Questions and Answers
What is the solution to the equation x + 2 = 13?
11
What is the solution to the equation x + 5 = 5?
0
What type of number was introduced to solve the equation x + 18 = 5?
Rational Number
What number is needed to solve the equation 2x = 3?
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Rational numbers are closed under division.
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Match the following properties with the correct type of numbers:
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Find using distributivity: $7 \times \left(-\dfrac{3}{12}\right) + 7 \times \left(\dfrac{5}{12}\right)$
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Find using distributivity: $9 \times 4 - 9 \times (-3)$
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What property allows you to compute $6 \times (3 \times 3)$ as $(6 \times 3) \times 3$?
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What is the result of $(-3) \times 1 - 6$?
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Name the property under multiplication used in the following equation: $13 \times (-2) = (-2) \times 13$
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Which property under multiplication allows you to compute $6 \times (3 \times 3)$ as $(6 \times 3) \times 3$?
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What is the result of $(-3) \times 1 - 6$?
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Name the property under multiplication used in the following equation: $13 \times (-2) = (-2) \times 13$
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What is the product of two rational numbers always?
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Study Notes
Introduction to Rational Numbers
- Rational numbers are used to solve simple equations, such as x + 2 = 13, where x = 11, a natural number.
- To solve equations like x + 5 = 5, we need to add the number zero to the collection of natural numbers, resulting in whole numbers.
- However, whole numbers are not sufficient to solve equations like x + 18 = 5, which requires the number -13, leading to the introduction of integers.
- Integers are not enough to solve equations like 2x = 3, which requires the number 3/2, leading to the concept of rational numbers.
Properties of Rational Numbers
Closure
- Whole numbers are closed under addition and multiplication, but not under subtraction and division.
- Integers are closed under addition, subtraction, and multiplication, but not under division.
- Rational numbers are closed under addition, subtraction, and multiplication, but not under division (unless we exclude zero).
Examples of Closure in Rational Numbers
- The sum of two rational numbers is always a rational number.
- The difference of two rational numbers is always a rational number.
- The product of two rational numbers is always a rational number.
- The division of two rational numbers is not always a rational number, unless we exclude zero.
Commutativity
Whole Numbers
- Addition and multiplication are commutative for whole numbers.
- Subtraction and division are not commutative for whole numbers.
Integers
- Addition and multiplication are commutative for integers.
- Subtraction and division are not commutative for integers.
Rational Numbers
- Addition and multiplication are commutative for rational numbers.
- Subtraction is not commutative for rational numbers.
- Division is not commutative for rational numbers.
Associativity
Whole Numbers
- Addition and multiplication are associative for whole numbers.
- Subtraction and division are not associative for whole numbers.
Integers
- Addition and multiplication are associative for integers.
- Subtraction and division are not associative for integers.
Rational Numbers
- Addition is associative for rational numbers.
- Subtraction is not associative for rational numbers.
- Multiplication is associative for rational numbers.
- Division is not associative for rational numbers.
Examples of Associativity in Rational Numbers
- Addition of three rational numbers can be performed in any order, and the result will be the same.
- Multiplication of three rational numbers can be performed in any order, and the result will be the same.### Properties of Rational Numbers
- Rational numbers are closed under the operations of addition, subtraction, and multiplication.
Commutative and Associative Properties
- The operations of addition and multiplication are commutative for rational numbers, meaning that the order of the numbers does not change the result.
- The operations of addition and multiplication are associative for rational numbers, meaning that the order in which the numbers are grouped does not change the result.
Additive Identity
- The rational number 0 is the additive identity for rational numbers, meaning that when 0 is added to any rational number, the result is the same rational number.
Multiplicative Identity
- The rational number 1 is the multiplicative identity for rational numbers, meaning that when 1 is multiplied by any rational number, the result is the same rational number.
Distributivity of Multiplication over Addition
- For all rational numbers a, b, and c, the following properties hold:
- a(b + c) = ab + ac
- a(b - c) = ab - ac
Examples of Distributivity
- -3 × (2 + (-5)) = -3 × 2 + (-3) × (-5)
- -3 × 2 - (-3) × (-5) = -6 - 15
- (-3 × 2) + (-3 × (-5)) = -6 + 15
Finding Rational Numbers between Two Given Rational Numbers
- The idea of mean helps us to find rational numbers between two rational numbers.
- There are countless rational numbers between any two given rational numbers.
Exercise 1.1
- Questions to practice the properties of rational numbers, including commutativity, associativity, and distributivity.
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Description
Introduction to rational numbers and solving simple equations, including the concept of variables and their values.
