Unit 1: Numbers, Quantification and Numerical Applications

Mathematics provides a powerful way of modelling real-world behavior. In this unit, we explore various quantitative tools and numerical applications essential for fields like finance, computing, and logical deduction.

1. Binary Numbers

A number system is a method for expressing quantities. The decimal system (Base 10) uses digits 0-9. The binary system (Base 2) is used extensively in computing and uses only two digits: 0 and 1.

Converting Decimal to Binary

To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. The binary representation is formed by reading the remainders from bottom to top.

Example: Convert 25 to binary.

• $25 \div 2 = 12$ remainder $1$
• $12 \div 2 = 6$ remainder $0$
• $6 \div 2 = 3$ remainder $0$
• $3 \div 2 = 1$ remainder $1$
• $1 \div 2 = 0$ remainder $1$

Reading bottom to top, $(25)_{10} = (11001)_2$.

Converting Binary to Decimal

Multiply each binary digit by $2^n$, where $n$ is the position of the digit starting from 0 on the right.

Example: Convert $(1011)_2$ to decimal. $$ (1011)2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 $$ $$ = 8 + 0 + 2 + 1 = (11){10} $$

2. Indices, Logarithm and Antilogarithm

Indices

Indices (or powers) tell us how many times to multiply a number by itself. Let $a^x = N$. Here $a$ is the base and $x$ is the index or exponent.

Logarithm

The logarithm is the inverse operation to exponentiation. If $a^x = N$, then the logarithm of $N$ to the base $a$ is $x$. $$ \log_a N = x $$ (where $a > 0, a \neq 1$)

Types of Logarithms:

• Common Logarithm: Base 10, often written as $\log x$.
• Natural Logarithm: Base $e$ (approx. 2.718), often written as $\ln x$.

Laws of Logarithms

1. Product Rule: $\log_a (mn) = \log_a m + \log_a n$
2. Quotient Rule: $\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n$
3. Power Rule: $\log_a (m^n) = n \log_a m$
4. Change of Base Rule: $\log_a m = \frac{\log_b m}{\log_b a}$

Antilogarithm

The antilogarithm of a number is the number for which the given value is the logarithm. If $\log_{10} x = y$, then $x = \text{Antilog}(y) = 10^y$.

3. Clock

A clock face is a circle of $360^\circ$ divided into 12 hour spaces ($30^\circ$ each) and 60 minute spaces ($6^\circ$ each).

Speeds of Hands

• Minute hand: Traverses $360^\circ$ in 60 minutes. Speed = $6^\circ$ per minute.
• Hour hand: Traverses $30^\circ$ in 60 minutes. Speed = $0.5^\circ$ per minute.

Angle between hands at time $H$ hours and $M$ minutes: $$ \theta = |30H - 5.5M|^\circ $$ (Note: If $\theta > 180^\circ$, the reflex angle is $360^\circ - \theta$.)

Example: Find the angle at 4:20. $$ \theta = |30(4) - 5.5(20)| = |120 - 110| = 10^\circ $$

4. Calendar

The calendar concept is based on finding the number of Odd Days – days remaining after grouping periods into full weeks.

• Ordinary Year: 365 days = 52 weeks + 1 odd day.
• Leap Year: 366 days = 52 weeks + 2 odd days.

Odd Days in Centuries

• 100 years: 5 odd days
• 200 years: 3 odd days
• 300 years: 1 odd day
• 400 years (and multiples): 0 odd days

Finding the Day of the Week

Count the total odd days up to the given date and map the result to:

• 0 = Sunday
• 1 = Monday
• 2 = Tuesday
• … 6 = Saturday

5. Time, Work and Distance

Time and Work

If a person can finish a piece of work in $n$ days, their 1-day’s work is $\frac{1}{n}$. If A does $\frac{1}{x}$ work in a day and B does $\frac{1}{y}$ work in a day, together their 1-day work is $\left(\frac{1}{x} + \frac{1}{y}\right)$.

Time and Distance

The fundamental relationship is: $$ \text{Distance} = \text{Speed} \times \text{Time} $$

6. Seating Arrangement

Logical deduction determines the exact position of a person based on given clues.

• Linear Arrangement: People sitting in a row. Left/Right tracking is crucial.
• Circular Arrangement: People sitting around a table. Left/Right depends on whether they face the center or outwards.

Competency-Based Questions

1. (Application/Analysis): A digital system tracks user login times using a secondary processor that registers time in binary format. If a user logged in at the binary equivalent of $105$ minutes past midnight, what is the binary representation of this minute? Furthermore, convert the binary time 1001011 to minutes to verify the logout time in decimal.

2. (Logarithms in Real Life): Certain virus spreads follow an exponential decay when quarantined, modeled by the equation $P = P_0 (0.8)^t$, where $t$ is the number of days. If the town had 10,000 cases initially ($P_0$), use logarithms to find out approximately how many days it will take for cases to drop below 100. (Given $\log 0.8 = -0.0969$).

3. (Clock/Angles): An automated telescope needs to align with a star tracking algorithm at exactly 8:15 PM. To ensure exact calibration, the software calculates the interior angle between the hour and minute hands of an analog clock at 8:15. What is this analytical angle?

4. (Calendar/Calculation): The historical independence day of a certain nation was celebrated on July 4, 1776. By counting the number of odd days starting from 0001 AD, determine the exact day of the week this event occurred.

5. (Time & Work): A construction project requires a bridge to be built. Company A can build it in 45 days. Company B can build it in 60 days. An external supervisor is tracking this project using project management software. If both companies work together for 12 days, what fraction of the work remains for the supervisor to allocate to a third company?

Answers to Competency-Based Questions

1. Application/Analysis:

• $105 = 64 + 32 + 8 + 1 = 2^6 + 2^5 + 2^3 + 2^0 \implies 1101001_2$.
• $1001011_2 = 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 1(2) + 1(1) = 64 + 8 + 2 + 1 = 75$ minutes.

2. Logarithms in Real Life: $$ 100 = 10000(0.8)^t \implies 0.01 = 0.8^t $$ Taking log on both sides: $$ \log(0.01) = t \log(0.8) \implies -2 = t(-0.0969) $$ $$ t = \frac{2}{0.0969} \approx 20.64 \text{ days} $$

3. Clock/Angles: At 8:15, $H=8$ and $M=15$. $$ \theta = |30(8) - 5.5(15)| = |240 - 82.5| = 157.5^\circ $$

4. Calendar/Calculation: Total Odd Days up to 1775:

• 1600 years = $0$ odd days.
• 100 years = $5$ odd days.
• 75 years = 18 leap years + 57 ordinary years = $(18 \times 2) + (57 \times 1) = 36 + 57 = 93 \equiv 2$ odd days. Odd days in 1776 up to July 4 (1776 is leap):
• Jan(3) + Feb(1) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(4) = $18 \equiv 4$ odd days.
• Total $= 5 + 2 + 4 = 11 \equiv 4$ odd days. 0=Sun, 1=Mon, 2=Tue, 3=Wed, 4=Thu. Thursday.

5. Time & Work: Work done by A and B in 1 day $= \frac{1}{45} + \frac{1}{60} = \frac{4+3}{180} = \frac{7}{180}$. Work done in 12 days $= 12 \times \frac{7}{180} = \frac{7}{15}$. Fraction of work remaining $= 1 - \frac{7}{15} = \frac{8}{15}$.