Mathematics Glossary 2

Notation

Less Than <

The symbol < is used to indicate that the quantity on the left of the < is smaller than the quantity to the right. The inequality a < b corresponds to the English statements:

  • a is less than b,
  • a is smaller than b. (Note that this means that a can not be equal to b.)

Less Than or Equal To \le

The symbol \le is used to indicate that the quantity on the left of the < is smaller than or equal to the quantity to the right. The inequality a\le b corresponds to the English statements:

  • a is less than or equal to b,
  • a is at most b.  (Note that this means that a can be equal to b.)

Greater Than >

The symbol > is used to indicate that the quantity on the left of the < is smaller than the quantity to the right. The inequality a > b corresponds to the English statements:

  • a is greater than b,
  • a is more than b. (Note that this means that a can not be equal to b.)

Greater Than or Equal To

The symbol \ge is used to indicate that the quantity on the left of the < is smaller than or equal to the quantity to the right. The inequality a\ge b corresponds to the English statements:

  • a is greater than or equal to b,
  • a is at least b.  (Note that this means that a can be equal to b.)

Bounded Intervals [a,b], (a,b), [a,b), (a,b]

The closed interval [a,b] is the section of the number line between a and b inclusive. It corresponds to the set values of x where a \le x \le b.

The open interval (a,b) is the section of the number line between a and b exclusive. It corresponds to the set of values of x where a < x < b.

The interval [a,b) is the section of the number line between a inclusive and b exclusive. It corresponds to the set of values of x where a \le x < b.

The interval (a,b] is the section of the number line between a exclusive and b inclusive. It corresponds to the set of values of x where a < x \le b.

Unbounded Intervals (-\infty, b], (-\infty, b), [a, \infty), (a, \infty), (-\infty, \infty)

The unbounded intervals (-\infty, b], (-\infty, b), [a, \infty), (a, \infty) correspond to the set of values of x given by the following inequalities:

(-\infty, b] corresponds to the set of values of x given by x \le b,

(-\infty, b) corresponds to the set of values of x given by x < b,

[a, \infty) corresponds to the set of values of x given by x \ge a,

(a, \infty) corresponds to the set of values of x given by x > a,

(-\infty, \infty) corresponds to the set of real numbers.

Vocabulary

Real Number Line

The real number line is a horizontal line with a zero at the centre where everything to the left of zero is negative numbers and everything to the right is positive numbers. Every real number has exactly one position on the line and every position on the line corresponds to a real number.

Number lines are useful for visualising set of real numbers, interval and values of x satisfying an inequality.

Interval

An interval is a subset of the real numbers corresponding to a section of the number line. The endpoints may or may not be included, for example [-3,2) is the interval from -3 (inclusive) to 2 (exclusive). THe square brackets [ indicates that the first number is included, and ( indicates that it is excluded. Similarly for ] and ) for the second number.

Inequalities

Inequalities are relationships between quantities or expressions where one is less than, less than or equal to, greater than or greater than or equal to the other.

Linear Inequalities

Linear inequalities involve one variable, and that variable does not occur as x^2 or any higher exponent.