Number : Real Numbers

Real Numbers

• The real number system contains all the rational and irrational numbers.
• Each real number has two attributes: size (magnitude) and sign.

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Rational Numbers

• ​Natural numbers, whole numbers, integers, some decimal numbers, as well as fractions and mixed numbers are subsets of rational numbers.
• A number that can be expressed as \(\frac{a}{b}\), where a and b are integers, and b​ ≠ 0. (For example: 4, 0, \(\frac{2}{3}\), \(\frac{-9}{5}\).)
• A number that can be expressed in decimal form, consisting of a finite sequence of digits to the right of the decimal point or an infinite and periodic sequence of digits. (For example: \(1.\overline{625}\), \(0.5\), \(4.75\).)

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Irrational Numbers

A number that cannot be represented as a fraction of integers and when expressed as a decimal, it does not terminate or repeat. (For example: \(3.13159…\), \(\sqrt{2}\), \(4.515 515 551…\), \(\pi \).)

There are two subgroupings: The first is rational numbers represented by the symbol “Q” and the second is irrational numbers represented by the symbol “Q prime”. Some examples of rational numbers are zero decimal zero three three repeating, negative 2 decimal five and seven ninths. Within the rational numbers is the subgrouping “Integers” represented by the symbol “Z”. Some examples of integer numbers are negative three, negative two, negative one, zero, one, two, three, et cetera. Within the Integer numbers is the subgrouping “Whole Numbers” represented by the symbol “W”. Some examples of whole numbers are: Zero, one, two, three, et cetera. Within the grouping of whole numbers is the subgrouping of “Natural numbers”, represented by the symbol “N”. Some examples of natural numbers are: One, two, three, et cetera. There are no subgroupings for Irrational numbers. Some examples are Pi, square root of 5, negative 1 decimal four five zero nine seven three two five continuing and square root of 2.

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Density, Infinity and Limit

• Density can be seen as the probability of choosing a number within a given subset. This concept can be used to compare the density of two subsets. For example, the set of square numbers between 0 and 100 is less dense than the set of odd numbers between 0 and 100 because there are more numbers in the second set.
• Infinity is a concept that helps characterize sets and certain decimal numbers. For example, the set of integers is infinite, and irrational decimal numbers have an infinite number of digits after the decimal point.
• The limit of a pattern or function, or the result as the number of terms increases, can be determined by observing its long-term behaviour. For example, if we observe the sequence 9, 3, 1, \(\frac{1}{3}\), \(\frac{1}{9}\), …, we can deduce that the limit will be 0.

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Numerical Representations

Numbers can be represented numerically in a variety of ways.

​Integers and positive and negative decimal fractions can be represented as a terminating decimal number.

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Operations

• The actions in a situation inform the choice of operation. The same operation can describe:
  • Situations that involve changing (joining or separating), combining or comparing can be represented with addition and subtraction.
  • Situations that involve equal groups (or rates), ratio comparisons or arrays can be represented with multiplication and division.
• Numbers can be decomposed into parts and can be composed by combining the parts. Decomposing and composing numbers may be used as a strategy to perform operations. For example: \(-3\frac{1}{4}+5\frac{3}{8}\) can be decomposed as \(-3+\left(-\frac{1}{8}\right)+\left(-\frac{1}{8}\right)+5+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\). To simplify operations, use the commutative property and opposites numbers to create zero pairs \(-3+5+\left[\left(-\frac{1}{8}\right)+\left(\frac{1}{8}\right)\right]+\left[\left(-\frac{1}{8}\right)+\left(\frac{1}{8}\right)\right]+\frac{1}{8}\), resulting in \(2\frac{1}{8}\).
• Factoring a number involves expressing it as a product of smaller numbers or factors. For example, 10 can be expressed as \(5 \ \times 2\) using the factors 5 and 2.
• The identity and zero elements, as well as the properties of operations such as commutativity, associativity, and distributivity, can be applied to any type of number.
  • Zero element: \(a \times 0 = 0,\, 0\,(x\, – a) = 0\).
  • Identity element: \(a + 0 = a,\, a - 0 = a,\, a \times 1 = a,\, a \div 1 = a\).
  • Associativity: \((a + b) + c = a + (b + c), (a \times b) \times c = a \times (b \times c)\).
  • ​​C​ommutativity: \(a + b = b + a,\, a \times b = b \times a\).
  • Distributivity: \(a \times (b + c) = (a \times b) + (a \times c)\).
• The order of operations property (BEDMAS) needs to be followed when given a numerical expression that involves multiple operations.
• A rational number can be used to estimate an irrational number

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Types of Numbers

The following is a summary of the types of numbers that students work with in each grade.