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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

Exponential expressions, such as 53, contain a base and an exponent. The exponent indicates how many times to use the base as a factor.  In the expression 53, 5 is the base and 3 is the exponent. The value of 53 is found by multiplying 5 by itself 3 times:

53 = 5 × 5 × 5 = 125

Rules for working with exponents are given below.

Operations with Exponents
Rule
Example
\(a^{0} = 1\)
\(5^{0} = 1\)
\(a^{−n} = \frac{1}{a^{n}}\)
\(5^{−3} = \frac{1}{5^{3}}\)
\(a^{m}a^{n}=a^{m+n}\)
\(5^{3}5^{4}=5^{3+4}=5^{7}\)
\((a^{m})^{n}=a^{mn}\)
\((5^{3})^{4}=5^{3(4)})\)
\(\frac{a^{m}}{a^{n}}=a^{m−n}\)
\(\frac{5^{4}}{5^{3}}=5^{4−3}=5^{1}\)
\((ab)^{n}=a^{n}b^{n}\)
\((5×6)^{3}=5^{3}6^{3}\)
\((\frac{a}{b})^{n}=\frac{a^{n}}{b^{n}}\)
\((\frac{5}{6})^{3}=\frac{5^{3}}{6^{3}}\)
\((\frac{a}{b})^{−n}=(\frac{b}{a})^{n}\)
\((\frac{5}{6})^{−3}=(\frac{6}{5})^{3}\)
\(\frac{a^{−m}}{b^{−n}}=\frac{b^{n}}{a^{m}}\)
\(\frac{5^{−3}}{6^{−4}}=\frac{6^{4}}{5^{3}}\)

Finding the root of a number is the inverse of raising a number to a power. Roots are named for the power on the base:

  • 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
\(\sqrt[b]{ac}=\sqrt[b]{a}\sqrt[b]{c}\)
\(\sqrt[3]{81}=\sqrt[3]{27}\sqrt[3]{3}=3\sqrt[3]{3}\)
\(\sqrt[b]{\frac{a}{c}}=\frac{\sqrt[b]{a}}{\sqrt[b]{ c}}\)
\(\sqrt{\frac{4}{81}}=\frac {\sqrt 4}{\sqrt 81}=\frac{2}{9}\)
\(\sqrt[b]{a^{c}}=(\sqrt[b]{a})^{c}=a^{\frac{c}{b}}\)
\(\sqrt[3]{6^{2}}=(\sqrt[3]{6})^{2}=6^{\frac{2}{3}}\)
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