Survey of Math - Section 5.4: The Irrational Numbers

Rationalized Note: To rationalize a denominator that contains only a square root, multiply both the numerator and denominator of the fraction by the square root of a number that will result in the radicand in the denominator becoming a perfect square.

A denominator of a fraction is ___ when it contains no radical expressions.

Irrational numbers

A real number whose decimal representation is a​ nonterminating, nonrepeating decimal number is known as

Irrational numbers

An ___ is a real number number that cannot be written as the ratio of two integers.

Perfect square

Any number that is the square of a natural number is said to be a ___.

To add or subtract two or more square roots with the same radicand, add or subtract their coefficients while keeping the common radicand.

Review!!! Addition and Subtraction of Radicals

Is 1/3 rational or irrational?? Answer: RATIONAL 1/3 - when written as a decimal​ number, is a repeating decimal.

Review!!! Determine whether the number is rational or irrational.

Square root of 25 The number is a RATIONAL because it CAN be expressed as a ratio of two integers.

Review!!! Determine whether the number is rational or irrational.

Square root of 29 The number is IRRATIONAL because it CANNOT be expressed as a ratio of two integers.

Review!!! Determine whether the number is rational or irrational.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100 ...

Review!!! Perfect Square root numbers

Quotient Rule for Radicals (see image)

Review!!! Simplifying DIVISION of Radicals See image for "Product Rule for Radicals"

Product Rule for Radicals (See image)

Review!!! Simplifying Radicals See image for "Product Rule for Radicals"

False This statement is false. A composite number can have an irrational square root.

Square root of c is a rational number for any composite number c. True or false?

Radicand

The number inside the radical sign is called ___.

Coefficient

The number that multiplies a radical is called the radical's ___.

Itself

The principal​ (or positive) square root of a number​ n, written sqr root n is the positive number​ that, when multiplied by _, given n.

False. The product of an irrational and a rational number can be a rational number.

The product of an irrational and a rational number is always an irrational number. True or False?

True The sum or difference of two rational numbers is always a rational number. All rational numbers can be expressed as fractions with integers in the numerator and​ denominator, and therefore their sum can also be expressed as a similar​ fraction, making it a rational number.

The sum of any two rational numbers is always a rational number. True or False?

Radical sign

The symbol (see image) is called the ___

Rationalized

When the denominator of a fraction contains no radical​ expressions, it is said to be ___.