Absolute Value and Integer Operations

Integer Operations

Integers are the positive and negative whole numbers, including zero:

Addition and Subtraction

• Same signs: add and keep the sign.
• Different signs: subtract and keep the sign of the larger absolute value.

For example:

Multiplication and Division

• Same signs give a positive result.
• Different signs give a negative result.

For example:

Absolute Value

The absolute value of a number is its distance from $0$ on the number line.

Absolute value is written using vertical bars:

For example:

and:

because both $5$ and $-5$ are $5$ units away from $0$.

In general:

Properties of Absolute Value

Product Rule

Example:

and:

Quotient Rule

Triangle Inequality

This means the distance traveled together cannot exceed the sum of the individual distances.

Example 1:

Find all solutions to the equation |x ^ 2 - 3x| = 4 .

Solution: We can split the problem into two cases: either

The first case yields the solutions 4 and -1; the second yields $\frac{3 \pm i \sqrt{7}}{2}$ If we restrict ourselves to real solutions, only 4 and -1 are acceptable.

Example 2: Solve the equation |(x + 2) / (3x - 1)| = 5

This gives two cases:

For the first case:

For the second case:

Since:

we must have:

Both solutions are valid.

Thus, the solutions are:

Sigma Notation

Sigma notation represents sums.

The Greek letter sigma is written as:

For example:

means:

which equals:

Example

means:

which equals:

Important Summation Formulae

Sum of the First (n) Natural Numbers

Proof

Let:

$$S = 1+2+3+\dots+n$$

Write the sum in reverse order:

$$S = n+(n-1)+(n-2)+\dots+1$$

Adding the two equations term by term:

$$2S=(n+1)+(n+1)+\dots+(n+1)$$

There are (n) terms, so:

$$2S=n(n+1)$$

Thus:

$$S=\frac{n(n+1)}{2}$$

Sum of the Squares of the First (n) Natural Numbers

Proof Sketch

One way to prove this formula is by mathematical induction.

Assume:

$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

Then:

$$\sum_{k=1}^{n+1} k^2 = \frac{n(n+1)(2n+1)}{6}+(n+1)^2$$

Factoring and simplifying gives:

$$\frac{(n+1)(n+2)(2n+3)}{6}$$

which matches the formula with (n+1).

Thus, the formula holds for all positive integers (n).

Sum of the Cubes of the First (n) Natural Numbers

Interesting Observation

The sum of cubes equals the square of the sum of the first (n) natural numbers.

That is:

$$1^3+2^3+\dots+n^3 = (1+2+\dots+n)^2$$

For example:

$$1^3+2^3+3^3 = 1+8+27 = 36$$

and:

$$(1+2+3)^2 = 6^2 = 36$$

This identity can also be proved using induction.

Pi Notation

Pi notation represents products.

The Greek capital pi is written as:

For example:

means:

which equals:

Another example:

means:

which equals:

Part 2

Great work if you read all the way till here! Due to File-Size Constraints, this module has been divided into 2 parts. Continue this Module in part 2. Good Luck!