Numbers
Numbers
TYPES OF NUMBERS
Numbers are classified based on their properties.
Types of Numbers | ||
|---|---|---|
Type of Number | Description | Examples |
Natural number | a number greater than zero and does not contain a decimal or fractional part | {1, 2, 3, 4, …} |
Whole numbers | natural numbers and the number zero | {0, 1, 2, 3, 4, …} |
Integers | all whole numbers and their opposites | {…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …} |
Rational number | can be represented as a quotient of two integers (fraction) with a denominator that is not zero; any decimal part must terminate or resolve into a repeating pattern | −12, \( −\frac{4}{5}\), 0.36, 7.777…, \( 26\frac{1}{2}\) |
Irrational number | cannot be represented as fraction; an irrational decimal never ends and never resolves into a repeating pattern | \(−\sqrt{7}\), π, 0.34567989135… |
Real number | can be represented by a point on a number line | All rational and irrational numbers |
If a real number is a natural number (e.g., 50), then it is also an integer, a whole number, and a rational number.
Every natural number (except 1) is either prime or composite. A prime number is a natural number greater than 1 that can only be divided evenly by 1 and itself. For example, 7 is a prime number because it can only be divided by the numbers 1 and 7.
A composite number is a natural number greater than 1 that can be evenly divided by at least one other number besides 1 and itself. For example, 6 is a composite number because it can be divided evenly by 1, 2, 3, and 6.
Composite numbers can be factored into prime factors using factor trees. For example, 54 factors as 2 × 27, then 27 factors as 3 × 9, and 9 factors as 3 × 3, as shown below. The prime factorization of 54 is 2 × 3 × 3 × 3.
Factors are numbers that are multiplied by each other.
Exponents and Radicals
53 = 5 × 5 × 5 = 125
Rules for working with exponents are given below.
Operations with Exponents | |
|---|---|
Rule | Example |
\(a^{0} = 1\) | \(5^{0} = 1\) |
\(a^{−n} = \frac{1}{a^{n}}\) | \(5^{−3} = \frac{1}{5^{3}}\) |
\(a^{m}a^{n}=a^{m+n}\) | \(5^{3}5^{4}=5^{3+4}=5^{7}\) |
\((a^{m})^{n}=a^{mn}\) | \((5^{3})^{4}=5^{3(4)})\) |
\(\frac{a^{m}}{a^{n}}=a^{m−n}\) | \(\frac{5^{4}}{5^{3}}=5^{4−3}=5^{1}\) |
\((ab)^{n}=a^{n}b^{n}\) | \((5×6)^{3}=5^{3}6^{3}\) |
\((\frac{a}{b})^{n}=\frac{a^{n}}{b^{n}}\) | \((\frac{5}{6})^{3}=\frac{5^{3}}{6^{3}}\) |
\((\frac{a}{b})^{−n}=(\frac{b}{a})^{n}\) | \((\frac{5}{6})^{−3}=(\frac{6}{5})^{3}\) |
\(\frac{a^{−m}}{b^{−n}}=\frac{b^{n}}{a^{m}}\) | \(\frac{5^{−3}}{6^{−4}}=\frac{6^{4}}{5^{3}}\) |
- 5 is the square root of 25 because 52 = 25
- 5 is the cube root of 125 because 53 = 125
- 5 is the fourth root of 625, because 54 = 625
The symbol for finding the root of a number is the radical: \(\sqrt{}\). The number under the radical is called the radicand. By itself, the radical indicates a square root. Other numbers can be included in front of the radical to indicate different roots.
\(\sqrt{36}\)
\(^{4}\sqrt{1296}=6\longleftrightarrow 6^{4}=1296\)
Rules for working with radicals are given below.
Operations with Radicals | |
|---|---|
Rule | Example |
\(\sqrt[b]{ac}=\sqrt[b]{a}\sqrt[b]{c}\) | \(\sqrt[3]{81}=\sqrt[3]{27}\sqrt[3]{3}=3\sqrt[3]{3}\) |
\(\sqrt[b]{\frac{a}{c}}=\frac{\sqrt[b]{a}}{\sqrt[b]{ c}}\) | \(\sqrt{\frac{4}{81}}=\frac {\sqrt 4}{\sqrt 81}=\frac{2}{9}\) |
\(\sqrt[b]{a^{c}}=(\sqrt[b]{a})^{c}=a^{\frac{c}{b}}\) | \(\sqrt[3]{6^{2}}=(\sqrt[3]{6})^{2}=6^{\frac{2}{3}}\) |