Saturday, 4 March 2017

The Number Spiral

The number spiral is all natural numbers arranged in a spiral.

If we remove all the numbers in this spiral that are not prime and leave only the prime numbers, we see that a pattern forms.

The prime numbers seem to line up along diagonal lines. Expanded, diagonal lines are clearly visible.

This type of spiral is known as the Ulam spiral, as it was devised by mathematician Stanislaw Ulam.

After many tests, the prime numbers are found to line up along diagonals with the function:

The number spiral has another interesting quality.

The numbers in the black boxes are square numbers. The even square numbers move in a diagonal to the top-left. The odd square numbers move in a diagonal to the bottom-right.

Trigonometric Proofs

Fundamental Law of Trigonometry

Let us assume that α > β > 0.
Consider a unit circle with centre O.
Let the terminal side of angles α and β cut the unit circle at A and B respectively. 
Evidently AOB = α - β. 
Take a point C on the unit circle so that XOC = AOB = α - β.

Join A with B and C with D.

The coordinates of A are ( cos α , sin α ). 
The coordinates of B are ( cos β , sin β ). 
The coordinates of C are [ cos(α - β) , sin(α - β) ].
The coordinates of D are ( 1 , 0 ) Now AOB and COD are congruent. 

How would you work out this question using the rule above?

You could break this down into cos(60 - 45), and put these numbers into the formula.

Friday, 24 February 2017

Spherical Geometry

In this sphere, the line D is the diameter. A diameter is a straight line drawn from the surface and after passing through the centre ending at the surface.
Every section made by a plane passed through a sphere is a circle. If the plane passes through the centre of a sphere, the plane section is a great circle; otherwise, the section is a small circle.

Frustum of a Sphere

The portion of the surface of a sphere included between two parallel planes, which intersect the sphere, is called a frustum. The distance between the two planes is called height or thickness of the frustum.

The volume of a frustum of a sphere can be found using the formula below:

The surface area of a frustum of a sphere can be found using the formula below:

Wednesday, 22 February 2017

The Volume of a Prism

The volume of a prism can be found by first finding the area of the cross-section, then multiplying the cross-section by the length of a prism.

In this prism, the red areas are the cross sections.

Take for example this question.

The cross section is the square with sides 20 cm. The area of this square is 20 cm x 20 cm = 400 cm2. The volume of the cuboid is thus 400 x 40 = 16000 cm3.

Take a more challenging question.

The cross section here is the triangle. The area of the triangle is 0.5 x 8 cm x 15 cm = 60 cm2. The volume of the prism is 60 x 10 = 600 cm3.

Try this question.

Wednesday, 15 February 2017

Logarithmic Tables

Before calculators, people learning logarithms had to use logarithmic tables of different bases. Here we will us base-10 logarithmic tables.

Say we wanted to find the log of 15. We then have to look on the first column of the logarithmic table and find 15.

As 15 lies between 10(101) and 100(102), the characteristic - the integer before the decimal place - will be 1 as the logarithm of 15 has to be between 1 and 2. The numbers after the decimal place is known as the mantissa.

In this case, 15 can be written as 15.0. The top row of numbers is the third digit of the number. As 0 is the third digit, we have to look at the cell where the column 0 and the row headed 15 meet. This cell is shown below.

The digits in the red box are the mantissa. Joining the mantissa with the characteristic show us that the log of 15 is 1.1761 (to 4 decimal places).

What if the number had two decimal places? If you wanted to find the log of a number like 25.29, you would follow the same steps, but you would also have to use the mean difference table.

The characteristic is 1, and as the digit after the decimal place is 2, we have to look at the cell where the row headed 25 meets the column headed 2. We do this because 2 and 5 are the first two digits of the number.

The mantissa in this case is .4014. However, as the fourth digit after the decimal place is 9, so we have to see where the row headed 25 meets the column headed 9 on the mean difference table. In some logarithmic tables, the mean difference table is next to the logarithmic table.

We see here that the mean difference is 15. We have to add the mean difference to the mantissa to find the digits behind the decimal place. 15 + 4014 is equal to 4029, so the log of 25.29 is 1.4029 (to 4 decimal places).

What if we had to find the log of 2.629? There does not seem to be any numbers less than 10 in the logarithmic table.
However, we can find the log of 2.629.

The first two digits are 2 and 6, so we have to look at the row headed 26. The third digit is 2, so we have to look at where the row and column meet.

The mantissa is .4183 and because 2.629 is between 0 and 10, its characteristic would be 0.
As the fourth digit is 9, we have to see where the row 26 meets with the column 9 in the mean difference table.

The mean difference is 15, so the digits after the decimal place is 15 + 4183, which equals 4198. The log of 2.629 is 0.4198.

A pattern emerges as we find the logs of numbers with the same digits, but the decimal place is in different places.
The log of 2.629 is 0.4198.
The log of 26.29 is 1.4198.
The log of 262.9 is 2.4198.
For numbers with the same digit order (to 4 significant figures) the mantissa is the same, but the characteristic increases by 1 for every shift left by the digits, and decreases by 1 for every shift to the right.

Try to solve these logarithms using the logarithmic tables above. Put your answers in the comments below.

Tuesday, 14 February 2017


Linear Inequalities

Consider this linear inequality:

To solve for x, we need to do what we do when we solve equations. We see that:

However, when we divide both sides of an inequality by a negative number, we also have to reverse the inequality sign, like below.

The graph for this inequality is x = -1, but every value less than -1 is shaded. As the inequality sign is 'less than', the line is dotted. However, if you have an inequality sign of greater than or equal to or less than or equal to, the line will not be dotted.

Quadratic Inequalities

Consider this quadratic inequality:

To solve for x, we need to first factorise the left hand side of the inequality.

Now we must sketch the graph to see where the values of y are greater than 0.

Here we see that y is greater than 0 where x is more than -1 and less than -2, so the answer to this inequality will be:

Monday, 13 February 2017

Sectors of circles

The formula to find the arc length of a sector is shown in Figure 1.
Figure 1

The formula to find the area of a sector is shown in Figure 2.
Figure 2

Take for example Figure 3.
Figure 3

The area of this sector can be found out like below.

The arc length of the sector can be found out like below:

The Quadratic Formula

Another way to solve tricky quadratic equations aside from completing the square is by using the quadratic formula.

For example, to solve the below equation, we would have to use the quadratic equation. Some questions will ask you to give your answer to 3 significant figures or in surd form. For the purposes of this example, we will write the answer in surd form.

If you are asked to give the answer to 3 significant figures, the answer would be like below.

Sunday, 12 February 2017

Completing the Square

Some quadratic equations cannot be factorised normally, so to factorise such equations, we must complete the square.

To complete the square, equations with an x squared coefficient of 1 should be written in this form:

To complete the square, the x coefficient has to be divided by two. This value would then become the value of p. The value of p has to then be squared and multiplied by -1. This value has to then be added or subtracted from the constant.
Consider this example:

In addition to this, equations with an x squared coefficient not 1 should be written in this form:

The coefficient of x squared has to be divided by the x squared coefficient as well as the x coefficient. Then the square can be completed.
An example is shown below.

Solving equations by completing the square

To solve equations by completing the square, the value of x has to be isolated, or left on its own.
Consider this, from the previous example:

Finding minimum and maximum points by completing the square

The value of x in the completed square which makes the equation in the bracket equal 0 is the x-coordinate of the point, and the value of y is the value of the constant. For example, the minimum point of the second equation is (-2, -13).

Saturday, 11 February 2017

Basic Integration

Integration is the reverse of differentiation.

The rule for integration is shown below.

This is the symbol of integration.

This is included because you are integrating with respect to x.

C is the constant of integration. When you differentiate, constants disappear, so lots of functions have the same derivative. When you integrate, the constant is unknown, so you have to write +c at the end.

Consider this equation.

Putting this into the above formula shows us this:

Similar to differentiation, with an equation with many different terms of x, each separate one is integrated separately. For instance: