Chapter 1 Numbers and Quantification (Concepts)

Binary Numbers

The number system we commonly use in our daily lives is the Decimal Number System (also known as Base-10 system). This system is based on powers of 10 and uses ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The value or magnitude of a digit within a decimal number is determined by its position, or place value. Each position represents a power of 10, increasing from right to left, starting with $10^0$. For instance, in the decimal number $754$:

$754_{10} = 7 \times 10^2 + 5 \times 10^1 + 4 \times 10^0 = 7 \times 100 + 5 \times 10 + 4 \times 1 = 700 + 50 + 4 = 754$.

In contrast to the decimal system, the Binary Number System (also known as Base-2 system) is a positional numeral system that uses only two digits: 0 and 1. It is the fundamental language of computers and digital systems because it aligns perfectly with devices that have only two stable states, such as switches being ON (1) or OFF (0), or electrical signals representing High (1) or Low (0) voltage. This two-state system makes digital circuits simpler and more reliable.

Place Value in Binary Numbers

Just like in the decimal system, the value of each digit in a binary number is determined by its position. However, in the binary system, the place values are powers of 2, not 10. Starting from the rightmost digit, which is the Least Significant Bit (LSB), the positions correspond to powers of 2 increasing as we move leftwards. The place values are $2^0, 2^1, 2^2, 2^3, 2^4$, and so on.

For a binary number represented as $(b_n b_{n-1} ... b_1 b_0)_2$, where $b_i$ is either 0 or 1 (a binary digit, also called a bit), its equivalent value in the decimal system is calculated as the sum of each digit multiplied by its corresponding power of 2. This can be expressed by the following formula:

Here, $n$ represents the highest position index (starting from 0 on the right). Each term $b_i \times 2^i$ contributes to the total decimal value. If $b_i$ is 0, the term is 0. If $b_i$ is 1, the term is $2^i$.

Conversion from Binary to Decimal

To convert a binary number into its decimal equivalent, we directly apply the formula mentioned above. We identify the position of each bit (starting from 0 for the rightmost bit), calculate the corresponding power of 2 for each position, multiply each bit by its respective power of 2, and finally, sum up all these products.

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Conversion from Decimal to Binary

To convert a decimal number to its binary equivalent, the most common method is the method of successive division by 2. In this method, we repeatedly divide the decimal number by 2 and record the remainder (which will always be either 0 or 1) at each step. We continue this process with the quotient until the quotient becomes 0. The binary representation is then formed by reading the remainders from bottom to top, where the last remainder is the Most Significant Bit (MSB) and the first remainder is the Least Significant Bit (LSB).

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Indices, Logarithm and Antilogarithm

Indices (Exponents)

Indices, also known as exponents or powers, are a mathematical notation used to represent repeated multiplication of a number by itself. If $a$ is a real number and $n$ is a positive integer, then $a^n$ (read as "$a$ raised to the power of $n$") is defined as the product of $n$ factors, each equal to $a$:

In the expression $a^n$, $a$ is called the base and $n$ is called the exponent or index. This notation provides a concise way to write very large or very small numbers involving repeated multiplication.

Laws of Indices (Recap)

For any non-zero real numbers $a$ and $b$, and rational exponents $m$ and $n$ (assuming the bases are such that the expressions are defined and real):

• Product Rule: When multiplying exponential terms with the same base, add the exponents.

  $a^m \times a^n = a^{m+n}$

  Example: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$. (Because $2^3 = 8$ and $2^4 = 16$, $8 \times 16 = 128$.)

• Quotient Rule: When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

  $$\frac{a^m}{a^n} = a^{m-n}$$, for $a \neq 0$

  Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$. (Because $5^6 = 15625$ and $5^2 = 25$, $15625 \div 25 = 625$.)

• Power Rule: When raising an exponential term to another power, multiply the exponents.

  $(a^m)^n = a^{mn}$

  Example: $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$. (Because $(3^2)^3 = 9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$.)

• Power of a Product: When raising a product of bases to a power, raise each base to that power.

  $(ab)^n = a^n b^n$

  Example: $(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000$. (Because $(2 \times 5)^3 = 10^3 = 1000$.)

• Power of a Quotient: When raising a quotient of bases to a power, raise the numerator and the denominator to that power.

  $$(\frac{a}{b})^n = \frac{a^n}{b^n}$$, for $b \neq 0$

  Example: $(\frac{4}{2})^3 = \frac{4^3}{2^3} = \frac{64}{8} = 8$. (Because $(\frac{4}{2})^3 = 2^3 = 8$.)

• Zero Exponent: Any non-zero base raised to the power of zero is equal to 1. The expression $0^0$ is typically considered undefined in elementary algebra, though in some contexts (like calculus or set theory), it might be defined as 1.

  $$a^0 = 1$$, for $a \neq 0$

  Example: $7^0 = 1$, $(\frac{1}{2})^0 = 1$.

• Negative Exponent: A base raised to a negative exponent is equal to the reciprocal of the base raised to the corresponding positive exponent.

  $$a^{-n} = \frac{1}{a^n}$$, for $a \neq 0$

  $$\frac{1}{a^{-n}} = a^n$$, for $a \neq 0$

  Example: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

• Fractional Exponent: A base raised to a fractional exponent $m/n$ represents the $n$-th root of the base raised to the power of $m$. If $m=1$, it represents the $n$-th root.

  $$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

  $$a^{1/n} = \sqrt[n]{a}$$

  Assuming $\sqrt[n]{a}$ is a real number. If $n$ is even, $a$ must be non-negative for the real root to exist. If $n$ is odd, $a$ can be any real number.

  Examples: $25^{1/2} = \sqrt[2]{25} = \sqrt{25} = 5$. $8^{2/3} = (\sqrt[3]{8})^2 = (2)^2 = 4$.

Logarithm

Logarithms are fundamentally linked to indices as they are the inverse operation to exponentiation. While exponentiation answers "What is $a$ raised to the power of $x$?", logarithms answer "To what power must the base $a$ be raised to get the number $N$?".

Formally, the logarithm of a positive number $N$ with respect to a positive base $a$ (where $a \neq 1$) is the exponent $x$ such that $a$ raised to the power of $x$ equals $N$. This relationship can be written in two equivalent forms:

These two forms are interchangeable and express the same mathematical relationship. In $\log_a N = x$:

• $a$ is the base of the logarithm. It must be a positive number and $a \neq 1$. The reason the base cannot be 1 is that $1^x = 1$ for any $x$, so you cannot uniquely determine $x$ if $N$ is not 1. If $N=1$, $\log_1 1$ could be any number, again not unique.
• $N$ is the argument or number for which the logarithm is being calculated. It must be a positive number ($N > 0$). This is because a positive base raised to any real power will always result in a positive number ($a^x > 0$).
• $x$ is the logarithm or the exponent. It is the value we are trying to find. The logarithm can be any real number (positive, negative, or zero).

Types of Logarithms

Although the base $a$ can be any positive real number other than 1, certain bases are used frequently and have special notation:

• Common Logarithms: These are logarithms with base 10. They are often written as $\log N$ or $\text{log}_{10} N$. The base 10 is convenient because our decimal number system is based on powers of 10. Common logarithms are useful in fields like engineering, physics, and chemistry, especially for calculations involving scaling and magnitude (e.g., Richter scale for earthquakes, pH scale for acidity).

  If $10^x = N$, then $\log_{10} N = x$ or $\log N = x$.

  Example: $\log_{10} 100 = 2$ because $10^2 = 100$. $\log_{10} 0.1 = -1$ because $10^{-1} = 0.1 = \frac{1}{10}$.

• Natural Logarithms: These are logarithms with base $e$, where $e$ is an important irrational mathematical constant approximately equal to $2.71828$. Natural logarithms are denoted by $\ln N$ or $\log_e N$. The base $e$ arises naturally in calculus and phenomena involving exponential growth or decay, such as compound interest, population growth, and radioactive decay.

  If $e^x = N$, then $\log_e N = x$ or $\ln N = x$.

  Example: $\ln e = 1$ because $e^1 = e$. $\ln 1 = 0$ because $e^0 = 1$.

Conversion between Exponential and Logarithmic Forms

Understanding the equivalence between $a^x = N$ and $\log_a N = x$ is crucial for working with logarithms. You can convert any equation from one form to the other.

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Antilogarithm

The antilogarithm is the inverse operation of the logarithm. Finding the antilogarithm of a number $x$ with respect to a base $a$ means finding the original number $N$ whose logarithm base $a$ is $x$.

In simpler terms, if you know the logarithm of a number ($\log_a N = x$), the antilogarithm operation helps you find the number ($N$) itself.

By definition, if $\log_a N = x$, then the exponential form is $N = a^x$. Therefore, the antilogarithm of $x$ to the base $a$ is simply $a^x$.

Commonly encountered antilogarithms are with base 10 and base $e$:

• For common logarithms (base 10): If $\log_{10} N = x$, then $N = \text{antilog}_{10}(x) = 10^x$. This is also called the inverse logarithm or the base-10 exponential function.
• For natural logarithms (base $e$): If $\log_e N = x$, then $N = \text{antilog}_e(x) = e^x$. This is also known as the exponential function.

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Laws and Properties of Logarithms

Logarithms are powerful tools for simplifying calculations, especially those involving multiplication, division, and exponentiation. Their utility comes from a set of fundamental laws and properties that are directly derived from the corresponding laws of indices (exponents). Understanding these laws is essential for solving problems involving logarithms.

Fundamental Laws of Logarithms

Let $a$ be a positive real number such that $a \neq 1$. This is the base of the logarithm. Let $M$ and $N$ be positive real numbers. These are the arguments of the logarithm. Let $p$ be any real number (can be positive, negative, or zero, rational or irrational). The following laws hold true:

1. Product Rule: The logarithm of a product is the sum of the logarithms.

This law states that the logarithm of the product of two positive numbers is equal to the sum of their individual logarithms, provided they have the same base.

Derivation:

2. Quotient Rule: The logarithm of a quotient is the difference of the logarithms.

This law states that the logarithm of the quotient of two positive numbers is equal to the difference between the logarithm of the numerator and the logarithm of the denominator, provided they have the same base.

Derivation:

3. Power Rule: The logarithm of a number raised to a power is the power multiplied by the logarithm of the number.

This law states that when the argument of a logarithm is raised to a power, that power can be brought down as a multiplier of the logarithm of the base number.

Derivation:

4. Change of Base Formula: Changing the base of a logarithm.

This is a very useful formula that allows us to convert a logarithm from any base $b$ to another base $c$ (where $c>0, c \neq 1$). This is particularly important when using calculators or software that only provide logarithms for specific bases, like base 10 (log) or base $e$ (ln).

Derivation:

Other Important Properties

Besides the fundamental laws, there are several other properties that directly follow from the definition of logarithms and the laws of indices. For any valid base $a$ ($a>0, a \neq 1$) and positive number $N$:

• Logarithm of the Base: The logarithm of the base itself is always 1.

  $$\log_a a = 1$$

  Explanation: This is true because, by definition, $\log_a a = x$ means $a^x = a$. Since $a = a^1$, we have $a^x = a^1$, which implies $x=1$.

• Logarithm of One: The logarithm of 1 to any valid base is always 0.

  $$\log_a 1 = 0$$

  Explanation: This is true because, by definition, $\log_a 1 = x$ means $a^x = 1$. From the laws of indices, we know that any non-zero number raised to the power of 0 is 1 ($a^0 = 1$). Therefore, $a^x = a^0$, which implies $x=0$.

• Inverse Property 1: Raising the base to the power of the logarithm of a number (with the same base) results in the number itself.

  $$a^{\log_a N} = N$$

  Explanation: This property highlights the inverse relationship between exponentiation and logarithm. If $\log_a N = x$, then by definition $a^x = N$. Substituting $x = \log_a N$ back into the exponential form gives $a^{\log_a N} = N$.

• Inverse Property 2: The logarithm of the base raised to a power is the power itself.

  $$\log_a (a^x) = x$$

  Explanation: This property also stems from the inverse relationship. Let $a^x = P$. By definition, $\log_a P = x$. Substituting $P=a^x$ back into the logarithmic form gives $\log_a (a^x) = x$. Alternatively, this can be seen as a special case of the power rule: $\log_a (a^x) = x \log_a a$. Since $\log_a a = 1$, we get $x \times 1 = x$.

• Logarithm with a Power in the Base: If the base of the logarithm is a power of some other number.

  $$\log_{a^k} N = \frac{1}{k} \log_a N$$

  Derivation: We can derive this using the change of base formula. Let the new base be $a$.

  $$\log_{a^k} N = \frac{\log_a N}{\log_a (a^k)}$$

  [Using change of base with base $a$]

  Apply the power rule to the denominator: $\log_a (a^k) = k \log_a a = k \times 1 = k$.

  $$= \frac{\log_a N}{k}$$

  $$= \frac{1}{k} \log_a N$$

  This property is useful when the base is a power, e.g., $\log_{16} 8 = \log_{2^4} 2^3$. Using the formula, $\log_{2^4} 2^3 = \frac{1}{4} \log_2 2^3 = \frac{1}{4} \times 3 = \frac{3}{4}$. Alternatively, using change of base: $\log_{16} 8 = \frac{\log_2 8}{\log_2 16} = \frac{3}{4}$.

• Reciprocal Property: The logarithm of a number to a certain base is the reciprocal of the logarithm of the base to that number.

  $$\log_a b = \frac{1}{\log_b a}$$

  Derivation: This can also be derived using the change of base formula. Use base $c=a$.

  $$\log_a b = \frac{\log_a b}{\log_a a} = \frac{\log_a b}{1}$$

  [Using change of base formula]

  Now, use base $c=b$ in the change of base formula for $\log_a b$:

  $$\log_a b = \frac{\log_b b}{\log_b a}$$

  [Using change of base with base $b$]

  Since $\log_b b = 1$ (logarithm of the base property), we get:

  $$= \frac{1}{\log_b a}$$

  This property shows how to swap the base and the argument of a logarithm.

Examples Applying Logarithm Laws

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Simple Applications of Logarithm and Antilogarithm

While the fundamental definition of logarithms relates them directly to exponents, their true power lies in their ability to simplify complex mathematical operations and model real-world phenomena. Historically, before the widespread availability of calculators and computers, logarithms were indispensable for performing tedious arithmetic calculations quickly and accurately. Today, they remain crucial in solving certain types of equations and representing data across vast scales in various scientific and practical applications.

Simplifying Calculations (Historical Context)

The invention of logarithms by John Napier and Joost Bürgi in the early 17th century revolutionized computation. They created extensive tables of logarithms (Napier's used a base related to $e$, while common logarithms, base 10, were later developed). By converting numbers to their logarithms, complicated operations could be replaced by simpler ones, as demonstrated by the laws of logarithms:

• Multiplication becomes addition ($\log_a (MN) = \log_a M + \log_a N$).
• Division becomes subtraction ($\log_a (\frac{M}{N}) = \log_a M - \log_a N$).
• Raising to a power becomes multiplication ($\log_a (M^p) = p \log_a M$).
• Finding a root becomes division ($\log_a (\sqrt[n]{M}) = \log_a (M^{1/n}) = \frac{1}{n} \log_a M$).

To perform a calculation like $P = M \times N$ using log tables (typically base 10):

1. Find the logarithm of each number from the log table: Find $\log_{10} M$ and $\log_{10} N$.

2. Use the Product Rule: $\log_{10} P = \log_{10} M + \log_{10} N$. Add the logarithms found in step 1.

3. Find the number whose logarithm is $\log_{10} P$. This is done using antilogarithm tables (which are essentially inverse log tables). $P = \text{antilog}_{10}(\log_{10} P)$.

Similarly, for division $Q = \frac{M}{N}$:

1. Find $\log_{10} M$ and $\log_{10} N$ from log tables.

2. Use the Quotient Rule: $\log_{10} Q = \log_{10} M - \log_{10} N$. Subtract $\log_{10} N$ from $\log_{10} M$.

3. Find the antilogarithm of the result: $Q = \text{antilog}_{10}(\log_{10} Q)$.

For finding powers $R = M^p$:

1. Find $\log_{10} M$ from log tables.

2. Use the Power Rule: $\log_{10} R = p \times \log_{10} M$. Multiply the logarithm by the power $p$.

3. Find the antilogarithm of the result: $R = \text{antilog}_{10}(\log_{10} R)$.

Characteristic and Mantissa (for Base 10 Logarithms)

When using log tables for base 10 logarithms ($\log_{10}$), it is helpful to understand the structure of the logarithm of a positive number. Any positive real number $N$ can be expressed in scientific notation (or standard form) as $N = m \times 10^c$, where $1 \leq m < 10$ and $c$ is an integer.

Taking the base-10 logarithm of both sides:

Using the Product Rule of logarithms ($\log_a (MN) = \log_a M + \log_a N$):

Using the Power Rule of logarithms ($\log_a (M^p) = p \log_a M$) on the second term:

Since $\log_{10} 10 = 1$:

Rearranging the terms, we get:

In this equation:

• $c$ is an integer and is called the characteristic of $\log_{10} N$. It determines the magnitude (number of digits) of the original number $N$.
• $\log_{10} m$ is the decimal part and is called the mantissa of $\log_{10} N$. Since $1 \leq m < 10$, taking $\log_{10}$ gives $\log_{10} 1 \leq \log_{10} m < \log_{10} 10$, which means $0 \leq \log_{10} m < 1$. The mantissa is always a non-negative decimal fraction less than 1. Log tables provide the mantissa for a sequence of digits, ignoring the decimal point.

The characteristic $c$ can be determined by the position of the decimal point in $N$:

• If $N \geq 1$, the characteristic is one less than the number of digits in the integer part of $N$. For example, if $N=123.45$, the integer part has 3 digits, so $c = 3-1=2$. $\log_{10} 123.45 \approx 2.0915$.
• If $0 < N < 1$, the characteristic is negative and its magnitude is one more than the number of zeros immediately after the decimal point. The negative characteristic is usually written with a bar above the digit, e.g., $\bar{1}, \bar{2}$. For example, if $N=0.0123$, there is one zero after the decimal point, so $c = -(1+1) = -2$, written as $\bar{2}$. $\log_{10} 0.0123 \approx \bar{2}.0899 = -2 + 0.0899 = -1.9101$. The mantissa ($0.0899$) is always positive.

Antilog tables work by using the mantissa to find the significant digits of the number $N$ and the characteristic to place the decimal point.

Solving Exponential and Logarithmic Equations

One of the most significant modern applications of logarithms is in solving equations where the unknown variable appears in the exponent (exponential equations) or as the argument of a logarithm (logarithmic equations). The key is to use the inverse relationship between exponentiation and logarithms, and the laws of logarithms to isolate the variable.

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Applications in Various Fields

Logarithms are widely used in many scientific, engineering, and financial disciplines, often because they help in handling very large or very small numbers by compressing the scale, or because the phenomena themselves follow logarithmic or exponential patterns.

Science and Engineering

• pH Scale: This scale measures the acidity or alkalinity of an aqueous solution. It is defined as the negative of the base-10 logarithm of the molar concentration of hydrogen ions ($[H^+]$) in moles per litre.

  $$pH = -\log_{10} [H^+]$$

  A difference of 1 on the pH scale corresponds to a tenfold difference in hydrogen ion concentration. For example, a solution with pH 3 is ten times more acidic than a solution with pH 4.

• Decibel (dB) Scale: Used in acoustics, electronics, and communication to measure ratios of power or intensity. It is a logarithmic unit that expresses the ratio of a value to a reference value.

  $$L_{I} (\text{in dB}) = 10 \log_{10} \left(\frac{I}{I_0}\right)$$

  Where $I$ is the intensity of the sound/signal and $I_0$ is a reference intensity (e.g., threshold of human hearing). A 10 dB increase represents a tenfold increase in intensity.

• Richter Scale: Measures the magnitude of earthquakes. Developed by Charles Richter, this scale assigns a single number to quantify the energy released by an earthquake. It is a base-10 logarithmic scale.

  $$M = \log_{10} A - \log_{10} A_0 = \log_{10} \left(\frac{A}{A_0}\right)$$

  Where $M$ is the magnitude, $A$ is the amplitude of the seismic waves measured by a seismograph, and $A_0$ is the amplitude of a reference small earthquake. A magnitude 7 earthquake is ten times larger in amplitude (and roughly $10^{1.5} \approx 32$ times more energetic) than a magnitude 6 earthquake.

• Star Brightness (Magnitude Scale): In astronomy, the apparent magnitude of a celestial body is a logarithmic measure of its brightness as seen by an observer on Earth. A difference of 1 magnitude corresponds to a brightness ratio of approximately 2.512 ($10^{0.4}$).

Finance and Growth Models

• Compound Interest: Logarithms are used to solve problems involving compound interest, particularly when you need to find the time period or the interest rate. The formula for the future value $A$ of an investment with principal $P$ compounded annually at an interest rate $r$ for $n$ years is:

  $$A = P(1 + r)^n$$

  ... (6)

  To find $n$, you can use logarithms.

• Exponential Growth and Decay: Phenomena like population growth, radioactive decay, and the charging/discharging of capacitors are often modeled by exponential functions of the form $N(t) = N_0 e^{kt}$ or $N(t) = N_0 a^t$. Logarithms (especially natural logarithms for base $e$ models) are used to solve for time $t$, initial amount $N_0$, or the rate constant $k$.

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