Math Formulas

Mathematical formulas are equations or expressions that establish a relationship between variables and constants used to solve problems in the real world. They provide us with concise methods to calculate in areas such as algebra, geometry, trigonometry, calculus, and so on.
 

Math formulas are essential tools that we use to solve problems faster and accurately. They help simplify problems that would otherwise seem impossible to solve. We use formulas in various fields. Here are some reasons why math formulas are very important:

• Breaks down complex problems into much simpler problems.
• Formulas guarantee accurate answers.
• We save a lot of time using formulas. By applying the correct formulas, we can solve problems much faster.
• Math formulas are also used in various fields like physics, engineering, robotics even in medicine. 

Around 2500 BC, Egyptians used formulas to measure land and build pyramids. Much later, mathematicians like Pythagoras introduced formulas for geometry. Another Greek mathematician Euclid, known as the father of Geometry, gave a set of principles called Euclid’s axioms. 

Soon after, Indian mathematicians like Brahmagupta and Aryabhatta created formulas for topics like algebra and trigonometry. Arabic scholars like Al-Khwarizmi advanced algorithms and algebraic methods. During the 17th century, scholars like Isaac Newton and Gottfried Leibniz developed formulas for calculus which we use to this day. Today, we continue to use formulas for various fields like physics, engineering, and robotics, making math formulas a vital part of technological progress.

We use formulas to make solving problems much easier and also to get accurate results. Here are some of the major categories of math formulas:

• Arithmetic Formulas
• Algebra Formulas
• Geometry Formulas
• Trigonometry Formulas
• Calculus Formulas
• Probability and Statistics Formulas
   

Let’s learn more about the formulas in these categories.

Arithmetic formulas are basic mathematical operations like addition, subtraction, multiplication, and division. 

The arithmetic operations are: 
Addition ( + ): a + b = c
Subtraction ( - ): a - b = c
Multiplication ( x ): a x b = c
Division ( ፥ ): a ፥ b = c or a/b = c

Let’s look at a few examples of arithmetic operations using these math formulas

Example 1: Jack has 4 Pokémon cards and you have 6 cards. If Jack decides to give his cards to you, how many cards will you have? 
Solution: a + b = c
    4 + 6 = 10 cards
So you will have a total of 10 pokemon cards.
Explanation: We use addition because we are combining two amounts. Therefore, 6 cards become 10 cards.

Example 2: Now Jack takes 2 cards back from your 10 cards. How many cards will you have left?
Solution: a - b = c
        10 - 2 = 8 cards
So now you will have a total of 8 cards.
Explanation: Jack took two of his cards back from your 10 cards. You will be left with only 8 cards.

Example 3: Jack now surprised you with 3 unopened packs of pokemon cards each with 5 cards inside. How many cards in total will you get from these packs?
Solution: a x b = c
       3 x 5 = 15
You will get a total of 15 cards from the 3 unopened pokemon packs.
Explanation: We use multiplication when we have groups of equal size. Adding would take more time, so we use multiplication instead. 

Example 4:  If you have a total of 30 cards, and you decide to split the cards equally between you and Jack. How many cards would each of you get?
Solution: a/b = c
        30/2 = 15
Jack and you would get 15 cards each. 
Explanation: Division helps us split things into equal parts. It's especially useful when we want to share or distribute something in equal amounts.
 

Algebra formulas help solve problems involving unknown variables, making them especially useful when some information is missing – for example, when solving for X.

Some important algebraic formulas to remember are: 
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
a2 - b2 = (a + b)(a - b)
a2 + b2 = (a - b)2 + 2ab
a3 + b3 = (a + b)(a2 - ab +b2)
a3 - b3 = (a - b)(a2 + ab +b2)
(a + b)3 = a3 + 3a2b = 3ab2 + b3
(a - b)3 = a3 - 3a2b = 3ab2 - b3

Quadratic formula: ax2 + bx + c = 0
x = -b  b2-4ac2a

Note: a, b, c are coefficients of real numbers where a ≠ 0

Example 1: Solve 2x2 - x - 1 = 0, using quadratic formula.
Solution: ax2 + bx + c = 0
Step 1: find x
    a = 2, b = -1, c = -1
    x = -(-1)  -12- 4(2)(-1)2(1)
    x = 1  1 - (-8)4
    x = 1  94
Step 2: Solve for both roots (+ and -)
     b2-4ac=9 = 3
    Now we solve for x:
Add first (for the 1st root):
x = 1 + 34= 44 = 1 
Subtract next (for the 2nd root):
    x = 1 - 34= -24 =-12
The roots are: x = 1 or x = -12
 

We use geometry to calculate the size or even the space of objects and shapes. Some formulas include area, volume, and perimeter.

Some major formulas of geometry are:

Perimeter Formulas:

Square: P = 4s (where s is the side of the square)
Rectangle: P = 2(l + b) 
Triangle: P = (a + b + c)
Circle: C = 2πr (where r is the radius)

Area Formulas:

Square: A = s2
Rectangle: A = L x B
Triangle: A = 12  b  h (where b is the base and h is the height of the triangle)
Circle: A = πr2

Volume formulas:

Cube: V = s3
Cuboid: V = L x B x H
Cylinder: V = πr2h
Cone: V = 13πr2h
Sphere: V = 43πr3

Some few examples using these math formulas: 

Example 1: Your garden is 6 meters long and 4 meters wide. What is the area of the garden?
Solution: To find area we use the formula: A = l x b
        A =  6 x 4 = 24 sq.m
 

We use trigonometry to understand relationships between angles and sides of triangles, mainly right angles.
The main formulas in trigonometry are:

Sine (sin): sinθ = OppositeHypotenuse
Cosine (cos): cosθ = AdjacentHypotenuse
Tangent: tanθ = OppositeAdjacent
Secant: secθ = HypotenuseAdjacent
Cosecant: cosec θ = HypotenuseOpposite
Cotangent: cot θ = AdjacentOpposite

Angles from 0° - 360° each have a special value which we use to solve problems.

Example 1:  You are standing 30 meters away from the base of a tree. The angle  of elevation to the top of the tree is 45° . Find the height of the tree. 
Solution: base of the tree is the adjacent side =  30m
We are trying to find the height of the tree which is the opposite side.

So we will use tanθ because we have the adjacent and the angle.
angle = 45° = 1 (Tan 45° = 1)
tanθ = OppositeAdjacent
tan(45°) = h30
1 = h30
h = 30m
After solving, we know that the height of the tree is 30 meters. 
 

Calculus is used to deal with problems that involve change, like speed, growth or even decay. It helps understand things that change over time. 

Differentiation: When we want to find how fast something is changing at a given moment in time, we use differentiation.

Formula: ddt (distance) = speed.
 

Integration: This helps us to calculate the total amount of area or distance during change of time.

Formula:  v(t) dt where v(t) is the velocity function
 

When we want to make predictions or analyze data, we use probability and statistics formulas. Probability tells us how likely an event is, and statistics helps us understand a collection of data.

Some important formulas to remember are:

Mean = Sum of given data / Total number of data
Median = For even numbers = sum of the middle two numbers/2 
For odd numbers = The middle number is the median.
Standard Deviation =√ ∑(xi - μ)²/n
Variance = ∑(xi -  x )²/n
Where,
x_i = the number in a list of numbers and i is the position of the number, in the data set.
x = mean or average
∑ = Sum of all the terms
(xi - μ)² = Squared difference
n = total number of data in the data set
Probability P(n) = number of ways n can occur/total number of possible outcomes

There are some rules and properties we follow when using math formulas:

Associative Law: This law says that, grouping numbers different when adding or multiplying will not change the result.
Example: (2+3) + 4 = 9 and 2 + (3 + 4) = 9

Commutative Law: This rule says that you can swap the numbers around when adding or multiplying and the result wont change.
Example: 2 x 4 = 8 and 4 x 2 = 8

Distributive Law: This rule helps simplify big problems, by multiplying each part separately, then we add the results together.
Example: 2 x (3 + 4) = (2 x 3) + (2 x 4) = 6 + 8 = 14

Rules of power and roots: Power is a shortcut to multiply the same number many times. Example: 23 = 2 x 2 x 2 = 8
Roots is the opposite of the power. It helps find the original number when you know its power. Example: Square root of 9 is 3. 3 x 3 = 9

Math can get tricky, but with a few simple tips and tricks, you can make solving problems faster. Here are a few strategies to guide you through:

Making use of mnemonics: Try to create catchy phrases to remember formulas. A very popular Mnemonic is “SOH - CAH - TOA” which stands for sine, cosine, and tangent. You can even make your own mnemonics to make learning much easier.
Visualization Techniques: Draw various diagrams, charts or graphs to better understand how formulas work. This is especially helpful for subjects like geometry and trigonometry.
Flashcards can be a great help: use flash cards to remember formulas, helpful for quick reviews.
 

Let’s talk about where we use Math formulas:

Engineering: Engineers constantly use formulas to design structures like buildings. They calculate the area and volume of the materials to see if it can handle weight safely.
 

Finance: Bankers and investors use math formulas like compound interest to manage loans
 

Physics: Scientists use math formulas to measure how fast something moves or how long it takes to travel from point A to point B.