A rational number is any number that can be expressed as the fraction of two integers, with the denominator y not equal to zero. Since y may be equal to 1, every integer is a rational number.
An irrational number is a number that cannot be expressed as a fraction. It is represented by a decimal that does not terminate or repeat.
Examples of irrational numbers are pi(π) = 3.142.. and = 1.4142…
The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line.
Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. Only the number 0 is neither negative nor positive.This video shows how to place rational and irrational numbers on the number line.
We can use an inequality to represent an interval with one endpoint. For example,
x < 5
x ≤ 5
x > 5
x ≥ 5
We can use a double inequality to represent an interval with two endpoints. For example,
1 < x < 3
1 ≤ x < 3
1 < x ≤ 3
1 ≤ x ≤ 3
The entire real number line is also considered to be an interval
The absolute value of a number describes the distance of the number on the number line from 0. It does not consider which direction from 0 the number lies. The absolute value of a number is always positive.
The absolute value of 3 is 3 which means that its distance from 0 is 3 units
The absolute value of −3 is also 3 which means that its distance from 0 is 3 units
The symbol for absolute value is two straight lines || (called bars) surrounding the number or expression for which you wish to indicate absolute value.
| 6 |
= 6 which means the absolute value of 6 is 6.
|| 3 − 7 | = | −4 | = 4
|| 6 +
3(−5) | = | 6 − 15 | = | −9 | = 9
It is important to note that the absolute value bars do NOT work in the
same way as parentheses. Recall that − (−5) = (−1) × (−5) = +5.
However, for the absolute value it is done by removing the absolute bar and then performing the sign operation.
−| −5 | = −(+5) = (−1) × (+5) = −5
There are several general properties of real numbers that are used frequently.
If a, b, and c are real numbers, then
a + b = b + a and ab = ba.
For example, 3 + 5 = 5 + 3 = 8 and (−4)(6) = (6)(−4) = −24
(a + b) + c = a + (b + c) and (ab)c = a(bc).
For example, (8 + 2) + 5 = 8 + (2 + 5) = 15 and (8) = 8() = 16
a (b + c) = ab + ac
For example, 7(4 + 16) = (7)(4) + (7)(16) = 140
a + 0 = a, (a)(0) = 0, and (a)(1) = a
If ab = 0, then either a = 0 or b = 0 or both.
Division by 0 is not defined; for example, 4 ÷ 0, 0 ÷ 0 are undefined.
If both a and b are positive, then both a + b and ab are positive.
If both a and b are negative, then a + b is negative and ab is positive.
If a is positive and b is negative, then ab is negative.
|a + b|≤ |a| + |b| This is known as the triangle inequality.
If a > 1, then a2 > a If 0 < b < 1 then b2 < b, for example,
42 = 16 > 4,
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