Podcast
Questions and Answers
Which of the following numbers is NOT a natural number?
Which of the following numbers is NOT a natural number?
- 1
- 100
- 5
- 0 (correct)
Which set includes zero and all positive integers?
Which set includes zero and all positive integers?
- Rational numbers
- Whole numbers (correct)
- Natural numbers
- Integers
Which of the following is an example of an integer?
Which of the following is an example of an integer?
- $\sqrt{2}$
- -5 (correct)
- $ rac{1}{2}$
- 3.14
Which of the following numbers is a rational number?
Which of the following numbers is a rational number?
Which set encompasses both rational and irrational numbers?
Which set encompasses both rational and irrational numbers?
Which of the following is an example of a non-real number?
Which of the following is an example of a non-real number?
How can a rational number be expressed?
How can a rational number be expressed?
Which of the following is a characteristic of irrational numbers?
Which of the following is a characteristic of irrational numbers?
Which type of decimal number is always a rational number?
Which type of decimal number is always a rational number?
What is the first step in converting a recurring decimal into a rational number?
What is the first step in converting a recurring decimal into a rational number?
When is a decimal form considered rational?
When is a decimal form considered rational?
What mathematical operation transforms an irrational number into a rational approximation?
What mathematical operation transforms an irrational number into a rational approximation?
What determines the rounding direction when rounding off a decimal number?
What determines the rounding direction when rounding off a decimal number?
What happens to the digit to be rounded up if it is 9?
What happens to the digit to be rounded up if it is 9?
What is a surd?
What is a surd?
Which of the following is NOT a surd?
Which of the following is NOT a surd?
What should you identify first when estimating surds?
What should you identify first when estimating surds?
What is the numerical factor in a term called?
What is the numerical factor in a term called?
What is a term without a variable called?
What is a term without a variable called?
What is an expression with two terms called?
What is an expression with two terms called?
Which of the following correctly represents the product of two binomials $(ax + b)(cx + d)$?
Which of the following correctly represents the product of two binomials $(ax + b)(cx + d)$?
What is the mathematical process of breaking down an expression into simpler expressions that, when multiplied together, give the original expression?
What is the mathematical process of breaking down an expression into simpler expressions that, when multiplied together, give the original expression?
Which of the following is the correct factorisation of the difference of two squares $a^2 - b^2$?
Which of the following is the correct factorisation of the difference of two squares $a^2 - b^2$?
Which of the following is the correct factorisation of the sum of two cubes $x^3 + y^3$?
Which of the following is the correct factorisation of the sum of two cubes $x^3 + y^3$?
What is the first step in simplifying algebraic fractions?
What is the first step in simplifying algebraic fractions?
What operation is equivalent to dividing by a fraction?
What operation is equivalent to dividing by a fraction?
Which of the following numbers is both rational and an integer?
Which of the following numbers is both rational and an integer?
Which of the following expressions represents the correct procedure for multiplying a binomial $(A + B)$ and a trinomial $(C + D + E)$?
Which of the following expressions represents the correct procedure for multiplying a binomial $(A + B)$ and a trinomial $(C + D + E)$?
Given that $x$ is a real number and $x^2 = -1$, what type of number is $x$?
Given that $x$ is a real number and $x^2 = -1$, what type of number is $x$?
What is the result of simplifying the following expression: $\frac{x^2 - 4}{x + 2}$?
What is the result of simplifying the following expression: $\frac{x^2 - 4}{x + 2}$?
How would you classify the number $0.12345678910111213...$?
How would you classify the number $0.12345678910111213...$?
If $a$ and $b$ are irrational numbers, which of the following statements is always true?
If $a$ and $b$ are irrational numbers, which of the following statements is always true?
Evaluate and simplify: $\frac{x^3 + y^3}{x + y}$
Evaluate and simplify: $\frac{x^3 + y^3}{x + y}$
Let $S = \sqrt{2} + \sqrt{3}$. Which of the following statements is correct?
Let $S = \sqrt{2} + \sqrt{3}$. Which of the following statements is correct?
To which set of numbers does $0$ belong, but $1$ does not?
To which set of numbers does $0$ belong, but $1$ does not?
Which of the following correctly describes the relationship between integers ($Z$) and whole numbers ($N_0$)?
Which of the following correctly describes the relationship between integers ($Z$) and whole numbers ($N_0$)?
Which of the following numbers is classified as an irrational number?
Which of the following numbers is classified as an irrational number?
Which of the following statements is true regarding real numbers?
Which of the following statements is true regarding real numbers?
What distinguishes an imaginary number from a real number?
What distinguishes an imaginary number from a real number?
Which of the following numbers can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$?
Which of the following numbers can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$?
Which statement accurately describes irrational numbers?
Which statement accurately describes irrational numbers?
Which of the following decimal forms is characteristic of rational numbers?
Which of the following decimal forms is characteristic of rational numbers?
When converting the recurring decimal $0.\overline{3}$ into a rational number, what is a crucial initial step?
When converting the recurring decimal $0.\overline{3}$ into a rational number, what is a crucial initial step?
Under what condition is a decimal form regarded as representing a rational number?
Under what condition is a decimal form regarded as representing a rational number?
Which operation is typically used to transform an irrational number into a rational approximation?
Which operation is typically used to transform an irrational number into a rational approximation?
What factor determines the direction to round when approximating a decimal number?
What factor determines the direction to round when approximating a decimal number?
What adjustment is made if the digit to be rounded up is a '9'?
What adjustment is made if the digit to be rounded up is a '9'?
What is a surd fundamentally?
What is a surd fundamentally?
Which of the following is not considered a surd?
Which of the following is not considered a surd?
What is the primary initial step in estimating surds?
What is the primary initial step in estimating surds?
In the algebraic term $5x^2$, what is the coefficient?
In the algebraic term $5x^2$, what is the coefficient?
What is the correct terminology for a term in an algebraic expression that does not contain a variable?
What is the correct terminology for a term in an algebraic expression that does not contain a variable?
What is the name given to an algebraic expression that consists of precisely two terms?
What is the name given to an algebraic expression that consists of precisely two terms?
What is the result of applying the distributive property to multiply a monomial $a$ with a binomial $(x + y)$?
What is the result of applying the distributive property to multiply a monomial $a$ with a binomial $(x + y)$?
What is the inverse operation of expanding algebraic expressions?
What is the inverse operation of expanding algebraic expressions?
What is the result of completely factorising the expression $4x^2 - 9$?
What is the result of completely factorising the expression $4x^2 - 9$?
How does the sum of two cubes, $a^3 + b^3$, factorise?
How does the sum of two cubes, $a^3 + b^3$, factorise?
What is the initial step in simplifying a complex algebraic fraction?
What is the initial step in simplifying a complex algebraic fraction?
Which operation is mathematically equivalent to division by a fraction?
Which operation is mathematically equivalent to division by a fraction?
Which expression accurately shows the product of a binomial $(P + Q)$ and a trinomial $(R + S + T)$?
Which expression accurately shows the product of a binomial $(P + Q)$ and a trinomial $(R + S + T)$?
Given that $x$ is a non-zero real number, how would you classify $\frac{1}{x}$?
Given that $x$ is a non-zero real number, how would you classify $\frac{1}{x}$?
Simplify the expression: $\frac{x^2 - 9}{x - 3}$
Simplify the expression: $\frac{x^2 - 9}{x - 3}$
How would you categorise the number $3.141592653589793...$?
How would you categorise the number $3.141592653589793...$?
If $p$ is a rational number and $q$ is an irrational number, what can be said about $p + q$?
If $p$ is a rational number and $q$ is an irrational number, what can be said about $p + q$?
Simplify the expression: $\frac{a^3 - b^3}{a - b}$
Simplify the expression: $\frac{a^3 - b^3}{a - b}$
Consider $T = \sqrt{5} - \sqrt{2}$. Which of the following statements accurately describes $T$?
Consider $T = \sqrt{5} - \sqrt{2}$. Which of the following statements accurately describes $T$?
Which of the following numbers is an element of the set of integers but not of the set of natural numbers?
Which of the following numbers is an element of the set of integers but not of the set of natural numbers?
How many integer solutions exist for the equation $x^2 = 5$?
How many integer solutions exist for the equation $x^2 = 5$?
Given the expression $P = (A + B)^2$, where $A$ is rational and $B$ is irrational, what can generally be said about $P$?
Given the expression $P = (A + B)^2$, where $A$ is rational and $B$ is irrational, what can generally be said about $P$?
Identify the correct factorisation of the quadratic trinomial $x^2 + 5x + 6$.
Identify the correct factorisation of the quadratic trinomial $x^2 + 5x + 6$.
Which number is both a whole number and an integer but not a natural number?
Which number is both a whole number and an integer but not a natural number?
Consider the number $N = 0.123123123...$ where the digits '123' repeat indefinitely. Is $N$ rational or irrational?
Consider the number $N = 0.123123123...$ where the digits '123' repeat indefinitely. Is $N$ rational or irrational?
Which set of numbers does $\sqrt[3]{-8}$ belong to?
Which set of numbers does $\sqrt[3]{-8}$ belong to?
Suppose $x$ and $y$ are two irrational numbers such that $x = \sqrt{a}$ and $y = \sqrt{b}$ where $a$ and $b$ are distinct prime numbers. Which of the following is true about their product $xy$?
Suppose $x$ and $y$ are two irrational numbers such that $x = \sqrt{a}$ and $y = \sqrt{b}$ where $a$ and $b$ are distinct prime numbers. Which of the following is true about their product $xy$?
Define $f(x) = x + \frac{1}{x}$ for all non-zero real numbers $x$. What type of number is $f(\sqrt{2})$?
Define $f(x) = x + \frac{1}{x}$ for all non-zero real numbers $x$. What type of number is $f(\sqrt{2})$?
Which set of numbers includes all integers and their fractional parts?
Which set of numbers includes all integers and their fractional parts?
Which of the following numbers is an element of the set of whole numbers?
Which of the following numbers is an element of the set of whole numbers?
Which of the following is considered a non-real number?
Which of the following is considered a non-real number?
Which of the following statements correctly describes the relationship between the sets of numbers?
Which of the following statements correctly describes the relationship between the sets of numbers?
If a decimal number is non-terminating and non-repeating, to which set does it belong?
If a decimal number is non-terminating and non-repeating, to which set does it belong?
What is the key characteristic that distinguishes rational numbers from irrational numbers?
What is the key characteristic that distinguishes rational numbers from irrational numbers?
What is the decimal representation of a rational number?
What is the decimal representation of a rational number?
When converting a recurring decimal to a rational number, what is the purpose of multiplying by a power of $10$?
When converting a recurring decimal to a rational number, what is the purpose of multiplying by a power of $10$?
What happens to an irrational number when it is rounded off to a certain number of decimal places?
What happens to an irrational number when it is rounded off to a certain number of decimal places?
In the process of rounding off a decimal number, what action is taken if the digit immediately following the desired decimal place is 4?
In the process of rounding off a decimal number, what action is taken if the digit immediately following the desired decimal place is 4?
During the rounding of a decimal number to a specific decimal place, what adjustment is made to the preceding digit if the digit to be rounded up is $9$?
During the rounding of a decimal number to a specific decimal place, what adjustment is made to the preceding digit if the digit to be rounded up is $9$?
Which of the following expressions represents a surd?
Which of the following expressions represents a surd?
Before estimating the value of $\sqrt{11}$, what should you identify?
Before estimating the value of $\sqrt{11}$, what should you identify?
In the algebraic expression $7x^3$, what is the coefficient?
In the algebraic expression $7x^3$, what is the coefficient?
What do you call a term in an algebraic expression that does not contain any variables?
What do you call a term in an algebraic expression that does not contain any variables?
An algebraic expression consisting of exactly two terms is called a what?
An algebraic expression consisting of exactly two terms is called a what?
What is the product of the monomial $3a$ and the binomial $(2x + y)$?
What is the product of the monomial $3a$ and the binomial $(2x + y)$?
What is the name of the process that reverses expansion, expressing an algebraic expression as a product of its factors?
What is the name of the process that reverses expansion, expressing an algebraic expression as a product of its factors?
What is the factorisation of the expression $9x^2 - 16$?
What is the factorisation of the expression $9x^2 - 16$?
How does the expression $x^3 + 8$ factorise?
How does the expression $x^3 + 8$ factorise?
When simplifying algebraic fractions, what is the first step?
When simplifying algebraic fractions, what is the first step?
What is the classification of $\sqrt{-9}$?
What is the classification of $\sqrt{-9}$?
What is the simplified form of the algebraic fraction $\frac{x^2 - 1}{x + 1}$?
What is the simplified form of the algebraic fraction $\frac{x^2 - 1}{x + 1}$?
Which of the following is true for any integer $n$?
Which of the following is true for any integer $n$?
If $p$ is a rational number and $q$ is an irrational number, then what is $p + q$?
If $p$ is a rational number and $q$ is an irrational number, then what is $p + q$?
Consider $X = \sqrt{3} + 1$. Which of the following is true?
Consider $X = \sqrt{3} + 1$. Which of the following is true?
Which of the following numbers belongs in the set of integers, but not in the set of natural numbers?
Which of the following numbers belongs in the set of integers, but not in the set of natural numbers?
How many real roots does the equation $x^2 + 1 = 0$ have?
How many real roots does the equation $x^2 + 1 = 0$ have?
Given the number $N = 0.33333...$ where the digit $3$ repeats indefinitely. What type of number is $N$?
Given the number $N = 0.33333...$ where the digit $3$ repeats indefinitely. What type of number is $N$?
To which set of numbers does the result of $\sqrt[3]{-1}$ belong?
To which set of numbers does the result of $\sqrt[3]{-1}$ belong?
Suppose $x$ and $y$ are irrational numbers. Which of the following must always be true?
Suppose $x$ and $y$ are irrational numbers. Which of the following must always be true?
Given that $a$ and $b$ are positive integers, and $a$ is a factor of $b$, what type of number is $\frac{b}{a}$?
Given that $a$ and $b$ are positive integers, and $a$ is a factor of $b$, what type of number is $\frac{b}{a}$?
Let $f(x) = x + \frac{1}{x}$ for all non-zero real numbers $x$. What type of number is $f(\sqrt{3})$?
Let $f(x) = x + \frac{1}{x}$ for all non-zero real numbers $x$. What type of number is $f(\sqrt{3})$?
Let's define a new operation $\star$ such that $a \star b = a^2 + b$, where $a$ is an irrational number and $b$ is a rational number. What can be said about the result of $a \star b$?
Let's define a new operation $\star$ such that $a \star b = a^2 + b$, where $a$ is an irrational number and $b$ is a rational number. What can be said about the result of $a \star b$?
Define an operation $\S$ such that for any two real numbers $x$ and $y$, $x \S y = \frac{x + y}{xy}$. If both $x$ and $y$ are irrational, is $x \S y$ necessarily irrational?
Define an operation $\S$ such that for any two real numbers $x$ and $y$, $x \S y = \frac{x + y}{xy}$. If both $x$ and $y$ are irrational, is $x \S y$ necessarily irrational?
For what real number $x$ is $\frac{1}{1 + \frac{1}{1 + x}}$ not defined?
For what real number $x$ is $\frac{1}{1 + \frac{1}{1 + x}}$ not defined?
What is the simplified form of the expression: $\frac{a^2-b^2}{a+b} - \frac{a^3+b^3}{a^2-ab+b^2}$?
What is the simplified form of the expression: $\frac{a^2-b^2}{a+b} - \frac{a^3+b^3}{a^2-ab+b^2}$?
Flashcards
Natural Numbers (N)
Natural Numbers (N)
Positive integers starting from 1, represented by the symbol N.
Whole Numbers (N0)
Whole Numbers (N0)
Natural numbers plus zero; symbol is N0.
Integers (Z)
Integers (Z)
Includes whole numbers and their negative counterparts; symbol is Z.
Rational Numbers (Q)
Rational Numbers (Q)
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Irrational Numbers (Q')
Irrational Numbers (Q')
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Real Numbers (R)
Real Numbers (R)
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Imaginary Numbers
Imaginary Numbers
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Rational Number (Q)
Rational Number (Q)
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Irrational Numbers (Q')
Irrational Numbers (Q')
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Rounding Off
Rounding Off
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Surd
Surd
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Term
Term
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Expression
Expression
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Coefficient
Coefficient
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Exponent
Exponent
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Base
Base
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Constant
Constant
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Variable
Variable
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Equation
Equation
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Monomial
Monomial
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Binomial
Binomial
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Trinomial
Trinomial
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Factorisation
Factorisation
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Common Factors
Common Factors
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Difference of Two Squares
Difference of Two Squares
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Factorising by Grouping
Factorising by Grouping
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Factorising a Quadratic Trinomial
Factorising a Quadratic Trinomial
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Sum of Two Cubes
Sum of Two Cubes
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Difference of Two Cubes
Difference of Two Cubes
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Simplification of Fractions
Simplification of Fractions
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Converting Recurring Decimals
Converting Recurring Decimals
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Perfect Squares
Perfect Squares
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Perfect Cubes
Perfect Cubes
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Surd Comparison
Surd Comparison
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Multiplying Two Binomials
Multiplying Two Binomials
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Binomial and Trinomial Product
Binomial and Trinomial Product
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Cancelling Common Factors
Cancelling Common Factors
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Division of Fractions
Division of Fractions
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Terminating Decimal
Terminating Decimal
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Repeating Decimal
Repeating Decimal
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Rounding Up
Rounding Up
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Rounding Down
Rounding Down
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Perfect Fourth Powers
Perfect Fourth Powers
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Term with Coefficient
Term with Coefficient
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Converting Recurring Decimals into Fractions
Converting Recurring Decimals into Fractions
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Irrational Number Approximation
Irrational Number Approximation
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Factorise First
Factorise First
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Distribution
Distribution
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Study Notes
The Real Number System
-
The real number system includes subsets of numbers commonly used in math.
-
Natural Numbers (N):
- Positive integers starting from 1.
- Symbol: ( N )
- Set: ( {1, 2, 3, \ldots} )
-
Whole Numbers ((N_0)):
- Includes natural numbers and zero.
- Symbol: ( N_0 )
- Set: ( {0, 1, 2, 3, \ldots} )
-
Integers (Z):
- Includes whole numbers and their negative counterparts.
- Symbol: ( Z )
- Set: ( {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots} )
-
Rational Numbers (Q):
- Numbers expressible as a fraction of two integers, where the denominator isn't zero.
- Symbol: ( Q )
- Examples: ( \frac{1}{2}, \frac{4}{5}, -\frac{3}{7}, 0.75, 0 )
-
Irrational Numbers (Q'):
- Numbers not expressible as a simple fraction of two integers.
- Decimal expansions are non-repeating and non-terminating.
- Symbol: ( Q' )
- Examples: ( \sqrt{2}, \pi, e )
-
Real Numbers (R):
- Includes all rational and irrational numbers.
- Symbol: ( R )
- Examples: ( \sqrt{2}, -3, 4.5, \pi )
Non-Real or Imaginary Numbers
-
Imaginary Numbers: Numbers that have a negative square root.
- Examples: ( \sqrt{-1}, \sqrt{-28}, \sqrt{-5} )
-
Hierarchy of number sets:
- Real Numbers (( R )) include:
- Rational Numbers (( Q )) which include:
- Integers (( Z )) which include:
- Whole Numbers (( N_0 )) which include:
- Natural Numbers (( N ))
- Whole Numbers (( N_0 )) which include:
- Integers (( Z )) which include:
- Irrational Numbers (( Q' ).
- Rational Numbers (( Q )) which include:
- Real Numbers (( R )) include:
Rational Numbers (Q)
- Rational Numbers (Q): Can be written in the form ( \tfrac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 ).
Irrational Numbers (Q')
- Irrational Numbers (Q'): Cannot be written as a fraction with integer numerator and denominator.
Decimal Numbers
- Rational numbers include:
- Terminating decimals
- Repeating single digit decimals
- Repeating pattern of multiple digits decimals
Converting Terminating Decimals into Rational Numbers
- For a decimal number ( x ):
- [ x = \text{integer part} + \frac{\text{first decimal digit}}{10} + \frac{\text{second decimal digit}}{100} + \frac{\text{third decimal digit}}{1000} + \ldots ]
Converting Recurring Decimals into Rational Numbers
- For a recurring decimal ( x ):
- Let ( x = \text{recurring decimal} ).
- Multiply ( x ) by a power of 10 to align the recurring pattern after the decimal point.
- Subtract the original equation from this new equation.
- Solve for ( x ).
Key Points on Rational and Irrational Numbers
- Rational numbers can be expressed as ( \frac{a}{b} ) with ( a, b \in \mathbb{Z} ) and ( b \neq 0 ).
- Irrational numbers cannot be written as simple fractions and have non-repeating, non-terminating decimal expansions.
- Decimal forms:
- Rational if terminating or repeating
- Irrational if non-terminating and non-repeating
- Decimal forms:
- Rounding off an irrational number converts it into a rational approximation.
Rounding Off Decimal Numbers
- Steps to Round Off a Decimal Number:
- Identify the required decimal place: Count the number of decimal places needed and mark the digit after this position.
- Determine the rounding direction:
- If the next digit is 5 or greater, round up the last digit of the required decimal place.
- If the next digit is less than 5, leave the last digit of the required decimal place unchanged.
- If the digit to be rounded up is 9, it becomes 0, and the preceding digit is rounded up by 1.
Key Points for Rounding
- For ( x ), rounded to ( n ) decimal places:
- Identify the ( n )-th decimal digit and the ( (n+1) )-th digit.
- Apply rounding rules based on the value of the ( (n+1) )-th digit.
- Adjust accordingly and present the rounded number.
Estimating Surds
- Surds: The (n)-th root of a number that cannot be simplified to a rational number.
- For example, (\sqrt{2}) and (\sqrt{6}) are surds, but (\sqrt{4}) is not because ( \sqrt{4} = 2 )
- Surds are often in the form (\sqrt[n]{a}) where (a) is any positive number. For example, (\sqrt{7}) or (\sqrt{5})
- For (n = 2), this is written as (\sqrt{a}) instead of (\sqrt[2]{a}).
Estimation Process for Surds
- Identify Perfect Powers: Determine the nearest perfect squares or cubes (or higher powers) that surround the given surd.
- Perfect Squares: Numbers obtained when an integer is squared (e.g., 9 from (3^2)).
- Perfect Cubes: Numbers obtained when an integer is cubed (e.g., 27 from (3^3)).
- Comparison:
- If ( a ) and ( b ) are positive whole numbers, and ( a < b ), then ( \sqrt[n]{a} < \sqrt[n]{b} ).
Products
- Term: A single mathematical entity.
- Expression: A combination of terms.
- Coefficient: The numerical factor in a term.
- Exponent: The power to which a base is raised.
- Base: The variable or number being raised to a power.
- Constant: A term without a variable.
- Variable: A symbol representing an unknown quantity.
- Equation: A statement that two expressions are equal.
- A monomial is an expression with one term. A binomial is an expression with two terms. A trinomial is an expression with three terms.
Multiplying Two Binomials
- The general formula for multiplying two linear binomials ((ax + b)(cx + d)) is:
- ((ax + b)(cx + d) = acx^2 + adx + bcx + bd)
Multiplying a Binomial and a Trinomial
- Use the following:
- ((A + B)(C + D + E) = A(C + D + E) + B(C + D + E))
Operations
- Multiplying a Monomial and a Binomial:
- [ a(x + y) = ax + ay ]
- Multiplying Two Binomials:
- [ (ax + b)(cx + d) = acx^2 + adx + bcx + bd ]
- Multiplying a Binomial and a Trinomial:
- [ (A + B)(C + D + E) = A(C + D + E) + B(C + D + E) ]
Factorisation
- Factorisation: Breaking down an expression into simpler expressions (factors) that, when multiplied together, give the original expression.
Common Factors
- Factorising by taking out a common factor: Identifying and extracting a factor common to all terms in the expression.
Difference of Two Squares
- A difference of two squares can be factorised using the identity:
- [ a^2 - b^2 = (a + b)(a - b) ]
Factorising by Grouping in Pairs
- Factorising by grouping: Grouping terms with common factors and then factorising each group.
Factorising a Quadratic Trinomial
- A quadratic trinomial of the form (ax^2 + bx + c) can be factorised by finding two binomials whose product is the original trinomial.
General Procedure for Factorising a Trinomial
- Identify any common factors.
- Write down two brackets with an (x) in each bracket:
- [ (x\ \ ) (x\ \ ) ]
- List the factors of (a) and (c).
- Generate possible pairs of factors.
- Expand the pairs to find the one that matches the middle term (bx).
Sum and Difference of Two Cubes
- Sum of Two Cubes:
- [ x^3 + y^3 = (x + y)(x^2 - xy + y^2) ]
- Difference of Two Cubes:
- [ x^3 - y^3 = (x - y)(x^2 + xy + y^2) ]
- These identities can be used to factorise expressions involving the sum or difference of cubes.
Simplification of Fractions
Multiplication and Division of Fractions
- ( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} ) where ( b \neq 0 ) and ( d \neq 0 )
- ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} ) where ( b \neq 0 ), ( c \neq 0 ), and ( d \neq 0 )
Addition of Fractions
- [ \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b} \quad \text{where } b \neq 0 ]
Simplification of Algebraic Fractions
- Factorise the numerator and the denominator: Apply factorisation techniques (common factors, difference of squares, quadratic trinomials).
- Cancel common factors: Reduce the fraction by cancelling common factors in the numerator and denominator.
Steps for Simplification
- Factorise the expression: Factorise both the numerator and the denominator to identify common factors.
- Cancel the common factors: Simplify the fraction by cancelling the common factors in the numerator and denominator.
General Procedures for Simplifying Complex Fractions
- Factorise all terms in the numerator and denominator.
- Rewrite the division as multiplication by the reciprocal.
- Combine fractions by finding a common denominator if necessary.
- Simplify the resulting expression.
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