Introduction to Proving

1. INTRODUCTION
TO PROVING
Bongabong, Miles G.
BSED 1C

2. Learning Objectives
At the end of the lesson, students should be able to:
1. Understand the concepts of algebra;
2. differentiate segments from rays; and
3. apply the concepts of algebra in proving.

3. Review of the Concepts
of Algebra

4. Arithmetic Operations
The real numbers have the following properties:

5. Fractions

7. Factoring
refers to the process of breaking down a mathematical
expression into its constituent factors, which are the numbers or
algebraic expressions that multiply together to give the original
expression.

8. Factoring difference of squares: Factoring difference of cubes:
Factoring sum of cubes:

9. Completing the Square

10. Quadratic Formula
The quadratic formula is used to solve quadratic equations of
the form , ax² + bx + c = 0 where a, b, and c are constants and
a≠0.

12. The Binomial Theorem

13. Radicals

14. Rationalizing refers to the process of restructuring a
mathematical expression so that its denominator does not
contain any radical or imaginary terms. This is often done to
simplify expressions or make them easier to work with in various
mathematical operations.

15. Exponents
Exponents represent repeated multiplication of a number by
itself. They are written as a superscript to the right of the base
number.

16. In words, these five laws can be stated as follows:
1. To multiply two powers of the same number, we add the exponents.
2. To divide two powers of the same number, we subtract the exponents.
3. To raise a power to a new power, we multiply the exponents.
4. To raise a product to a power, we raise each factor to the power.
5. To raise a quotient to a power, we raise both numerator and denominator to
the power.

17. Inequalities
In mathematics, inequalities are statements that compare two
quantities or expressions, indicating that one is greater than, less
than, or not equal to the other. They are often represented using
symbols such as < (less than), > (greater than), ≤ (less than or
equal to), and ≥ (greater than or equal to).

18. Rule 1 says that we can add any number to both sides of an inequality,
and Rule 2 says that two inequalities can be added. However, we have
to be careful with multiplication.
Rule 3 says that we can multiply both sides of an inequality by a
positive number, but Rule 4 says that if we multiply both sides of an
inequality by a negative number, then we reverse the direction of the
inequality. For example, if we take the inequality 3 < 5 and multiply by
2, we get 6 < 10, but if we multiply by -2, we get -6 > -10.
Finally, Rule 5 says that if we take reciprocals, then we reverse the
direction of an inequal- ity (provided the numbers are positive).

19. Absolute Value
The absolute value of a number , denoted by , is the distance
from to on the real number line.
Distances are always positive or , so we have;
| a | ≥ 0 for every number a
In general, we have

20. For solving equations or inequalities involving absolute values, it’s
often very helpful to use the following statements.

21. Properties of
Equality

22. • Properties of equality are fundamental rules that apply to
equations and express the idea that both sides of an equation are
equal. If an arithmetic operation has been used on one side of the
equation, then the same should be used on the other side.
Properties of equality are all about balance.
• These properties allow us to manipulate equations, perform
operations, and solve for unknown variables in a consistent and
reliable manner.

23. List of Different Properties of Equality
1. Addition Property of Equality
When you add the same value to both sides of an equation, the equation remains
true. This concept is known as the addition property of equality.
Mathematically, for real numbers a, b and c, we have
If a=b, then a + c = b + c
2. Subtraction Property of Equality
When you subtract a real number from both sides of an equation, the equation
remains true. This principle is known as the subtraction property of equality.
To understand this mathematically, let’s assume that p, q and r are real numbers.
If q=p, then p - r = q - r

24. 3. Multiplication Property of Equality
When you multiply both sides of an equation by the same real number, the
equation remains balanced. This is known as the multiplication property of
equality.
Let’s understand this mathematically. Consider any three real numbers, a, b and
c.
If a = b, then a × b = b × c
4. Division Property of Equality
If you divide both sides of an equation by the same non-zero real number, the
equation remains equal. This is known as the division property of equality.
Let’s assume three real numbers, x, y and z.
If x = y and z not equal to 0 then according to the division property of equality,
x/z = y/z

25. 5. Reflexive Property of Equality
The reflexive property of equality states that each real number is always equal to
itself.
For any real number x, x = x.
6. Symmetric Property of Equality
By symmetric property of equality, the order of equality does not matter.
For all real numbers x and y, if x = y , then y = x.
7. Transitive Property of Equality
By the transitive property of equality, two numbers that are equal to the same
number are also equal.
If p, q and r are real numbers such that p = q and q = r, then p = r.

26. 8. Substitution Property of Equality
According to the substitution property of equality, if we have two real numbers
equal to each other, then we can substitute the value of any of them in any
algebraic equation.
For all real numbers x and y, if x = y, then we can substitute y for x in any
expression.
9. Square Root Property of Equality
This property of equality states that if two real numbers are equal, their square
roots will also be equal.
To better understand, let’s see it mathematically
Let x and y be two real numbers
If x = y, the √x = √y

27. Properties of
Inequality

28. • An inequality is a mathematical correlation between the two expressions
that can be expressed using any of the following symbols:
< - less than
≤ - less than or equal to
> - greater than
≥ - greater than or equal to

29. If a, b, and c are real numbers, then:
• Transitive Property if a<b and b<c then a<c
• Addition Property if a<b then a+c<b+c
• Subtraction Property if a<b then a−c<b−c
• Multiplication Property (Multiplying by a positive number) if a<b and c>0 then ac<bc
• Multiplication Property (Multiplying by a negative number) if a<b and c<0 then ac>bc
• Division Property (Dividing by a positive number) if a<b and c>0 then a/c<b/c
• Division Property (Dividing by a negative number) if a<b and c<0 then a/c>b/c
• Comparison Property if a=b+c and c>0 , then a>b
Properties of Inequalities

30. Absolute Value

31. • The absolute value of a number refers to the distance of a number from
the origin of a number line. It is represented as | a |, which defines the
magnitude of any integer ‘a’. The absolute value of any integer, whether
positive or negative, will be the real numbers, regardless of which sign it
has. It is represented by two vertical lines |a|, which is known as the
modulus of a.
We can define the absolute values like the following:
{ a if a ≥ 0 }
|a| = { -a if a < 0 }

32. SEGMENTS AND
RAYS

33. Line Measurement
The linear measurement is the distance between the two given points
or objects. Thus, we can define length as:
“Total gap measured between the leftmost and rightmost end of an
object in the mentioned system of units.”

34. Linear Measurement Instruments
The following are examples of common linear measuring instruments:
1. The most common type of tape measure is a metal or plastic strip
with linear markings on one side. It’s a standard tool for measuring
length and distance in the building and woodworking industries.
2. A ruler is a straight, flat piece of material that has been marked along
its length. Its usefulness in measuring short distances makes it a
standard tool in classrooms, offices, and garages.
3. The Vernier caliper is a highly accurate tool for measuring both
external and internal dimensions. A calibrated scale and two jaws (one
fixed and one movable) make up this measuring tool.

35. 4. The micrometer is a very precise tool for measuring distances
between 0.001 and 1 inch. Common in the fields of manufacturing
and engineering, this tool consists of a calibrated screw and a
measuring scale.
5.A meter stick is a measuring tool used for measuring the length of
objects using the metric scale.
6. Digital Caliper: A digital caliper is an electronic version of the
vernier caliper. It provides quick and accurate measurements, often
with a digital display showing the readings.

37. Distance is measured in linear units, such as:
inches- in
feet- ft
yards- yd
miles- mile
meters- m
centimeters- cm
millimeters- mm
kilometers- km

38. Line Segment
In geometry, a line segment is bounded by two distinct points on a
line. Or we can say a line segment is part of the line that connects
two points. A line has no endpoints and extends infinitely in both the
direction but a line segment has two fixed or definite endpoints.

39. Rays
Ray is another part of a line. It is a combination of a line and a line
segment that has an infinitely extending end and one terminating
end. As its one end is non-terminating, its length cannot be
measured.

40. THANK YOU!