1. ALGEBRA
Math 10

2. Arithmetic Abstraction Algebra
🍎 + 🍎 = 2🍎
🍊 + 🍊 = 2🍊
🍕 + 🍕 = 2🍕
x + x = 2x
𝑥−1 𝑥+2 = 𝑥 2 +𝑥−2
Arithmetic Abstraction Algebra

3. One goal is to solve equations. 𝑥 2 +𝑥−2=0
𝑥−1 𝑥+2 =0
𝑥=1 𝑜𝑟 𝑥=−2
Elementary Algebra

4. Mathematical models of real- world scenarios are often in the form of equations.
Example in cooking:
3 𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛 𝑜𝑓 𝑠𝑎𝑙𝑡 5 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 = 10 𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛 𝑜𝑓 𝑠𝑎𝑙𝑡 𝑥 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
𝑥= 50 3
Elementary Algebra

5. Find the value of the following as fast as possible: 90 2 − 89 2 =179
Challenge:
Find the value of the following as fast as possible:
90 2 − 89 2
= −89
=179
Elementary Algebra

6. Using geometry:
𝑥 2 −𝑥=0
Elementary Algebra

7. Using geometry:
𝑥 2 −𝑥=0
Elementary Algebra

8. Mathematical models of real- world scenarios are often in the form of equations or inequalities.
Example in shopping:
Jose has PhP25. He wants to buy chocos each worth PhP10 and cookies each worth PhP1.
10𝑥+𝑦≤25
𝑥,𝑦∈𝕎
Elementary Algebra

9. Be careful with your algebra! −10𝑥≤25−2 𝑥≥ 23 10
−10𝑥+2≤25
−10𝑥≤25−2
𝑥≥ 23 10
Elementary Algebra

10. Diet Problem: Animals in an experiment are to be kept under a strict diet. Each animal should receive 20 grams of protein and 6 grams of fat. The laboratory technician is able to purchase two food mixes: Mix A has 10% protein and 6% fat; mix B has 20% protein and 2% fat. How many grams of each mix should be used to obtain the right diet for one animal?
Matrices

11. Diet Problem: x: grams of mix A y: grams of mix B 0. 1𝑥+0. 2𝑦=20 0
Matrices

12. Matrices

13. Abstract Algebra: a peek
<{0,1},×>
Abstract Algebra: a peek x 1

14. Abstract Algebra: a peek
<{F,T},∧>
Abstract Algebra: a peek ∧ F T

15. Abstract Algebra: a peek
<{off,on},∧>: series circuit
Abstract Algebra: a peek ∧
off on

16. Abstract Algebra: a peek
<{A,R,B,L},fb>
Abstract Algebra: a peek fb A R B L

17. Abstract Algebra: a peek
<{1,i,-1,-i},×>
Abstract Algebra: a peek × 1 i -1 -i

18. Abstract Algebra: a peek
<{0,1,2,3}, + 4 >: addition modulo 4
Abstract Algebra: a peek
+ 𝟒 1 2 3

19. Abstract Algebra: a peek
<{0,1,2,3,…,11}, + 12 >:
addition modulo 12
Abstract Algebra: a peek

20. Abstract Algebra: a peek
Group
A group is defined as a set of elements, together with an operation performed on pairs of these elements (<F,∘>) such that:
The operation, when given two elements of the set, always returns an element of the set as its result. It is thus fully defined, and closed over the set.
One element of the set is an identity element. Thus, if we call our operation ∘, there is some element of the set (e) such that for any other element of the set (x), e ∘ x = x ∘ e = x.
Every element of the set has an inverse element. If we take any element of the set (p), there is another element (q) such that p ∘ q = q ∘ p = e.
The operation is associative. For any three elements of the set (a,b,c), (a ∘ b) ∘ c always equals a ∘ (b ∘ c).
Abstract Algebra: a peek

21. Abstract Algebra: a peek
Group or not?
<ℕ,+>
<𝕎,+>
<ℤ,−>
<ℤ,÷>
Abstract Algebra: a peek

22. Abstract Algebra: a peek
Group theory and symmetry
Abstract Algebra: a peek

23. Abstract Algebra: a peek
Wallpaper groups: 17 possible plane symmetry groups
Abstract Algebra: a peek

24. Abstract Algebra: a peek
Field
<F,+,×>
F is a commutative (Abelian) group under +
F – {0} is an Abelian group under ×
note: 0 is the additive identity
Abstract Algebra: a peek