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Quick Review Flashcards - Click to flip and test your knowledge!
Question
Into which two main categories is the complete number system divided?
Answer
Imaginary numbers and Real numbers.
Question
How is a square root of a negative number (e.g. $\sqrt{-4}$) classified?
Answer
As an imaginary number.
Question
What is the mathematical definition of a rational number?
Answer
A number that can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
Question
In the rational number $\frac{a}{b}$, what condition must $a$ and $b$ meet regarding their common factors for it to be in simplest form?
Answer
They must have no common factor other than $1$ (they must be co-primes).
Question
Which symbol is used to denote the set of all rational numbers?
Answer
The letter $Q$.
Question
According to the source, what is the usual sign requirement for the denominator $b$ in a rational number?
Answer
It is usually positive.
Question
Under what condition are two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ considered equal?
Answer
If and only if $a \times d = b \times c$.
Question
If $\frac{a}{b} > \frac{c}{d}$ for positive denominators, what must be true of the products $a \times d$ and $b \times c$?
Answer
$a \times d > b \times c$.
Question
What formula can be used to find a single rational number that lies exactly between any two rational numbers $a$ and $b$?
Answer
$\frac{a+b}{2}$
Question
Beyond the mean method, what alternative fraction using numerators $a, c$ and denominators $b, d$ always lies between $\frac{a}{b}$ and $\frac{c}{d}$?
Answer
$\frac{a+c}{b+d}$
Question
When inserting $n$ rational numbers between $x$ and $y$ ($x < y$), what is the formula for the common difference $d$?
Answer
$d = \frac{y - x}{n + 1}$
Question
In the method for finding a large number of rationals, what are the first three terms inserted between $x$ and $y$?
Answer
$x + d$, $x + 2d$, and $x + 3d$.
Question
To find 4 rational numbers between two fractions using the 'alternative method', by what factor should you multiply the numerators and denominators after finding a common denominator?
Answer
Multiply by $n + 1$, which is $5$.
Question
What property of rational numbers states that the sum, difference, and product of any two rational numbers is always a rational number?
Answer
Closure property.
Question
Under what specific condition is the division of one rational number by another guaranteed to be a rational number?
Answer
When the divisor is a non-zero rational number.
Question
What name is given to decimals where the division ends and no remainder is left?
Answer
Terminating decimals.
Question
How are decimals described when a digit or a set of digits repeats continually in the decimal part?
Answer
Non-terminating recurring decimals (or periodic/circulating decimals).
Question
What term refers to the specific repeating digit or set of digits in a recurring decimal?
Answer
The period of the recurring decimal.
Question
In recurring decimal notation, where is a dot or a bar placed to indicate repetition?
Answer
Above the repeating digit or over the entire set of repeating digits.
Question
What is the period of the decimal representation of $\frac{4}{7} = 0.\overline{571428}$?
Answer
571428
Question
What distinguishes a 'mixed recurring decimal' from a 'pure recurring decimal'?
Answer
In a mixed recurring decimal, at least one digit in the decimal part is not repeating.
Question
What is the shortcut formula for the numerator when converting a recurring decimal to a fraction?
Answer
(All digits in the decimal part) minus (all non-recurring digits in the decimal part).
Question
What is the shortcut rule for determining the denominator when converting a recurring decimal to a fraction?
Answer
A number of nines equal to the repeating digits, followed by a number of zeros equal to the non-repeating digits.
Question
Without performing division, how can you identify if a rational number is convertible into a terminating decimal?
Answer
The prime factors of the denominator must only be $2$ and/or $5$ (expressed as $2^m \times 5^n$).
Question
If the prime factors of the denominator of a rational number in simplest form include a factor other than $2$ or $5$, what type of decimal is produced?
Answer
A non-terminating recurring decimal.
Question
What defines an irrational number in terms of its decimal representation?
Answer
A non-terminating and non-recurring decimal.
Question
Why is $\pi$ classified as an irrational number despite often being used as $\frac{22}{7}$ in calculations?
Answer
Because its decimal representation is non-terminating and non-recurring; $\frac{22}{7}$ is only an approximate value.
Question
Under what condition is the square root of a natural number $m$ ($\sqrt{m}$) considered an irrational number?
Answer
If $m$ is not a perfect square.
Question
How can you find one irrational number between two positive rational numbers $a$ and $b$ if $ab$ is not a perfect square?
Answer
$\sqrt{ab}$
Question
What is the result of the operation: (a rational number) + (an irrational number)?
Answer
An irrational number.
Question
Is the sum of two irrational numbers always an irrational number?
Answer
No, it may or may not be irrational (e.g. $(3 + \sqrt{5}) + (6 - \sqrt{5}) = 9$).
Question
What is the result of multiplying a non-zero rational number by an irrational number?
Answer
An irrational number.
Question
How do you compare two irrational numbers with different indices, such as $\sqrt[3]{4}$ and $\sqrt{3}$?
Answer
Convert them to like surds by finding the L.C.M. of their indices to make the indices the same.
Question
How is the set of Real Numbers ($R$) defined in relation to rational and irrational numbers?
Answer
The union of the set of rational numbers ($Q$) and the set of irrational numbers ($\bar{Q}$).
Question
What is a 'surd' or 'radical'?
Answer
An irrational root of a positive rational number.
Question
In the expression $\sqrt[n]{x}$, what is the term '$n$' called?
Answer
The order of the surd.
Question
Is every irrational number a surd?
Answer
No; for example, $\pi$ is irrational but not a surd.
Question
Is every surd an irrational number?
Answer
Yes, by definition a surd must be an irrational root.
Question
What are 'rationalising factors'?
Answer
Two surds whose product results in a rational number.
Question
What is the least rationalising factor of $\sqrt{27}$?
Answer
$\sqrt{3}$ (since $\sqrt{27} = 3\sqrt{3}$ and $3\sqrt{3} \times \sqrt{3} = 9$).
Question
What is the rationalising factor for a denominator of the form $a + \sqrt{b}$?
Answer
$a - \sqrt{b}$
Question
What is the rationalising factor for a denominator of the form $\sqrt{x} + \sqrt{y}$?
Answer
$\sqrt{x} - \sqrt{y}$
Question
In the context of the number system tree, what sub-categories make up the set of Integers?
Answer
Negative Integers, Zero, and Positive Integers (Natural Numbers).
Question
What set of numbers is formed by combining Zero and Positive Integers?
Answer
Whole Numbers ($W$).
Question
If $x$ and $y$ are rational and $\sqrt{z}$ is irrational, what does $x + \sqrt{z} = y + \sqrt{z}$ imply?
Answer
$x = y$
Question
If $a + b\sqrt{x} = c + d\sqrt{x}$ where $a, b, c, d$ are rational and $\sqrt{x}$ is irrational, what are the values of $a$ and $b$?
Answer
$a = c$ and $b = d$.
Question
What is the order of the surd $\sqrt[3]{10}$?
Answer
Order 3.
Question
Determine the rationalising factor of $2\sqrt{125}$.
Answer
$\sqrt{5}$
Question
Identify the type of number: $\frac{22}{7}$.
Answer
Rational number.
Question
Identify the type of number: $3.01\dots$ (non-terminating, non-recurring).
Answer
Irrational number.
Question
Is the difference of two irrational numbers always irrational?
Answer
No, it may be rational (e.g. $(8 - \sqrt{10}) - (3 - \sqrt{10}) = 5$).
Question
What is the decimal representation of $\frac{1}{11}$?
Answer
$0.\overline{09}$
Question
Define the term 'pure arithmetic' as presented in the unit heading.
Answer
The study of the properties and relations of numbers, specifically rational and irrational numbers in this context.
Question
Why is $\sqrt[3]{64}$ not a surd?
Answer
Because its value is $4$, which is a rational number.
Question
If $x = 3$ and $y = 5$, what is the first rational number inserted between them using $d = \frac{y-x}{n+1}$ for $n=3$?
Answer
$3.5$ (or $3\frac{1}{2}$).
Question
What characterises 'non-integral rationals'?
Answer
Rational numbers that are not integers, such as fractions like $\frac{5}{8}$ or $-\frac{3}{7}$.
Question
How is the number 0 classified within the real number system tree?
Answer
As an integer, a whole number, and a rational number.