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.
… −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.
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:
7 is a rational number because all whole numbers can be expressed as the number over 1:
These are numbers that cannot be expressed as a quotient. For example:
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 = y means that x equals y, 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 ≃.
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:
Given the mathematical expression:
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.
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.
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:
Note: you must treat the absolute a symbol of grouping, much like parentheses.