The World of Numbers — Notes

The Dawn of Mathematics

Mathematics, at its core, is a system born from necessity. Long before formal schooling, the need to quantify, to count, and to measure drove early human ingenuity. This fundamental drive laid the groundwork for what we now recognize as mathematics.

A History Written in Bone

The earliest evidence of humans recording numbers doesn't involve sophisticated symbols or writing systems, but rather tally marks carved into bone. These artefacts demonstrate an early understanding of natural numbers and the desire to track quantities.

• Lebombo Bone: Discovered in the Lebombo Mountains, this bone dates back approximately 35,000 years. It features 29 distinct notches. It is believed to have served as a tool for counting, potentially tracking lunar cycles or menstrual periods, illustrating the early connection between natural numbers and timekeeping.
• Ishango Bone: Found near the Nile River and dating to around 20,000 BCE, this bone shows more complex groupings of notches.
• One column features tallies corresponding to prime numbers between 10 and 20 (11, 13, 17, 19). This suggests an awareness of number properties beyond simple counting.
• Another column appears to demonstrate multiplication by 2 (doubling).

These archaeological finds prove that the abstract concept of a 'number' is tens of thousands of years old, predating written language as we know it.

The Indian Context: Trade and Astronomy

As human societies developed, so did their need for more advanced mathematical concepts, particularly for trade and astronomy.

• Trade and Accounting: In ancient Indian civilizations like the Indus Valley, standardized weights and measures were essential for trade. Merchants required robust systems for accounting and exchange, necessitating a clear understanding of numbers.
• Fascination with Large Numbers: During the Vedic period, Indian thinkers were deeply interested in large quantities. The Vedas themselves mention names for powers of 10 up to ${10}^{12}$ (parārdha). Later texts, like the Lalitavistara, extended this to ${10}^{53}$ (tallakṣhaṇa). This explicit use of powers of 10 laid the foundation for the decimal place-value system that India would later develop and which is now used globally.
• Foundation for Zero: The development of a number system based on powers of 10, coupled with the philosophical exploration of concepts like Śhūnyatā (emptiness or void), created fertile ground for the invention of zero. This invention, a cornerstone of modern mathematics, would revolutionize arithmetic and algebra.

Natural Numbers and One-to-One Correspondence

The most basic form of counting arises from the human need to match items. This is known as one-to-one correspondence.

• Concept: For every item in one group, there is exactly one corresponding item in another group.
• Application: Early humans used this to ensure no livestock were lost. For each cow that left, a pebble was placed in a pot. For each cow that returned, a pebble was removed. If the pot was empty, all cows had returned.
• Result: This process naturally leads to the set of Natural Numbers, denoted by $\mathbb{N}=\{1,2,3,4,\dots \}$. These are the positive whole numbers used for counting.

Example 1

Given: A herder has 5 sheep. Each sheep is tagged with a unique number from 1 to 5. To find: How can the herder use one-to-one correspondence to ensure all sheep return?

Solution: The herder can create a set of markers, for example, 5 pebbles. When the 5 sheep leave to graze, the herder places one pebble for each sheep in a basket. When the sheep return, the herder removes one pebble for each sheep that enters the enclosure. If all 5 pebbles are removed and no sheep are missing, the correspondence is maintained. If a sheep is missing, the herder will have pebbles left in the basket. If more sheep return than left (unlikely, but mathematically possible for demonstration), the herder would run out of pebbles. Answer: The herder uses 5 pebbles to represent the 5 sheep, establishing a one-to-one correspondence.

Example 2

Given: The Ishango bone column shows the numbers 11, 13, 17, 19. To find: What do these numbers have in common? List the next three numbers that fit this pattern.

Solution: These numbers are all greater than 1. Each of these numbers is only divisible by 1 and themselves. Therefore, these numbers are prime numbers. The prime numbers following 19 are 23, 29, and 31. Answer: The numbers are prime numbers. The next three are 23, 29, and 31.

Example 3

Given: Natural Numbers are closed under addition, meaning for any two natural numbers $a$ and $b$, $a+b$ is also a natural number. To find: Are natural numbers closed under subtraction? Justify with examples.

Solution: Consider two natural numbers, $a=5$ and $b=3$. Their subtraction is $a-b=5-3=2$. The result, 2, is a natural number. This example supports closure. However, consider two natural numbers, $a=3$ and $b=5$. Their subtraction is $a-b=3-5=-2$. The result, -2, is an integer, but not a natural number. Since we found at least one case where the difference of two natural numbers is not a natural number, the set of natural numbers is not closed under subtraction. Answer: Natural numbers are not closed under subtraction, as shown by $3-5=-2$.

Example 4

Given: Ancient Indians used finger joints for counting. Each finger has 3 joints, and the thumb is used to count them. To find: How many can be counted on one hand using this method? How does this relate to ancient base-12 counting systems?

Solution: A hand has 4 fingers. Each finger has 3 joints. Using the thumb to point to each joint on one finger, you can count 3 joints. Repeating this for all 4 fingers: $4\text{ fingers}\times 3\text{ joints/finger}=12\text{ joints}$. This method allows counting up to 12 distinct items on one hand. This system directly relates to a base-12 counting system (also known as duodecimal), where numbers are grouped in twelves rather than tens. Answer: Up to 12 can be counted on one hand. This relates to ancient base-12 counting systems.

Remember

The natural numbers ($\mathbb{N}$) start from 1. The concept of one-to-one correspondence is the foundational idea behind counting. 🎯 Exam-smart: Questions might ask you to identify patterns in number sequences (like prime numbers), explain the concept of closure for number sets, or relate historical counting methods to modern mathematical concepts like number bases.

The Revolution of Śhūnya: When Nothing Became Something

From Philosophy to Mathematics: The Concept of Śhūnyatā

The journey of zero from a philosophical idea to a mathematical concept is deeply rooted in ancient Indian thought. The concept of Śhūnyatā, meaning emptiness or nothingness, was a central theme in Upanishadic and Buddhist literature, predating the 7th century BCE. It represented a profound state of mental tranquility achieved through meditation, where the mind is emptied of all fluctuations (vṛttis).

• Patanjali's Yoga Sutras (around 3rd century BCE) also highlighted śhūnyatā as a means to gain control over one's mind, body, and senses.
• This philosophical reverence for emptiness created a fertile ground for accepting 'nothingness' as a concept in various fields, including mathematics.
• The philosophical concept of emptiness thus paved the way for the mathematical concept of zero, a transformation championed by Indian mathematicians like Āryabhaṭa and Brahmagupta.

Remember

The acceptance of zero as a number was significantly influenced by India's rich philosophical traditions surrounding emptiness and nothingness.

Exam-smart

This concept might appear in questions asking to explain the historical development of mathematical ideas, emphasizing the interdisciplinary nature of knowledge.

The Bakhśhālī Manuscript and Brahmagupta’s Rules

The practical representation of zero evolved over time. The Bakhśhālī Manuscript, dated to the early centuries CE, shows a dot (bindu) being used to represent zero, indicating a visual placeholder.

However, it was Brahmagupta, in his seminal work Brāhmasphuṭasiddhānta (628 CE), who formally established zero as a mathematical number with operational rules.

Brahmagupta's Rules for Zero:

• Addition: Adding zero to any number leaves the number unchanged.

• Subtraction: Subtracting zero from any number leaves the number unchanged.

• Multiplication: Multiplying any number by zero results in zero.

Brahmagupta defined zero as the result of subtracting a number from itself:

Example 1

Given: A temperature of 4 °C drops by 15 °C. To find: The midnight temperature. Solution: The initial temperature is $4$ °C. A drop of $15$ °C means we subtract $15$ from $4$. We need to calculate $4-15$. Using Brahmagupta's rules, we can think of this as adding a negative number.

The rules for integer arithmetic (discussed next) will help us solve this.

Answer: The midnight temperature is $-11$ °C.

Example 2

Given: A spice trader takes a loan of 850, makes a profit of 1,200, and then incurs a loss of `450. To find: The trader's final financial standing. Solution: Represent the loan (debt) as a negative number: $-850$. Represent the profit (fortune) as a positive number: $+1200$. Represent the loss as a negative number: $-450$. The sequence of transactions can be written as an equation:

First, combine the debt and the loss:

Using Brahmagupta's rule for adding debts (negative numbers):

Now, combine this total debt with the profit:

This is equivalent to subtracting a smaller number from a larger one, with the result taking the sign of the larger number.

Answer: The spice trader's final financial standing is a debt of `100.

Example 3

Given: Calculations involving Brahmagupta’s laws. To find: The results of the given operations. Solution: (i) $(-12)\times 5$ Using Brahmagupta's rule for the product of a debt and a fortune:

(ii) $(-8)\times (-7)$ Using Brahmagupta's rule for the product of two debts:

(iii) $0-(-14)$ Subtracting a negative number is equivalent to adding its positive counterpart. This concept is formally defined in the arithmetic of integers.

(iv) $(-20)÷4$ This operation falls under the arithmetic of integers. Division by a positive number of a negative number results in a negative number.

Answer: (i) $-60$, (ii) $56$, (iii) $14$, (iv) $-5$.

Remember

Brahmagupta's rules are the foundational laws for operating with zero.

Exam-smart

Expect questions requiring the application of these rules to solve simple arithmetic problems involving zero, often presented as word problems related to temperature, money, or quantities.

Integers: Expanding the Horizon

Brahmagupta's innovation did not stop at zero. He considered what happens when a smaller number is subtracted from a larger one, a situation that arises naturally in commerce. He conceptualized two states:

• Fortunes (Dhana): Represented by positive numbers (natural numbers).
• Debts (Ṛiṇa): Represented by negative numbers.

By formally introducing numbers to the "left" of zero on the number line, Brahmagupta expanded the number system to include negative numbers.

The set of Integers ($Z$) is formed by the combination of:

• Positive Integers (Natural Numbers: 1, 2, 3, ...)
• Negative Integers (-1, -2, -3, ...)
• Zero (0)

Visually, this is represented on the number line: Negative Integers $\leftarrow$ Debt (Ṛiṇa) $$ Zero (Śhūnya) $$ Fortunes (Dhana) $\to$ Positive Integers ... –5 –4 –3 –2 –1 $$ 0 $$ 1 2 3 4 5 ...

Example 4

Given: The idea of subtracting a negative number. To find: An explanation using a real-world example of debt. Solution: Consider the operation $10\text{–}(-5)$. Subtracting a negative number is equivalent to removing a debt. If you have $10andsomeone\ast takesaway\ast$(-5)$fromyou,itmeanstheyareremovingadebtof$5 that you owe. When a debt is removed, your financial situation improves. So, removing a debt of $5isthesameas\ast gaining\ast$5. Therefore, $10\text{–}(-5)=10+5=15$. Answer: Subtracting a negative number is the same as adding a positive number because removing a debt increases your assets.

Remember

Integers include all whole numbers, both positive and negative, along with zero. The number line visually represents their order.

Exam-smart

Questions may ask to identify whether a given scenario (like temperature, altitude, or financial transactions) can be represented by positive or negative integers.

The Arithmetic of Integers

Brahmagupta established the fundamental rules for arithmetic operations involving integers, which remain the same today.

Addition of Integers:

1. Fortune + Fortune = Fortune: Adding two positive numbers results in a positive number.

2. Debt + Debt = Debt: Adding two negative numbers results in a negative number. $(-5)+(-4)=-9$ (If you owe $5andowe$4 more, your total debt is $9.)

Subtraction Involving Zero:

3. Fortune – Zero = Fortune: Subtracting zero from a positive number leaves it unchanged.

Debt – Zero = Debt: Subtracting zero from a negative number leaves it unchanged.

Multiplication of Integers:

4. Debt × Fortune = Debt: The product of a negative and a positive number is negative. $(-3)\times 4=-12$ (If you incur 4 debts of $3each,yourtotaldebtis$12.)
5. Debt × Debt = Fortune: The product of two negative numbers is positive.

Understanding Debt × Debt = Fortune

Think of multiplication as repeated addition or subtraction. Multiplying by $-1$ can be seen as reversing the sign. If we want to find $(-3)\times (-4)$: We can see it as $(-3)\times (4\times -1)$. We know $(-3)\times 4=-12$. So, we have $-12\times -1$. Multiplying $-12$ by $-1$ reverses the sign of $-12$, making it positive $12$. Alternatively, consider "removing debts": if $(-3)$ represents a debt of $3$, then multiplying by $(-4)$ means "removing 4 of these debts". Removing debts makes you richer, hence the positive result.

Try this:

Calculate: $(-15)+(-8)$ Answer: $-23$

Try this:

Calculate: $12\times (-3)$ Answer: $-36$

Try this:

Calculate: $(-6)\times (-9)$ Answer: $54$

Remember

The product of two negatives is a positive. This is a key rule that often causes confusion.

Exam-smart

Questions will involve applying these rules to solve arithmetic problems with integers. They might be direct calculations or embedded in word problems describing scenarios like temperature changes or financial gains/losses. For example, a question might describe an investment losing $500 each month for 3 months and ask for the total change.

Filling the Spaces: Fractions and Rational Numbers

Fractions and Rational Numbers

Fractions are numbers that represent parts of a whole. They arise naturally in situations involving division, such as dividing a quantity or a measurement. For example, if a field is divided among three children, each child receives a fraction of the field.

Just as every positive integer has a negative counterpart (its additive inverse), positive fractions also have negative counterparts. For example, the additive inverse of $\frac{1}{3}$ is $-\frac{1}{3}$, and the additive inverse of $\frac{19}{5}$ is $-\frac{19}{5}$. A negative fraction's minus sign can be placed with the numerator, the denominator, or in front of the fraction itself, e.g., $-\frac{1}{5}=\frac{-1}{5}=\frac{1}{-5}$.

When we combine all integers (positive, negative, and zero) and all fractions (positive and negative), we get the set of Rational Numbers. This set is denoted by the symbol $\mathbb{Q}$, derived from the word "quotient".

A rational number is formally defined as any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers, and importantly, $q\mathrm{\ne }0$.

Why $q\mathrm{\ne }0$?

The condition $q\mathrm{\ne }0$ is crucial because division by zero is undefined in mathematics. If $q$ were 0, the expression $\frac{p}{0}$ would not represent a specific numerical value.

Key Properties of Rational Numbers:

• Inclusion of Integers: All integers are rational numbers. Any integer $n$ can be written as $\frac{n}{1}$. For example, $5$ can be written as $\frac{5}{1}$, and $-10$ can be written as $\frac{-10}{1}$.
• Non-unique Representation: Rational numbers do not have a single, unique representation in the form $\frac{p}{q}$. For instance, $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$, $\frac{-5}{-10}$, and $\frac{10}{20}$ all represent the same rational number. These are called equivalent rational numbers or equivalent fractions.
• Simplest Form: When we commonly refer to a rational number like $\frac{2}{3}$ or represent it on a number line, we often mean its simplest form, where the numerator ($p$) and the denominator ($q$) have no common factors other than 1. In other words, $p$ and $q$ are co-prime. For example, $\frac{2}{3}$ is in simplest form, but $\frac{4}{6}$ is not, as both 4 and 6 share a common factor of 2. We can simplify $\frac{4}{6}$ to $\frac{2}{3}$ by dividing both numerator and denominator by 2.

Arithmetic of Rational Numbers

The rules for addition, subtraction, multiplication, and division developed by mathematicians like Brahmagupta apply to all rational numbers, including both positive and negative fractions.

1. Equality of Rational Numbers: Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal if and only if $ad=bc$.

2. Addition and Subtraction: To add or subtract rational numbers, they must first be expressed with a common denominator. If the denominators are already the same ($b$), then:

If the denominators are different, find their Least Common Multiple (LCM) to create equivalent fractions with the common denominator.

3. Multiplication: The product of two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ is given by:

This rule applies provided that $b\mathrm{\ne }0$ and $d\mathrm{\ne }0$.

4. Division: To divide a rational number $\frac{a}{b}$ by a non-zero rational number $\frac{c}{d}$, we multiply $\frac{a}{b}$ by the reciprocal (or multiplicative inverse) of $\frac{c}{d}$:

This rule applies provided that $b\mathrm{\ne }0$, $d\mathrm{\ne }0$, and $c\mathrm{\ne }0$.

Closure Properties for Rational Numbers:

• Closure under Addition, Subtraction, and Multiplication: If you add, subtract, or multiply any two rational numbers, the result is always another rational number.
• Closure under Division: If you divide one rational number by another, the result is also a rational number, as long as you are not dividing by zero.

Example: Adding Rational Numbers

Example 1

Given: Calculate the sum: $\frac{2}{5}+\frac{3}{10}$ To find: The sum of the two rational numbers. Solution:

• Identify the denominators: 5 and 10.
• Find the LCM of 5 and 10, which is 10.
• Convert $\frac{2}{5}$ to an equivalent fraction with denominator 10:

• Now add the fractions with the common denominator:

• Simplify the numerator:

Answer: $\frac{7}{10}$

Example 2

Given: Calculate the difference: $\frac{5}{6}-\frac{1}{4}$ To find: The difference between the two rational numbers. Solution:

• Identify the denominators: 6 and 4.
• Find the LCM of 6 and 4, which is 12.
• Convert $\frac{5}{6}$ to an equivalent fraction with denominator 12:

• Convert $\frac{1}{4}$ to an equivalent fraction with denominator 12:

• Now subtract the fractions with the common denominator:

• Simplify the numerator:

Answer: $\frac{7}{12}$

Example 3

Given: Find the product: $-\frac{2}{3}\times \frac{3}{10}$ To find: The product of the two rational numbers. Solution:

• Apply the multiplication rule $\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$:

• Calculate the numerator and denominator:

• Simplify the fraction by dividing numerator and denominator by their greatest common divisor, 6:

Answer: $-\frac{1}{5}$

Example 4

Given: Find the quotient: $\frac{2}{3}÷\frac{1}{8}$ To find: The result of dividing the first rational number by the second. Solution:

• Identify the reciprocal of the divisor $\frac{1}{8}$, which is $\frac{8}{1}$.
• Multiply the first rational number by the reciprocal of the second:

• Apply the multiplication rule:

Answer: $\frac{16}{3}$

Example 5

Given: Simplify the expression: $(\frac{1}{2}+\frac{1}{4})\times \frac{2}{3}$ To find: The simplified value of the expression. Solution:

• First, add the fractions inside the parentheses. Find the LCM of 2 and 4, which is 4.

• Add:

• Now, multiply the result by $\frac{2}{3}$:

• Calculate the numerator and denominator:

• Simplify the fraction by dividing numerator and denominator by their greatest common divisor, 6:

Answer: $\frac{1}{2}$

Remember

Always ensure that the denominator of a rational number is not zero. When performing operations, reduce fractions to their simplest form whenever possible.

Exam-smart

Questions might involve real-world scenarios like sharing resources or calculating costs involving parts of a whole. You'll need to identify the rational numbers involved and apply the correct arithmetic operations. For example, a question might ask about a price increase or decrease expressed as a fraction, requiring you to add or subtract rational numbers.

Representation of Rational Numbers on the Number Line

We are familiar with representing integers on a number line. We start with an origin (0), and move one unit to the right for positive integers (1, 2, 3, ...) and one unit to the left for negative integers (-1, -2, -3, ...). The distance between any two consecutive integers is uniform.

Rational numbers, unlike integers, can lie between two integers. For example, $\frac{1}{2}$ is exactly halfway between 0 and 1, and $-\frac{2}{3}$ lies between -1 and 0.

How to Represent a Rational Number $\frac{p}{q}$ on the Number Line:

1. Identify the unit interval: This is the distance between two consecutive integers (e.g., the distance between 0 and 1, or 1 and 2).
2. Divide the unit interval: Divide the chosen unit interval into $q$ equal parts.
3. Locate the point:

• If the rational number $\frac{p}{q}$ is positive, start from 0 and move $p$ of these equal parts to the right.
• If the rational number $\frac{p}{q}$ is negative, start from 0 and move $p$ of these equal parts to the left.

Example: Representing $\frac{3}{4}$

Given: Represent $\frac{3}{4}$ on the number line. To find: The position of $\frac{3}{4}$ on the number line. Solution:

• We are representing a positive rational number, so we will focus on the interval to the right of 0.
• The denominator is $q=4$. We need to divide the unit interval between 0 and 1 into 4 equal parts.
• The numerator is $p=3$. From 0, we count 3 of these equal parts to the right.
• The point reached after moving 3 parts is the location of $\frac{3}{4}$.
• The 4 equal parts between 0 and 1 represent $\frac{1}{4}$, $\frac{2}{4}$ (or $\frac{1}{2}$), and $\frac{3}{4}$.

Example: Representing $\frac{5}{4}$

Given: Represent $\frac{5}{4}$ on the number line. To find: The position of $\frac{5}{4}$ on the number line. Solution:

• This is an improper fraction. $\frac{5}{4}=1\frac{1}{4}$. This means it is 1 whole unit plus $\frac{1}{4}$ of the next unit.
• First, locate the integer 1 on the number line.
• Now, consider the unit interval between 1 and 2.
• Divide this interval (from 1 to 2) into $q=4$ equal parts.
• From the integer 1, move $p=1$ of these equal parts to the right.
• The point reached is the location of $\frac{5}{4}$.

Absolute Value of a Rational Number

The absolute value of a rational number $x$, denoted by $\mathrm{\mid }x\mathrm{\mid }$, represents its distance from 0 on the number line. Distance is always non-negative.

• If $x$ is positive, $\mathrm{\mid }x\mathrm{\mid }=x$.
• If $x$ is negative, $\mathrm{\mid }x\mathrm{\mid }=-x$ (which is the positive value of $x$).
• $\mathrm{\mid }0\mathrm{\mid }=0$. So, $\mathrm{\mid }x\mathrm{\mid }\ge 0$ for any rational number $x$.

The distance between two rational numbers $a$ and $b$ on the number line is given by $\mathrm{\mid }a-b\mathrm{\mid }$.

Example: Absolute Value

Given: Calculate $\mathrm{\mid }-\frac{5}{3}\mathrm{\mid }$ and $\mathrm{\mid }\frac{5}{3}\mathrm{\mid }$. To find: The absolute values. Solution:

• For $\mathrm{\mid }-\frac{5}{3}\mathrm{\mid }$: Since $-\frac{5}{3}$ is negative, its absolute value is its positive counterpart.

• For $\mathrm{\mid }\frac{5}{3}\mathrm{\mid }$: Since $\frac{5}{3}$ is positive, its absolute value is the number itself.

Answer: $\mathrm{\mid }-\frac{5}{3}\mathrm{\mid }=\frac{5}{3}$ and $\mathrm{\mid }\frac{5}{3}\mathrm{\mid }=\frac{5}{3}$.

Remember

To represent $\frac{p}{q}$, you divide the unit interval into $q$ parts. If the number is positive, count $p$ parts from 0 to the right; if negative, count $p$ parts from 0 to the left.

Exam-smart

You might be given a number line with points marked and asked to identify the rational number represented by a point, or to mark a given rational number. You might also get questions about the distance between two rational numbers, which involves calculating their absolute difference.

The Density of Rational Numbers

A remarkable property of rational numbers is their density. This means that between any two distinct rational numbers, no matter how close they are, there always exists another rational number.

How to find a rational number between two given rational numbers: One common method is to find their average (or mean). If $a$ and $b$ are two rational numbers, then $\frac{a+b}{2}$ is a rational number that lies between $a$ and $b$.

Why is the average a rational number between $a$ and $b$?

• It's a rational number: If $a$ and $b$ are rational, then $a+b$ is rational (closure property). Dividing a rational number ($a+b$) by a non-zero integer (2) results in another rational number.
• It's between $a$ and $b$:
• If $a<b$, then $a+a<a+b$, so $2a<a+b$, which means $a<\frac{a+b}{2}$.
• Also, if $a<b$, then $a+b<b+b$, so $a+b<2b$, which means $\frac{a+b}{2}<b$.
• Thus, $a<\frac{a+b}{2}<b$. The same logic applies if $b<a$.

Example: Finding Rational Numbers Between Two Given Numbers

Example 1

Given: Find a rational number between $\frac{1}{4}$ and $\frac{1}{2}$. To find: One rational number that lies strictly between $\frac{1}{4}$ and $\frac{1}{2}$. Solution:

• Calculate the average of $\frac{1}{4}$ and $\frac{1}{2}$. First, find a common denominator for $\frac{1}{4}$ and $\frac{1}{2}$, which is 4.

• Now add:

• Divide the sum by 2:

Answer: $\frac{3}{8}$ is a rational number between $\frac{1}{4}$ and $\frac{1}{2}$.

Example 2

Given: Find three distinct rational numbers between 0.3 and 0.4. To find: Three rational numbers strictly between 0.3 and 0.4. Solution:

• We can write 0.3 as $\frac{3}{10}$ and 0.4 as $\frac{4}{10}$.
• First rational number: Find the average of $\frac{3}{10}$ and $\frac{4}{10}$.

So, $\frac{7}{20}$ (or 0.35) is between 0.3 and 0.4.

• Second rational number: Now find a rational number between $\frac{3}{10}$ (0.3) and $\frac{7}{20}$ (0.35).

Convert $\frac{3}{10}$ to $\frac{6}{20}$:

So, $\frac{13}{40}$ (or 0.325) is between 0.3 and 0.35.

• Third rational number: Find a rational number between $\frac{7}{20}$ (0.35) and $\frac{4}{10}$ (0.4).

Convert $\frac{4}{10}$ to $\frac{8}{20}$:

So, $\frac{15}{40}$ (or 0.375) is between 0.35 and 0.4.

Answer: Three distinct rational numbers are $\frac{7}{20}$, $\frac{13}{40}$, and $\frac{15}{40}$ (or 0.35, 0.325, and 0.375).

The density property implies that the rational numbers, though infinite, are packed so closely together that they appear to fill the entire number line. However, as we will see in later sections, there are still "gaps" that are filled by irrational numbers.

Remember

The average of two rational numbers is always a rational number between them. This property is key to finding multiple rational numbers between any given pair.

Exam-smart

Questions might ask to insert a specified number of rational numbers between two given numbers (which can be fractions or decimals). You'll need to repeatedly apply the averaging method or use equivalent fractions with a larger common denominator to find distinct numbers. A case might also involve a scenario where a quantity needs to be divided into smaller parts, requiring you to find rational numbers.

Irrational Numbers

Numbers that cannot be expressed as a ratio of two integers $\frac{p}{q}$, where $p$ and $q$ are integers and $q\mathrm{\ne }0$, are called irrational numbers.

The Proof of Irrationality of $\sqrt{2}$

The irrationality of $\sqrt{2}$ is proven using a technique called Proof by Contradiction. This method assumes the opposite of what we want to prove and shows that this assumption leads to a logical inconsistency.

Proof Steps:

1. Assumption: Assume $\sqrt{2}$ is a rational number. Therefore, it can be written as $\sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are co-prime integers ($q\mathrm{\ne }0$) and their simplest form.
2. Square both sides: $(\sqrt{2}{)}^2=(\frac{p}{q}{)}^2$, which gives $2=\frac{p^2}{q^2}$.
3. Rearrange: $2q^2=p^2$.
4. Deduction for $p$: Since $p^2=2q^2$, $p^2$ is an even number. If the square of an integer is even, the integer itself must be even. So, $p$ is an even integer. We can write $p=2k$ for some integer $k$.
5. Substitute $p=2k$: Substitute this into the equation from Step 3: $2q^2=(2k{)}^2$. This simplifies to $2q^2=4k^2$.
6. Simplify: Divide both sides by 2: $q^2=2k^2$.
7. Deduction for $q$: Since $q^2=2k^2$, $q^2$ is also an even number. Therefore, $q$ must also be an even integer.
8. Contradiction: We have deduced that both $p$ and $q$ are even. This means they share a common factor of 2. However, this contradicts our initial assumption in Step 1 that $\frac{p}{q}$ was in its simplest form and $p$ and $q$ were co-prime.
9. Conclusion: Since our initial assumption leads to a contradiction, the assumption must be false. Therefore, $\sqrt{2}$ cannot be expressed as a ratio of two integers and is irrational.

Example

Given: Prove that $\sqrt{2}$ is an irrational number. To find: The proof of irrationality of $\sqrt{2}$. Solution: We will use proof by contradiction. Assume $\sqrt{2}$ is rational. Then $\sqrt{2}=\frac{p}{q}$, where $p,q$ are co-prime integers and $q\mathrm{\ne }0$. Squaring both sides, $2=\frac{p^2}{q^2}$. Multiplying by $q^2$, we get $2q^2=p^2$. This implies $p^2$ is even, so $p$ must be even. Let $p=2k$ for some integer $k$. Substituting $p=2k$ into $2q^2=p^2$:

Dividing by 2: $q^2=2k^2$. This implies $q^2$ is even, so $q$ must be even. Since both $p$ and $q$ are even, they have a common factor of 2. This contradicts our assumption that $p$ and $q$ are co-prime. Therefore, our initial assumption that $\sqrt{2}$ is rational is false. Answer: $\sqrt{2}$ is irrational.

Remember

The key to the proof of irrationality of $\sqrt{2}$ is showing that if we assume it's rational ($\frac{p}{q}$ in simplest form), it inevitably leads to both $p$ and $q$ being even, meaning they have a common factor of 2, thus contradicting the "simplest form" assumption.

Exam-smart

For a "show that" question, lay out the steps of the proof clearly, stating the assumption and highlighting the contradiction. For multiple choice, recall that $\sqrt{2},\sqrt{3},\sqrt{5}$ are irrational.

Construction of Length

We can construct line segments with irrational lengths using geometric methods, often involving the Pythagorean theorem.

Constructing $\sqrt{2}$ on the Number Line:

1. Start with a number line. Mark a point O at the origin (0) and A at 1 unit.
2. Draw a perpendicular line segment AB of length 1 unit at point A.
3. Join the origin O to point B. By the Pythagorean theorem in right triangle OAB, $O{B}^2=O{A}^2+A{B}^2=1^2+1^2=2$. So, $OB=\sqrt{2}$.
4. With O as the center and OB as the radius, draw an arc that intersects the number line at point P.
5. The point P represents the irrational number $\sqrt{2}$.

Generalizing Construction: To construct $\sqrt{n}$ (where $n$ is a positive integer), we can use a sequence of right-angled triangles. For $\sqrt{3}$, we could construct a triangle with base $\sqrt{2}$ and height 1. For $\sqrt{5}$, we could use a base of 2 and a height of 1, or a base of $\sqrt{4}$ and a height of 1.

Example

Given: Construct a line segment of length $\sqrt{5}$ on the number line. To find: The representation of $\sqrt{5}$ on the number line. Solution: We can use the Pythagorean theorem $a^2+b^2=c^2$. We need $c=\sqrt{5}$, so $c^2=5$. We can choose $a=1$ and $b=2$, since $1^2+2^2=1+4=5$. Step 1: On the number line, mark a point O at 0 and A at 2 units. Step 2: Draw a perpendicular line segment AB of length 1 unit at point A. Step 3: Join the origin O to point B. By the Pythagorean theorem, $O{B}^2=O{A}^2+A{B}^2=2^2+1^2=4+1=5$. So, $OB=\sqrt{5}$. Step 4: With O as the center and OB as the radius, draw an arc that intersects the number line at point P. Step 5: Point P represents the irrational number $\sqrt{5}$. Answer: Point P on the number line represents $\sqrt{5}$.

Remember

The construction of irrational lengths relies on repeatedly applying the Pythagorean theorem to create right-angled triangles where one side's length is a previously constructed irrational number, and the hypotenuse becomes the new irrational length.

Exam-smart

You might be asked to describe the steps to construct a specific irrational length (like $\sqrt{3}$ or $\sqrt{5}$) or to identify which point on a given number line represents a particular irrational number based on its construction.

The Story of Pi ($\pi$) and Madhava’s Infinite Series

$\pi$ (Pi) is a famous irrational number. It is defined as the ratio of a circle's circumference ($C$) to its diameter ($d$):

This means no fraction can exactly represent $\pi$. Even approximations like $\frac{22}{7}$ or $3.1416$ are not exact values of $\pi$.

Madhava of Sangamagrama (14th century) discovered that irrational numbers like $\pi$ can be represented as an infinite series – a sum of infinitely many terms. He found the following series for $\pi$:

Adding more terms of this series brings us closer and closer to the actual value of $\pi$. This concept of infinite series is crucial for understanding irrational numbers that cannot be simply defined by geometric constructions or simple ratios.

Remember

$\pi$ is an irrational number, meaning it cannot be written as a simple fraction $\frac{p}{q}$. Its decimal expansion is non-terminating and non-repeating.

Exam-smart

Questions might involve understanding that $\pi$ is irrational and therefore cannot be perfectly represented by $\frac{22}{7}$ or any finite decimal. You might also see questions about historical approximations of $\pi$.

Real Numbers: Decimals and Cyclic Patterns

When we combine the set of all rational numbers and all irrational numbers, we get the set of real numbers ($\mathbb{R}$). The real number line is considered a continuous line with no gaps.

The decimal expansion is a key way to distinguish between rational and irrational numbers.

Rational Decimals: Terminating and Repeating

When a rational number $\frac{p}{q}$ is converted into a decimal, one of two patterns emerges:

1. Terminating Decimals: The decimal representation ends after a finite number of digits. This happens when the denominator $q$ of the rational number (in its simplest form) has only prime factors of 2 and 5.

• Example: $\frac{3}{8}$. When you divide 3 by 8, you get $0.375$. The remainder becomes 0, and the decimal terminates.

2. Repeating Decimals (or Recurring Decimals): The decimal representation has a sequence of digits that repeats infinitely. This happens when the denominator $q$ of the rational number (in its simplest form) has prime factors other than 2 or 5.

• Example: $\frac{1}{3}$. When you divide 1 by 3, you get $0.333\dots$. The digit '3' repeats infinitely. This is written as $0.\bar{3}$.
• Example: $\frac{5}{11}$. Dividing 5 by 11 gives $0.454545\dots$, which is written as $0.\overline{45}$.

The reason for repetition is that in long division, there are only a finite number of possible remainders. Once a remainder repeats, the sequence of quotients (the decimal digits) must also repeat.

Example

Given: Find the decimal expansion of $\frac{7}{8}$ and $\frac{5}{6}$. To find: Whether the decimal expansions terminate or repeat. Solution:

For $\frac{7}{8}$:

The remainder is 0, so the decimal terminates.

For $\frac{5}{6}$:

The remainder 2 repeats, so the digits after the decimal point will repeat. The digit '3' repeats.

Answer: $\frac{7}{8}$ has a terminating decimal expansion ($0.875$). $\frac{5}{6}$ has a repeating decimal expansion ($0.8\bar{3}$).

Remember

A rational number's decimal expansion terminates if the denominator's prime factors are only 2 and 5. It repeats if the denominator has any other prime factors.

Exam-smart

You might be given a rational number and asked to determine if its decimal expansion will terminate or repeat without actually performing the division. You would analyze the prime factors of the denominator. You might also be given a terminating or repeating decimal and asked to express it as a fraction $\frac{p}{q}$.

Predicting the Type of Decimal Expansion

We can determine if a rational number's decimal expansion will terminate or be repeating without performing long division. This is achieved by examining the prime factors of the denominator.

A rational number $\frac{p}{q}$ (in its lowest terms, meaning $p$ and $q$ share no common factors other than 1) will have a terminating decimal expansion if and only if the prime factorization of the denominator ($q$) consists only of powers of 2 and/or 5.

Why does this work? If the denominator $q$ has only prime factors of 2 and 5, we can multiply both the numerator and denominator by a suitable power of 2 or 5 to make the denominator a power of 10 (like 10, 100, 1000, etc.). Numbers with denominators that are powers of 10 have terminating decimal representations.

For example, consider $\frac{3}{20}$. The prime factorization of the denominator 20 is $2^2\times 5^1$. Since it only contains factors of 2 and 5, its decimal expansion will terminate. To convert it to a decimal, we can make the denominator a power of 10: $\frac{3}{20}=\frac{3}{2^2\times 5^1}$. To get a power of 10 (which requires equal powers of 2 and 5), we need to multiply by another 5. $\frac{3}{2^2\times 5^1}=\frac{3\times 5}{2^2\times 5^1\times 5^1}=\frac{15}{2^2\times 5^2}=\frac{15}{(2\times 5{)}^2}=\frac{15}{{10}^2}=\frac{15}{100}=0.15$.

Example 1: Determine the type of decimal expansion for $\frac{7}{8}$.

Given: The rational number $\frac{7}{8}$. To find: Whether its decimal expansion terminates or repeats.

Solution: We examine the denominator, which is 8. The prime factorization of 8 is $2\times 2\times 2=2^3$. Since the prime factorization of the denominator consists only of the prime factor 2, the decimal expansion of $\frac{7}{8}$ will terminate.

To verify: $\frac{7}{8}=\frac{7}{2^3}$. We need to make the denominator a power of 10. To do this, we need five factors. We currently have three 2s, so we multiply by $5^3$: $\frac{7}{2^3}=\frac{7\times 5^3}{2^3\times 5^3}=\frac{7\times 125}{(2\times 5{)}^3}=\frac{875}{{10}^3}=\frac{875}{1000}=0.875$.

Example 2: Determine the type of decimal expansion for $\frac{13}{15}$.

Given: The rational number $\frac{13}{15}$. To find: Whether its decimal expansion terminates or repeats.

Solution: We examine the denominator, which is 15. The prime factorization of 15 is $3\times 5$. Since the prime factorization of the denominator includes a prime factor other than 2 or 5 (in this case, 3), the decimal expansion of $\frac{13}{15}$ will repeat.

Remember

The key is that the prime factors of the denominator must only be 2s and 5s for the decimal to terminate. Even one other prime factor guarantees a repeating decimal.

Exam-smart

You might be given a fraction like $\frac{27}{125}$ and asked to predict its decimal type. You'd factor the denominator: $125=5^3$. Since only 5s are present, it terminates. You might then be asked to convert it to decimal form, which involves multiplying by $2^3$ to get $\frac{27\times 8}{1000}=\frac{216}{1000}=0.216$.

Converting Rational Decimals into the Form p/q

This involves converting decimals back into their fractional form.

Case 1: Terminating decimals

Terminating decimals are those that end after a finite number of digits. They can be directly converted by writing the digits after the decimal point as the numerator and a power of 10 as the denominator.

Example 3: Convert 0.75 into the form $\frac{p}{q}$.

Given: The terminating decimal 0.75. To find: The equivalent fraction $\frac{p}{q}$.

Solution: 0.75 has two digits after the decimal point. We write the digits '75' as the numerator. We use ${10}^2$ (which is 100) as the denominator because there are two decimal places. So, $0.75=\frac{75}{100}$. Now, we simplify this fraction by dividing the numerator and denominator by their greatest common divisor, which is 25. $\frac{75÷25}{100÷25}=\frac{3}{4}$.

Answer: $0.75=\frac{3}{4}$.

Case 2: Pure repeating decimals

A pure repeating decimal is one where the repeating block starts immediately after the decimal point.

Example 4: Convert 0.666... (or $0.\bar{6}$) into the form $\frac{p}{q}$.

Given: The pure repeating decimal $0.\bar{6}$. To find: The equivalent fraction $\frac{p}{q}$.

Solution: Step 1: Let $x$ be equal to the decimal. $x=\mathrm{0.666...}$ (Equation 1)

Step 2: Count the number of digits in the repeating block. Here, there is one repeating digit (6). Multiply $x$ by ${10}^1$ (which is 10) to shift the decimal point one place to the right. $10x=\mathrm{6.666...}$ (Equation 2)

Step 3: Subtract Equation 1 from Equation 2. This eliminates the repeating decimal part.

Step 4: Solve for $x$.

Simplify the fraction: $x=\frac{6÷3}{9÷3}=\frac{2}{3}$.

Answer: $0.\bar{6}=\frac{2}{3}$.

Example 5: Convert $\mathrm{0.454545...}$ (or $0.\overline{45}$) into the form $\frac{p}{q}$.

Given: The pure repeating decimal $0.\overline{45}$. To find: The equivalent fraction $\frac{p}{q}$.

Solution: Step 1: Let $x=\mathrm{0.454545...}$ (Equation 1)

Step 2: There are two repeating digits (45). Multiply $x$ by ${10}^2$ (which is 100). $100x=\mathrm{45.454545...}$ (Equation 2)

Step 3: Subtract Equation 1 from Equation 2.

Step 4: Solve for $x$.

Simplify the fraction by dividing by 9: $x=\frac{45÷9}{99÷9}=\frac{5}{11}$.

Answer: $0.\overline{45}=\frac{5}{11}$.

Case 3: General repeating decimals

These decimals have some non-repeating digits after the decimal point, followed by a repeating block.

Example 6: Convert $\mathrm{0.1666...}$ (or $0.1\bar{6}$) into the form $\frac{p}{q}$.

Given: The general repeating decimal $0.1\bar{6}$. To find: The equivalent fraction $\frac{p}{q}$.

Solution: Step 1: Let $x=\mathrm{0.1666...}$ (Equation 1)

Step 2: Shift the decimal so the repeating part is immediately after it. There is one non-repeating digit (1). Multiply by ${10}^1$ (which is 10). $10x=\mathrm{1.666...}$ (Equation 2)

Step 3: Now, consider this new equation ($10x$). To get the repeating part aligned for subtraction, we need to multiply $10x$ by the power of 10 corresponding to the length of the repeating block. The repeating block is '6' (1 digit long). So, multiply Equation 2 by ${10}^1$ (which is 10).

$100x=\mathrm{16.666...}$ (Equation 3)

Step 4: Subtract Equation 2 from Equation 3 to eliminate the repeating decimals.

Step 5: Solve for $x$.

Simplify the fraction by dividing by 15: $x=\frac{15÷15}{90÷15}=\frac{1}{6}$.

Answer: $0.1\bar{6}=\frac{1}{6}$.

Example 7: Convert $\mathrm{2.35777...}$ (or $2.35\bar{7}$) into the form $\frac{p}{q}$.

Given: The general repeating decimal $2.35\bar{7}$. To find: The equivalent fraction $\frac{p}{q}$.

Solution: Step 1: Let $x=\mathrm{2.35777...}$ (Equation 1)

Step 2: Shift the decimal to place the repeating part immediately after it. There are two non-repeating digits ('35'). Multiply by ${10}^2$ (which is 100). $100x=\mathrm{235.777...}$ (Equation 2)

Step 3: Now, align the repeating part. The repeating block is '7' (1 digit long). Multiply Equation 2 by ${10}^1$ (which is 10).

$1000x=\mathrm{2357.777...}$ (Equation 3)

Step 4: Subtract Equation 2 from Equation 3.

Step 5: Solve for $x$.

Simplify the fraction by dividing by 2: $x=\frac{2122÷2}{900÷2}=\frac{1061}{450}$.

Answer: $2.35\bar{7}=\frac{1061}{450}$.

Summary Table for Conversion of Repeating Decimals to p/q

Remember

The number of 9s in the denominator of the resulting fraction corresponds to the number of repeating digits, and the number of 0s corresponds to the number of non-repeating digits after the decimal point.

Exam-smart

A question might ask you to convert $\mathrm{0.123123...}$ to $\frac{p}{q}$. You'd identify three repeating digits (123), so you'd multiply by ${10}^3=1000$. $1000x=\mathrm{123.123...}$. Subtracting $x$: $999x=123$, so $x=\frac{123}{999}$. Simplify by dividing by 3 to get $\frac{41}{333}$.

The Magic of Cyclic Numbers

Certain rational numbers have remarkable repeating decimal expansions. One such fascinating example is $\frac{1}{7}$.

When you perform the long division of 1 by 7, you get:

The repeating block is 142857. This block is known as a cyclic number because when you multiply it by digits 1 through 6, the digits rearrange themselves in a cycle without changing the order of the sequence:

• $142857\times 1=142857$
• $142857\times 2=285714$
• $142857\times 3=428571$
• $142857\times 4=571428$
• $142857\times 5=714285$
• $142857 \times 6 = 857142

This cyclic property is a special characteristic of the reciprocal of 7.

Example 8: Verify the cyclic property for $\frac{2}{7}$.

Given: The cyclic number 142857 from $\frac{1}{7}$. To find: The decimal for $\frac{2}{7}$ and verify its cyclic nature.

Solution: We know that $\frac{2}{7}=2\times \frac{1}{7}$. Using the decimal expansion of $\frac{1}{7}=0.\overline{142857}$: $2\times 0.\overline{142857}=0.\overline{285714}$.

The repeating block is 285714. This is indeed a cyclic permutation of the original block 142857.

Remember

Cyclic numbers are a beautiful mathematical curiosity found in the repeating decimals of certain fractions, most notably the reciprocals of prime numbers.

Exam-smart

You might be asked to show that $\frac{3}{7}$ has a repeating block that is a cyclic permutation of $\frac{1}{7}$'s block. You would calculate $3\times 0.\overline{142857}=0.\overline{428571}$ and observe that the digits 428571 are a cyclic shift of 142857.

Irrational Decimals: Chaos and Infinity

In contrast to rational numbers, irrational numbers have decimal expansions that are infinite and non-repeating. This means there is no pattern, no block of digits that repeats indefinitely.

• No Termination: The decimal goes on forever.
• No Repetition: There is no sequence of digits that repeats in a cycle.

Examples of irrational numbers include:

• $\sqrt{2}$: Its decimal expansion begins $\mathrm{1.4142135623730950488...}$ and continues infinitely without repeating.
• $\pi$ (Pi): Approximately $\mathrm{3.1415926535897932384...}$, it also continues infinitely without any repeating pattern.
• $e$ (Euler's Number): Approximately $\mathrm{2.7182818284590452353...}$, it too is non-terminating and non-repeating.

These decimal expansions appear chaotic because they lack the predictable structure of repeating decimals. However, they are not random; they represent specific, fixed values.

Example 9: Classify $\sqrt{3}$ based on its decimal expansion.

Given: The number $\sqrt{3}$. To find: Whether its decimal expansion is terminating, repeating, or non-terminating and non-repeating.

Solution: We know that $\sqrt{3}$ is an irrational number. By definition, irrational numbers have decimal expansions that are infinite and non-repeating. Therefore, the decimal expansion of $\sqrt{3}$ is non-terminating and non-repeating. Its approximate value starts as $\mathrm{1.7320508075688772935...}$ and continues infinitely without repeating.

Remember

A decimal expansion is either terminating, repeating (if rational), or infinite and non-repeating (if irrational). There are no other possibilities.

Exam-smart

You might be given a number like $\mathrm{0.10110111011110...}$ and asked to classify it. You would observe that there is no repeating block of digits, as the sequence of 0s between the 1s keeps increasing. This indicates it is an irrational number with a non-terminating, non-repeating decimal expansion.

Conclusion

The journey through the "World of Numbers" has encompassed a vast historical and conceptual landscape. We've moved from rudimentary counting methods to the sophisticated understanding of the real number system. This chapter serves as a culmination, reinforcing the interconnectedness of number systems and their evolution to meet human needs.

Evolution of Number Systems

The development of numbers has been a gradual process, driven by practical requirements and philosophical inquiry. Each expansion of the number system addressed limitations and opened new possibilities in mathematics and science.

Non-uniqueness of Decimal Representations

While most numbers have a unique decimal representation, there's a special case for terminating decimals. A terminating decimal can also be represented by an equivalent decimal with repeating nines.

• Example:
• $1$ can be written as $\mathrm{1.000...}$ or $\mathrm{0.999...}$
• $2.47$ can be written as $\mathrm{2.47000...}$ or $\mathrm{2.46999...}$

This arises because:

And also:

This concept highlights that the decimal representation isn't always a one-to-one mapping for rational numbers.

Remember

Terminating decimals have an alternative representation ending in an infinite sequence of nines. This is a crucial distinction when considering the uniqueness of decimal forms.

Exam-smart

Questions might present a number like $\mathrm{0.999...}$ and ask if it's rational or irrational, or to convert it to its fractional form. Understanding this non-uniqueness is key.

The Frontier Beyond Real Numbers: Imaginary Numbers

The real number line, while comprehensive for many physical measurements, cannot accommodate certain mathematical operations. Specifically, finding the square root of a negative number, like $\sqrt{-1}$, is impossible within the real number system because no real number multiplied by itself yields a negative result.

• $1\times 1=1$
• $(-1)\times (-1)=1$

To address this, mathematicians introduced Imaginary Numbers. The fundamental imaginary unit is denoted by '$i$', where $i=\sqrt{-1}$. This concept extends the number system into a new dimension, leading to the development of Complex Numbers. Although termed "imaginary," these numbers are vital in fields like electrical engineering and quantum mechanics.

Remember

Real numbers are on a one-dimensional line. Imaginary numbers require stepping "off" this line into a new dimension to solve equations that have no real solutions.

Exam-smart

While a deep dive into complex numbers is for later years, the concept is introduced to show the continuous evolution of numbers. Be prepared to identify situations where real numbers are insufficient, such as $\sqrt{-5}$.