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.

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.

Inequalities

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:

Indices

Indices are ways to write multiplications of the same number. For example:

The number 4 is called the power or index.

The first rule of indices is shown below:

This shows that two powers of the same base multiplied together is equal to the powers added together.
If you were asked to answer the below expression, the answer would be found using the first rule of indices.

The second rule of indices is shown below:

This shows us that the power of a power is equal to the powers multiplied together.
An example is shown below:

The third rule of indices is this:

This shows us that an index divided by another index is equal to the two indexes subtracted.
An example is shown below.

The fourth rule of indices is here:

Anything to the power of 0 is 1.

The fifth rule of indices is shown here.

For example, a to the power of a half is equal to the square root of a.

Logarithms

To understand the rules of logarithms we have to understand indices.

In this example, the number 4 is the power or the exponent. The number 2 is the base.

An alternative way to write this expression is illustrated below.

This shows us the first rule of logarithms.

We know that anything to the power of one is itself, like the expression below.

The version of the above expression logarithmically is shown below.

We have now found the second rule of logarithms.

The third rule of logarithms is shown below.

Consider this question.

Following the third rule of logarithms where x is 2 and y is 4, we see that

The fourth rule of logarithms is with regards to powers in log expressions.

Consider this expression.

Following the fourth rule of logarithms, we see that

The fifth rule of logarithms goes as follows:

Consider this expression.

The expression can be rewritten as shown below.

as 6 divided by 3 is 2.



The last log rule you need to remember is shown below.

as