Chapter 6: Real Numbers and Their Representations

This set of flashcards covers key concepts related to real numbers, rational numbers, and their properties, operations, and representations.

What is a rational number?

A rational number is a quotient of two integers with the denominator not equal to zero.

How can rational numbers be expressed?

Rational numbers can be expressed as either a terminating or a repeating decimal.

What is the difference between terminating and repeating decimals?

A terminating decimal stops, while a repeating decimal has reoccurring sets of numbers.

What determines whether a rational number has a terminating decimal?

A rational number has a terminating decimal if the only prime factors of the denominator are 2, 5, or both.

What results in a repeating decimal?

A rational number results in a repeating decimal if prime numbers other than 2 or 5 appear in the prime factorization of the denominator.

What does it mean for a rational number to be in lowest terms?

A rational number is in lowest terms if the greatest common factor of the numerator and the denominator is 1.

How do you check if two fractions are equivalent?

You can check using the Cross-product Test for Equality of Rational Numbers.

What is the Cross-product Test for Equality of Rational Numbers?

For fractions a/b and c/d, it states that they are equal if and only if ad = bc.

What is the least common multiple used for in operations with rational numbers?

It is used when adding or subtracting rational numbers.

How do you add two rational numbers?

Convert to a common denominator, then add the numerators.

What is the formula for multiplying two rational numbers?

If a/b and c/d are rational numbers, then (a/b) * (c/d) = (ac)/(bd).

What is the formula for dividing two rational numbers?

Dividing is equivalent to multiplying by the inverse of the denominator: (a/b) ÷ (c/d) = (a/b) * (d/c).

What is the Density Property of Rational Numbers?

Between any two rational numbers, there always exists another rational number.

How is the arithmetic mean of two numbers calculated?

It is found by adding the two numbers and dividing the sum by 2.

What is a common method to express decimals as quotients of integers?

You can convert by expressing a decimal as a fraction or using multiplication for repeating decimals.

How do you write 0.85 as a fraction?

0.85 = 85/100 = 17/20 after simplification.

What is an irrational number?

An irrational number cannot be expressed as a quotient of integers and is represented by a nonrepeating, nonterminating decimal.

Give an example of a common irrational number.

Common examples include √2 and π.

What is the Product Rule for Square Roots?

For nonnegative real numbers, √a * √b = √(a*b).

What conditions exist for a square root to be in simplified form?

The radicand has no factors that are perfect squares, contains no fractions, and no radical appears in the denominator.

When can square root radicals be added or subtracted?

They can be combined if they have the same radicand, using the distributive property.

How do you simplify √18 - √32?

√18 - √32 = 3√2 - 4√2 = -√2.

How do you perform operations involving square roots?

Use the product and quotient rules and ensure radicals are similar for addition or subtraction.

Explain how to write repeating decimals as quotients of integers.

Set x equal to the decimal, multiply by 10^n (where n is the number of repeating digits), and solve the resulting equations.

What is the simplified form of 0.858585…?

It can be simplified to 85/99.

What is the significance of the greatest common factor in rational numbers?

It is used to determine if a fraction can be simplified to its lowest terms.

What defines rational numbers as dense?

For any two rational numbers, there are infinitely many rational numbers that can exist between them.

How do you convert fractions to decimals?

Divide the numerator by the denominator.

What is the result of 3/5 ÷ 7/15?

The result is 9/7.

When do you need to use least common denominators in operations?

You use it when adding or subtracting rational numbers that have different denominators.

What is a nonnegative real number regarding square roots?

A nonnegative real number is any real number that is zero or positive.