Real Numbers and Rational Numbers

12 Questions

How do you find the sum of two rational numbers?

Add the fractions or decimals

What is the identity property for multiplication?

The result of multiplying any number by one is the original number

Which property states that for multiplication and addition, the order of the numbers being multiplied and added does not matter?

Commutative property

What happens if the denominator in a rational number division is zero?

The answer is undefined

Which type of number cannot be expressed as a ratio of integers?

Irrational numbers

What are irrational numbers examples of?

$\sqrt{2}$ and $\pi$

Which type of numbers can be expressed as a ratio between two integers?

Rational numbers

What property states that the sum of two rational numbers is always a rational number?

Closure under addition

Which of the following is an example of a rational number?

0.75

What is the subset of real numbers that includes all integers, fractions, and decimals with repeating patterns?

Rational numbers

Which property ensures that the order of addition or multiplication does not matter for rational numbers?

Commutative properties

What distinguishes irrational numbers from rational numbers?

Irrational numbers cannot be expressed as ratios of integers.

Study Notes

Real Numbers: Rational Numbers

In mathematics, real numbers are the set of numbers that include all rational numbers and all irrational numbers. Rational numbers are numbers that can be expressed as a ratio between two integers, while irrational numbers cannot be expressed as a ratio of integers.

Rational Numbers

Rational numbers are a subset of real numbers. They are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. This includes all integers (whole numbers, positive and negative), as well as fractions and decimal numbers that terminate or have a repeating pattern.

Examples of Rational Numbers

3/2 (as a fraction) or 1.5 (as a decimal)
-5 (as an integer) or -5/1 (as a fraction)
2.2222... (a decimal that repeats forever) or 2.2 (as a fraction)
0.5 (as a decimal or as the fraction 1/2)

Properties of Rational Numbers

Closure under addition: The sum of two rational numbers is always a rational number.
Closure under subtraction: The difference between two rational numbers is always a rational number.
Closure under multiplication: The product of two rational numbers is always a rational number.
Closure under division: The quotient of two rational numbers is always a rational number (except for division by zero).
Associative properties: The order of addition and multiplication does not matter.
Commutative properties: The order of the numbers being added or multiplied does not matter.
Identity properties: For addition, any rational number added to zero results in the original number. For multiplication, any number multiplied by one results in the original number.
Distributive property: For multiplication and addition, the order of the numbers being multiplied and added does not matter.

Finding Rational Numbers

Addition: To find the sum of two rational numbers, just add the corresponding fractions or decimals.
Subtraction: To find the difference between two rational numbers, subtract the smaller from the larger (or vice versa).
Multiplication: To find the product of two rational numbers, multiply the numerators and denominators separately, then simplify if possible.
Division: To find the quotient of two rational numbers, divide the numerator by the denominator. If the denominator is zero, the quotient is undefined.

Real Numbers vs. Rational Numbers vs. Irrational Numbers

Real numbers include both rational and irrational numbers. Rational numbers are in the form of p/q, where p and q are integers, and q is not equal to zero. Irrational numbers, on the other hand, cannot be expressed as a ratio of integers. Examples of irrational numbers include √2, π, and e.