In mathematics, you will have trouble understanding the material unless you clearly understand the *language of the words* that are being used. Here are some commonly used words and phrases that you need to know, and that will be repeated throughout the duration of this course.

## Integers

The values:

… −4, −3, −2, −1, 0, 1, 2, 3, 4 …

are called **integers **because *integers *are **positive** or **negative** whole numbers, including zero. The three dots on the ends indicate that the sequence of numbers continues indefinitely in both directions.

## Rational Numbers

The rational numbers include the integers and all other numbers that can be expressed as the quotient (the result of division) of two integers. In other words, a rational number is any number that can be expressed as a fraction. Some rational numbers are:

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7 is a *rational number* because all whole numbers can be expressed as the number over 1:

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## Irrational Numbers

These are numbers that cannot be expressed as a quotient. For example:

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All of these equations can never be the quotient of a fraction because they’re never ending numbers. Evaluating the √2 gives you 1.414213562 …

## Symbols of Equality and Inequality

Several symbols are used to show the relative positions of two quantities on the **number line**.

x = ymeans that xequalsy, and that x and y occupy the same position on the number line.

**x ≠ y **means that x and y are **not equal** and have different locations on the number line.

**x > y **means that x is **greater** than y, and x lies to the right of y on the number line.

**x < y** means that x is **less than** y, and x lies to the left of y on the number line.

**x ≈ y **means that x is **approximately equal** to y, and that x and y are near each other on the number line. Other symbols sometimes used for approximately equal to are ≅ and ≃.

## Reciprocal

The **reciprocal** is a function that takes any number and flips it (reciprocates). For example, the reciprocal of 5 is 1/5. Generally, this is summarized as:

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More examples:

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## Exponents

Given the mathematical expression:

**5**^{2}

This statement is the same as taking the **base** (positive 5) and multiply it in itself 2 times (**exponent of 2**). This can be read as *5 raised to the power of 2* or *5-squared*.

More examples:

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Notice the significance of the brackets in **#2**: by not enclosing the base in brackets, the **sign** of the answer is changes from being positive in **#1** to negative in **#2**.

## Absolute Value

The absolute value of a number is its **magnitude** regardless of its sign. It is written |n|, where n is any number. It is the distance between *n* and *zero* on the number line, without regard to direction. For example:

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Note: you must treat the absolute a *symbol of grouping*, much like parentheses.