of the number is 0 or 5.
5670; 735
CHAPTER
 I: The Numeral System and the Real Number

27. DIVISIBILITY RULES
Divisible
by Conditions Examples

6

The sum of all the digits of the
number is divisible by 3 and the
last digit is an even number or
zero.
7854 -> 7+8+5+4=24/3=8 & the
last digit is even
(7854/3 = 2618)
28230 -> 2+8+2+3+0=15
??????15/3=5 & the last digit is 0.
(28230/3 = 9410)
CHAPTER
 I: The Numeral System and the Real Number

28. DIVISIBILITY RULES
Divisible
by
Conditions Examples
7
Delete the last digit from the number
then subtract 2 times the deleted
digit from the remaining number,
repeat the process until it becomes 0
or divisible by 7.
630 -> 63-0=63-> 6-(2x3)=0
(630/7 = 90)
518->51-(2x8) = 35->35/7 =5
(518/7 = 74)
CHAPTER
 I: The Numeral System and the Real Number

29. DIVISIBILITY RULES
Divisible
by
Conditions Examples
8
The last three digits of a number is 3
zeros or divisible by 8
11000-> 11000/8=1375
1824-> 824/8=103
(1824/8 = 228)
CHAPTER
 I: The Numeral System and the Real Number

30. DIVISIBILITY RULES
Divisible
by
Conditions Examples
9
The sum of all the digits of the
number is divisible by 9.
873 -> 8+7+3 = 18 -> 18/9 = 2
(873/9 =97)
CHAPTER
 I: The Numeral System and the Real Number

31. DIVISIBILITY RULES
Divisible
by
Conditions Examples
10 The last digit is 0. 7130 -> 7130/10=713
CHAPTER
 I: The Numeral System and the Real Number

32. DIVISIBILITY RULES
Divisible
by
Conditions Examples
11
Alternately subtract and add the
digits from left to right, the result is 0
or divisible by 11.
13574-> 1-3+5-7+4=0
506 -> 5-0+6=11 ->11/11=1
CHAPTER
 I: The Numeral System and the Real Number

33. DIVISIBILITY RULES
Divisible
by
Conditions Examples
12
The number is both divisible by 3
and 4.
2808 -> 2+8+0+8=18 ->
18/3=6 and 08 is divisible by
4
CHAPTER
 I: The Numeral System and the Real Number

34. DIVISIBILITY RULES
Divisible
by
Conditions Examples
13
Delete the last digit from the number,
and then subtract 9 times the deleted
digit from the remaining number,
repeat the process until it becomes 0
or divisible by 13.
3185 -> 318-(9x5) = 273 -
>27-(9x3)=0
221 -> 22 - (9x1) = 13 ->
13/13 =1
CHAPTER
 I: The Numeral System and the Real Number

35. DIVISIBILITY RULES
Divisible
by
Conditions Examples
15
The number is both divisible by 3
and 5.
1200 -> 1+2+0+0=3 divisible
by 3 and divisible by 5 since
it ends in zero.
CHAPTER
 I: The Numeral System and the Real Number

36. Tree
Method
100
25
5
5
4
2
2
CHAPTER
 I: The Numeral System and the Real Number
100
10
5
2
10
5
2
100
50
5
10
2
100
50
4
20
2
100
50
2
25
2
100
20
5
4
5
2
2
5
2
5
5
5
4
2
2
100 = 2x2x5x5 =
2
2
x5
2

37. Greatest Common Factor
and
Least Common Multiple
Number Factors Common factors GCF Multiples Common multiples LCM
8
 
1, 2, 4, 8
1, 2, 44
8, 16, 24, 32,
40, 48,
56, 64, 72, …
 
24, 48, 72,
… 24
12
1, 2, 3,
4, 6, 12
12, 24, 36, 48,
60, 72, …
CHAPTER
 I: The Numeral System and the Real Number

38. REFERENCES
https://en.wikipedia.org/wiki/Numeral_system
https://en.wikipedia.org/wiki/Real_number
https://jamesbrennan.org/algebra/numbers/real_number_system.htmlonrealnumbers...
https://mathbitsnotebook.com/Algebra1/RealNumbers/RNProp.html
http://mathforum.org/library/drmath/sets/select/dm_lcm_gcf.html
https://medium.com/i-math/using-factor-trees-to-find-gcfs-and-lcms-64c0d4ef594
https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-greatest-common-divisor/v/lcm-and-gcf-greatest-co
mmon-factor-word-problems?

https://www.kullabs.com/classes/subjects/units/lessons/notes/note-detail/6067
https://www.mathsisfun.com/numbers/real-numbers.html
https://www.purplemath.com/modules/lcm_gcf.htm
CHAPTER
 I: The Numeral System and the Real Number

39. Thank you
CHAPTER
 I: The Numeral System and the Real Number