Let's Do Basic Differentiation.

 One of the basics that used in the calculus. 

• Power Differentiation
• Find the Gradient of the Curve
• Trigonometric Differentiation

What is differentiation? Differentiation is to find the rate of change, for example when in the graph, there is x plane and y plane. Differentiation for that graph is rate of change of x over y, also known as dy/dx. 

The usage of differentiation also happened in finding the velocity (speed with a direction) over time, that is acceleration (the unit would be m/s²). Displacement divided by time is velocity, while velocity divided by time is acceleration.

Power Differentiation

This power differentiation means that a number with a variable (usually x), and the power would be on the variable, then the number with powered variable is differentiated. The example is:

y=5x³

dy/dx= 15x² 

Why the 5x³ becomes 15x² after differentiation? The explanation is in the formula below:

If y= axⁿ , then dy/dx= anxⁿ⁻¹

For the formulas above, a is any number, x is a variable, and n is the power number. Let's do another example:

y= 6x⁴+ 5x²

dy/dx= (6)(4)x⁴⁻¹ + (5)(2)x²⁻¹ = 24x³+10x

Another example is below: 

The picture above shows the differentiation process. The first step is to eliminate the x^(4/3) at the denominator. Division for the normal number is as subtraction for the powers. Thus, the power 10 is minus by power 4/3 to get power 26/3, while power 3 is minus by power 4/3 to get power 5/3. Thus, y is 12x power of 26/3, plus 5x power of 5/3. When in differentiation, the number is multiplied by the power number, and power numbers are decreased by 1. Thus, the differentiation of y is 104x power of 23/3, plus 25/3 multiplied by x power of 2/3.

Find the Gradient of the Curve

What is the gradient of the curve? The gradient is a straight line on a point of the curve, and gradient is the tangent of that point. Gradient is also known as slope. The example is the picture below (The picture is from BestMaths Website about gradient): 

The gradient of the line at the curve, given at the point P is 1/2 or 0.5. Let's see how to find the gradient on a curve, below:

y=2x² <- The formula of the curve.

dy/dx= 4x <-gradient

Let the point of gradient= (2,8). The gradient would be 4x= (4)(2)= 8. 

Insert the gradient to line formula:

y=mx+c

y=8x+c

Use the point of gradient to find y intersection (c)

8=8(2)+c

c=8-16=-8

Thus, the gradient line is:

y=8x-8

The picture of the gradient touch the curve at (2,8) is shown below:

Based on the picture above, the blue line is gradient line, while red line is curve line.

Trigonometric Differentiation

The trigonometric differentiation is to find the rate of change for the formulas that involving trigonometric onto variables, such as sine, cosine and tangent. Below is the basic differentiation that involving the three trigonometry:

y= sin x -> dy/dx= cos x

y= cos x -> dy/dx= -sin x

y= tan x -> dy/dx= sec² x

What happened when the angle component (x in sin x or cos x or tan x) has power on it? Differentiate it on the outside. Example:

y= cos x³ -> dy/dx= (3x²)(-sin x³)= -3x²sin x³

Another example on the trigonometric differentiation:

y= 2sin x+ 3cos x⁵- 5tan x

dy/dx= 2cos x+ (5x⁴)(-3sin x⁵)- 5sec² x 

        = 2cos x- 15x⁴sin x⁵- 5sec² x   #

The differentiation table is below, provided by Galway Math Grinds:

 

From the table above, f(x) is y, while f'(x) is same as dy/dx. 

To know more about differentiation, please refer here.

The blog of basic differentiation is finished here, if any comment or idea please comment below. Thank you!

 

How to Calculate for 2D Shapes.

The shapes that can be drawn on a paper. 

• Before 2D
• What is 2D shape?
• Circle
• Triangle
• Square
• Rectangle

Before 2D

Before 2D shape, let's go to 0 Dimensional shape and 1 Dimensional shape. The 0 Dimensional shape is less likely a shape when we look at it. Example of 0D (0 Dimensional) shape:

 

Yes, that only dot is a 0D shape. It is not like 1D, 2D, 3D shape or so on. The picture below shows the transition from 0D to 1D shape:

The 1D shape is like a line, a straight line or squiggly line or any form of line. It is 1D shape because the beginning of the line did not meet with the end of line. Below is the transition from 1D to 2D shape:

It is not 2D shape until the beginning of the line meet the end of the line. 

What is 2D shape?

The 2D shape, also called as polygon, is made out of line or lines that connected to each other until it is appeared in closed shape. Means that 2D shape comprised of line or lines that connected from the beginning of line(s) to end of line(s). 

The 2D shape can be smooth with no point(s) like a circle, and also can be pointy like triangle, square, parallelogram, quadrilateral, rectangle,  pentagon, hexagon and so on. Below are the example of 2D shapes:

 

Circle

Circle is the shape that we find at the objects such as tyre, coin, bottle cap and the letter o. The circle did not have points and no sides. The circle has circumference around it, and the centre of the circle is called origin. The length between origin and circumference line is called radius, while 2 radius are called diameter. Below is the picture of tyre from this website about Toyota Raize by Blauda:

Next, below is the picture that explains origin, circumference, radius and diameter:

The black circle is the circumference, blue line is radius, red dot is origin and green line is diameter. 

Next, let's go to the pi (π). What is π? π is the ratio of circumference length of a circle (C) by the diameter of the circle (d or 2r, r is radius). Every circle has the same π value, which is 3.142.

The circumference (C) of the circle equals to π multiply by 2 radius. The circumference considered as the perimeter of the circle. 

 

The image above shows the example of calculation of circumference. The circumference for circle above is 12.568 cm. 

Next, to know the area of circle. Area (A) is the calculated size for the shape, or the space that available inside the shape. The area for the circle is π multiply by radius, then multiply by radius.     

Above is the circle with the radius of 3 cm. The area of the circle (A) is 28.278 cm².  

Triangle

Triangle is the shape that has 3 sides and 3 points. It also has width or breadth, and height. Below is the image of triangles:

To calculate the perimeter of a triangle, we must consider the total length of all sides. 

The perimeter (P) of the triangle above is 21 cm. Each side of the triangle is 7 cm. 

The area (A) of triangle is height multiply by breadth multiply by half. Why multiply by half? it is because triangle is considered half to the area of four-sides shape, be it square or rectangle. 

Based on the image above, the breadth is length of left to right point of the triangle. The area (A) of the triangle is 27.41 cm². 

Square

A square is a shape that has 4 sides and 4 points, and each side is in same length. The square also looks straight, because all angles are 90 degrees. Below is a picture of square:

To calculate the parameter, add all length of sides.

The parameter (P) of the square above is 20 cm. 

To calculate the area of the square, multiply breadth (b) with height (h). 

The area (A) of the square above is 25 cm². 

Rectangle

The rectangle is same as square, but the different is the breadth of rectangle is longer than the height (or length) of rectangle. Rectangle has 4 sides and 4 points. Below is the image of rectangle:

To calculate the parameter of the rectangle, add all sides of rectangle.

The perimeter (P) of rectangle above is 38 cm.

To calculate the area (A), the height (length) of rectangle is multiplied by breadth of rectangle. 

The area (A) of the rectangle above is 78 cm². 

Here is more information on 2D shapes.

That's all for the 2D shapes. Any idea, please comment it here. Thank you! 

 

 

Calculate using Matrices.

 This is how you calculate, but with rows and columns. 

• What is Matrix
• Matrices Addition
• Matrices Subtraction
• Matrix multiplication by scalar
• Matrices Multiplication

What is Matrix

Matrix is a condition where numbers are arranged to rows and columns.

(rows x columns).

Let's call matrix A:

As we can see, the matrix A has numbers such as 6, 3, -4 and 10. The columns of matrix A are two, the first column has number 6 and -4. While the second column has number 3 and 10. How about the rows? The first row has number 6 and 3, while second row has number -4 and 10. 

How to determine matrix element? Below is how.

Continue on looking on matrix A above, let's say that Ali want to know what is a₁,₁. The first 1 subscript means the number at the first row, the second 1 subscript means the number at first column. So, a₁,₁ is 6. Below is the answer for each matrix A element:

a₁,₁=6

a₁,₂=3

a₂,₁=-4

a₂,₂=10

For more information, look here to know matrix introduction.

Matrices Addition

The addition between two matrices is processed according to the elements' position of two matrices. See the example in the picture below:

The picture above shows the addition of two matrices, B and C. Both of the matrices are 3x2, means 3 rows and 2 columns. The B matrix has elements such as 3, 3, 5, 12, 7 and 6, while C matrix has elements (numbers) like 5, 0, 5, 8, 6 and 1. Then, the addition process taking place, where the elements are added according to their positions. For example, b₃,₁ (7) perform addition with c₃,₁ (6). Then, solve the addition as the picture above. 

Matrices Subtraction

The subtraction process for matrices is the similar process with addition for matrices in terms of elements' position. The difference are the operation involved. See the example of matrices subtraction below:

Above is the subtraction between matrix D and matrix E. Below is the result of subtractions based on matrices' elements:

d₁,₁- e₁,₁= 1-7=-6

d₁,₂- e₁,₂= 2-8=-6

d₁,₃- e₁,₃= 3-9=-6

d₂,₁- e₂,₁= 4-10=-6

d₂,₂- e₂,₂= 5-11=-6

d₂,₃- e₂,₃= 6-12=-6 #

Look here to find out addition and subtraction of matrices.

Matrix multiplication by scalar

This process is also called "multiplicative identity property". The usage of this multiplication by scalar can be applicate to estimate the 300 persons on value of average income per month, average number of children and average value of houses. By the another example below, we should understand the application further:

The above picture shows the scalar multiplication of matrix between 5 and F. The number 5 is expanded to each of F elements to multiply to each of them. Below are the multiply results of 5 and F elements:

5 ⸳ f₁,₁ = 5 ⸳ 3 = 15

5 ⸳ f₁,₂ = 5 ⸳ 6 = 30

5 ⸳ f₂,₁ = 5 ⸳ 9 = 45

5 ⸳ f₂,₂ = 5 ⸳ 11 = 55 #

Matrices Multiplication

The matrices multiplication is the process when two matrices are multiplying each other. Remember that rows from first matrix are multiplying with columns from second matrix. The example is shown below:

Based on the picture above, the matrix G is multiplied by H. The upper left corner of the answer (20) is result of multiplication of G's first row and H's first column. 28 is result of multiplication of G's first row and H's second column. 46 is result of multiplication of G's second row and H's first column. Then, 76 is result of multiplication of G's second row and H's second column. 

Note that the dot in picture above represents multiplication. And for matrices multiplication, the number of rows and columns of matrix answer determines by number of rows of first matrix (like G matrix) and number of columns of second matrix (like H matrix).     

Here is more information on matrices multiplication.

That's all for matrix. Feel free to share ideas in the comment below. Thank you!

 

Four Operation Of Mathematics.

The operations that become the basis in the mathematics, in solving numbers. 

• Addition
• Subtraction
• Multiplication
• Division

In the mathematics and in calculating using formula, you will use these always because these operations can be considered as fundamental of mathematics.

Addition

Addition is when you combine the numbers of stuffs to one or more numbers of the same stuffs. For example:

The picture above shows how addition or plus operation is working. The first box shows 3 cars (🚗🚗🚙) available. Then, the addition come in place when the usage of + sign, that means add or plus. This symbol will add 3 cars to number after the + sign, that is 2 cars (🚗🚙). After the = sign (equal sign), write the answer of 3 plus 2. Thus, there is 5 cars (🚗🚗🚙🚗🚙).   

Even after swapping the 3 cars and 2 cars, it makes no difference to addition process. The answer is still 5 cars. Look at the picture below:

Let's look on another example:

From the example in the picture above, we can see that there are 2 planes (✈️✈️), then there is add sign (+). After the add sign there are 4 planes (✈️✈️✈️✈️). Then there is equal sign, which after that will give us the answer to 2 planes add with 4 planes. The answer is 6 planes (✈️✈️✈️✈️✈️✈️). 

The table below shows the addition processes from 0 to 9 with 0 to 9:

From the table above, we can calculate for addition. For example, take 5 from the upper side and 8 from the left side. Thus, they meet on 13, which is 5 plus 8 (5+8=13). Let's try another numbers. Take 3 from the upper side and take 6 from the right side. They meet on 9, which is 3 plus 6 (3+6=9). 

Let's add two digit numbers with two digit numbers.

The picture above shows the addition process between 25 and 19. First, we must add 5 and 9. It will be 14. Why 1 in 14 is blue, you may ask. It is because the number 1 will be carry forward to ten place, that has 2 and 1. Then, 2 plus 1 is 3, then 3 plus 1 (carry forward number) is 4. Thus, 25 plus 19 is 44 (25+19=44). 

Here is more information on addition process.

Subtraction

The subtraction is the opposite of addition. Instead of making the numbers more bigger, subtraction makes the numbers more smaller. Subtraction also called as additive inverse. Subtraction takes some numbers away from original numbers.  For example: 

From the picture above, we can see 4 apples. On the left of 4 apples, it is subtraction sign (-). After the subtraction sign, there are 2 apples. After the equal sign (=) it shows 2 apples, because 4 apples subtract with 2 apples equal 2 apples (or 4-2=2). 

What if 4 apples and 2 apples change place?

The picture above shows another subtraction process. at the front, there is 2 apples. Besides that, there is a subtraction sign or minus sign (-). At the left of minus sign, there are 4 apples. Then, after the equal sign, it shows the result of 2 apples minus 4 apples. But the answer is "need 2 apples more". Why did this happen?

Imagine you have 2 apples. Ahmad asks you for 4 apples. Then, you give Ahmad 2 apples and you say "I will give you 2 apples tomorrow". Ahmad agrees. Thus, that is why the answer is "need 2 apples more", or in mathematical terms: 

2-4=-2. 

Below is the table on subtraction (upper numbers subtracted by left numbers):

Let's take 6 from upper numbers and 2 from left numbers in the table. They meet on 4 because 6 subtracted by 2 is 4 (or 6-2=4). Then, take 4 from upper numbers and 8 from left numbers. They meet on -4, because 4 minus 8 is -4 (or 4-8=-4).

Below is the subtraction between 2 digit number to 2 digit number:

 

As we see the image above, 28 is minus by 15. The first step is to minus 8 by 5, the answer will be 3. Then, look at the ten place, that are 2 and 1. So, 2 subtracted by 1 is 1. Thus, the answer to 28 subtracted by 15 is 13 (28-15=13).

Let's look on another example of subtraction or minus operation:

From the image above, it shows 25 subtract by 18. First, we must subtract numbers in one place, that are subtract 5 with 8. But, it is not enough to subtract 5 with 8. So, we borrow 10 from ten place. Then, 10 plus 5 is 15, then 15 subtract with 8 is 7. At the ten place, there are two numbers, 1 and 1. 2 had become 1 because one place borrows 1 from 2. Thus, 1 minus 1 is 0. Thus, 25 subtracted with 18 is 7 (25-18=7).   

For more information regarding subtraction, look here.

Multiplication

Multiplication is how many times do you add the number. The process of multiplication called multiply or times. For example, 2 times 1 is 2, means 2 for 1 times. But 2 times 2 is 4, means 2 for 2 times, also 2+2. What is 2 times 3? It is same as 2+2+2, means 6. 

Let's look at the example below:

The picture above is a multiplication between 5 and 3. The multiplication symbol is x. 5 multiply by 3 equals to 15. Note that 5 times 3 (5x3) is same as 5 plus 5 plus 5 (5+5+5), which is 15. Thus, 5 times 3 is same as adding 5 for three times. What for 3 times 5 (3x5)? The answer would be the same, it is 15, because multiplication is commutative (eg 5x3=3x5). 

Next, below is the table on multiplication 0 to 9 with 0 to 9:

From the table above, we can see the results of multiplication. For example, take 4 from the upper numbers and 5 from the left numbers. They meet on 20, which is the result of 4 times 5 (4x5=20). From this table, it shows that 5 times 4 is also 20 (5x4=20). Let's look at another example. Take 7 from upper numbers and 8 from left numbers. They meet on 56, which is the answer to 7 times 8 (7x8=56). 

Below is the multiplication between two digit numbers and two digit numbers:

The picture above shows the multiplication between 17 and 34. First, multiply 17 to 4. 7 multiply by 4 is 28, then 4 multiply by 1 is 4. 4 will be added to 2 from 28 earlier. Thus, 17 times 4 is 68. Then, multiply 17 by 3. 7 times 3 is 21 and 1 times 3 is 3. 3 will be added to 2 from 21. Thus, 17 multiply by 3 is 51. Then, add 68 to 51, but put 1 in the same level as 6, like the picture above. The final answer to 17 times 34 is 578 (17x34=578). 

A lot of information regarding multiplication are present here.

Division

Division is how you divide the total stuffs in fair numbers. For example:

From the picture above, we can see that there are 9 pencils. Beside the division sign (÷), there are three persons. We must divide the pencils fairly. To be fair, 9 pencils had divided to 3 so each person has 3 pencils, which are the same for all 3 persons. From here we should know that 9 divided by 3 equals 3 (9÷3=3, or 9/3=3).

We know that 9÷3=3, but what does the name for each position of number? Look below:

The one that is divided (9) is dividend, the divider (3 after ÷) is divisor and the answer (3 after =) is quotient. The same case when 9/3=3, just change the symbol ÷ into /. 

Sometimes, there are remains that could not be divided. Let's look at the image below:

Let's say we want to divide 10 pencils fairly to each person. There are 3 persons. So, each person would get 3 pencils. But, 1 pencil is remaining. That pencil is what we called as remainder (R in the picture). Thus, 10 divided by 3 is 3 with the remainder 1. 

Division is the opposite of multiplying. Thus in division, everyone use multiplication table to solve it. Let's look at the picture below to solve long dividend:

From the picture above, it shows the process of 9405 divided by 9. First, we should solve the number in front, that is 9. 9x1 is 9, so write 1 on top of 9 in division, then minus 9 by 9. Then, bring down 4. There is no 4 in 9 multiplication table, so put 0 at the top of 4, then minus 4 by 0, equal 4. Then, bring down 0 to put it besides 4, becomes 40. 9 times 4 is 36, thus put 4 at the top of 0, then minus 40 by 36. The answer would be 4. Then, bring down 5 besides 4. 9x5 is 45, so write 5 on the top of 5 and 45 minus 45. Two dots below 45 represent zero. Thus, 9405 divided by 9 is 1045 (9405÷9=1045). 

There are a lot of information about division here.

The four basics operation of mathematics had been covered.

For the info regarding four operations, go here.

If you want online math class for 5 Ringgit per class, contact me on the email or in the comment. 

That is all for this blog. Any comments or anything to correct, please comment below. Thank you!

 

What are base numbers and how to change them?

Different base numbers have different types of counting. 

Normal type of counting, base 10 numbers

This is a usual type of counting that used by everyone, also called Decimal Number System. Base 10 numbers can be represented by our hands

The example of number "seven" in base 10 is shown below:

Or, it could be written as:

The "7" number is used for counting, but "10" number is indication that 7 is base 10 number. The full counting of base 10 is:

Table above shows that number of objects compared to base 10 numbers. You may ask, where is 10? 10 is actually exists in base 10 numbers even the numbers after 10, it depends on the number of objects you want to count. Thus, 10 is after 9, 11 is after 10 and so on.  

Base 2 numbers, computer numbers

This number is usually used by the computers and logic gates in the microchips, and another name for "base 2" is "binary number". The digits is only two: 0 and 1. Wait, how to count from 1 to 10, you may ask. Just repeat the 0s and 1s to complete the counting. The table below shows how to count:

Then, the number ten in binary would be:

"1010" is base 2 number for 10 in usual number (base 10). The "2" subscript means "1010" is base 2 number. 

Base 8

The base 8 seems to have digits less than 10, but it is also can be used for counting. Base 8 is also called "octal numbers". The base 8 numbers would be used if octopus can somehow counts. 

For base 8 numbers, look at the table below:

What about ten? The notation of ten in base 8 number is 12. 

Base 16 numbers

The base 16 numbers, also called hexadecimal numbers. The hexadecimal numbers have multiple uses in technological spectrum, such as HTML colour codes, MAC address (your PC has this) and memory address inside memory of computer. Below is how to count using base 16:

 

Honourable mention

One of the base numbers that can be mention is base 3 numbers. This number system is also called "ternary numbers". This type of counting is possible when you use lines on fingers or phalanx on fingers (one finger has three segments). 

The counting of base 3 numbers will be like the table below:

Base 10 to base 2 and vice versa

To change from base 10 (normal number) to base 2, you can do the following (eg. base 10 number is 45):

Thus, the base 2 number for 45 is 101101. The process consists of division, with remainder put beside the number below. 45 divided by 2 is 22, remainder 1. This 1 is least significant, so it is the back of base 2 number. Next, 22 divided by 2 is 11, with 0 remainder. So 0 would be placed in tens unit (would be 01 of base 2). The division process goes on until 1 had divided by 2, causing remainder 1.

Next, how to turn base 2 number: 101101, to base 10:

The 1 on the front (left) is most significant, thus the units is 2 power of 5. After discovering the units, each base 2 numbers will multiply with respective units. After multiply all of them, add them together to get base 10 number (45). 

Base 10 to base 8 and vice versa

Below is how to change base 10 number to base 8:

The base 10 number is 46. When undergone division, 46 divided by 8 equal 5, and the remainder (6) is put beside 5. Then 5 divided by 8, but remainder is still 5. Thus, 5 is carry to below 5 with 8. Then, 5 is most significant because it is below, and 6 is least significant because it is in upper position. So, the base 8 is 56. 

Next, lets we look on how to change base 8 number to base 10:

The 5 is most significant and at unit tens of base 8, so 5 is multiplied by 8 power of 1. Next, 6 is least significant, thus 6 is multiplied by 8 power of 0 (equals 1). After each base number is multiplied by respective units, add them together. So, base 10 number of 56 is 46.  

Base 16 to base 10 and vice versa

How to change from base 10 number to base 16? Look below:

Base 10 number is 47. When 47 is divided by 16, it is equal 2 and the remainder is 15. So, 15 is put beside 2 and 15 when using base 16 notation, 15 equals to F. Then, 2 is smaller than 16, so 2 is written below 2 that with 15. 2 is most significant because it is at the bottom and F is less significant because it is higher than 2. So, hexadecimal number is 2F. 

Lets look on how to change base 16 number to base 10 number: 

The base 16 number is 2F. To turn to base 10, we should know which is most significant and which is least significant. 2 is most significant and positioned in tens unit, thus 2 is multiplied by 16 power of 1. Then, F is least significant and positioned in a unit (not tens, not hundreds), thus F (F is 15) is multiplied by 16 power of 0. After multiply all of them, add all of them together. So, the base 10 number is 47. 

There are more info on base numbers here.

Feel free to comment any ideas. Thank you!