Comparison and Ordering of Numbers

Comparison Methods for Different Number Types

Comparing numbers is a fundamental skill in mathematics, allowing us to determine the relative size or value of two or more numbers. Comparison involves stating whether one number is greater than, less than, or equal to another. Ordering involves arranging a set of numbers based on their values, typically in ascending (smallest to largest) or descending (largest to smallest) order.

The standard mathematical symbols used for comparison are:

• $a > b$: $a$ is greater than $b$.
• $a < b$: $a$ is less than $b$.
• $a = b$: $a$ is equal to $b$.
• $a \ge b$: $a$ is greater than or equal to $b$.
• $a \le b$: $a$ is less than or equal to $b$.

The method used for comparing and ordering numbers often depends on the type of numbers involved, such as integers, fractions, or decimals.

Comparing Integers ($\mathbb{Z}$)

Integers ($\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$) are discrete points on the number line. Their order is clearly defined by their position relative to the origin (0) and to each other.

The most intuitive way to compare integers is to visualize them on the number line:

For any two distinct integers, the integer that appears further to the right on the standard horizontal number line is the greater number.

• Comparing positive integers: A positive integer with a larger value is greater. E.g., $7 > 3$, $100 > 99$.
• Comparing positive and negative integers: Any positive integer is always greater than any negative integer. E.g., $5 > -100$, $1 > -1$.
• Comparing zero with other integers: Zero is greater than any negative integer and less than any positive integer. E.g., $0 > -50$, $0 < 20$.
• Comparing two negative integers: For two negative integers, the one with the smaller absolute value is greater. This is because it is closer to zero on the number line, and numbers increase as you move right towards zero from the negative side. E.g., compare $-2$ and $-5$. The absolute value of $-2$ is $|-2|=2$. The absolute value of $-5$ is $|-5|=5$. Since $2 < 5$, $-2$ is closer to 0 than $-5$. On the number line, $-2$ is to the right of $-5$. Therefore, $-2 > -5$. Similarly, $-1 > -10$.

Consider the integers $-5, -2, 0, 3$. Their positions on the number line clearly show their order:

(Note: The image shows a number line with marks at -5, -2, 0, and 3. The visual arrangement confirms their order.)

Based on their positions from left to right, we can order them:

Answer:

Comparing Fractions ($\mathbb{Q}$ in fractional form)

Comparing fractions involves determining which fraction represents a larger or smaller part of a whole. This can be done effectively by expressing the fractions with respect to a common reference.

Method 1: Finding a Common Denominator

This is a reliable method, especially for comparing more than two fractions. To compare fractions like $\frac{a}{b}$ and $\frac{c}{d}$ (where $b \neq 0, d \neq 0$), we convert them into equivalent fractions that share the same denominator.

1. Find the Least Common Multiple (LCM) of the denominators: The LCM of the denominators $b$ and $d$ is the smallest positive integer that is a multiple of both $b$ and $d$. This LCM will serve as the common denominator. For multiple fractions, find the LCM of all denominators.
2. Convert each fraction to an equivalent fraction with the common denominator: For each fraction, multiply both its numerator and its denominator by the factor that makes the denominator equal to the LCM.

   For a fraction $\frac{a}{b}$, the multiplying factor is $\frac{\text{LCM}(b,d)}{b}$. The equivalent fraction is $\frac{a \times \left(\frac{\text{LCM}(b,d)}{b}\right)}{b \times \left(\frac{\text{LCM}(b,d)}{b}\right)}$.

   For a fraction $\frac{c}{d}$, the multiplying factor is $\frac{\text{LCM}(b,d)}{d}$. The equivalent fraction is $\frac{c \times \left(\frac{\text{LCM}(b,d)}{d}\right)}{d \times \left(\frac{\text{LCM}(b,d)}{d}\right)}$.

3. Compare the numerators: Once all fractions have the same denominator, their magnitudes are directly proportional to their numerators. The fraction with the larger numerator is the greater fraction.

   If you are comparing $\frac{N_1}{D}$ and $\frac{N_2}{D}$ (where $D > 0$), then:

   • If $N_1 > N_2$, then $\frac{N_1}{D} > \frac{N_2}{D}$.
   • If $N_1 < N_2$, then $\frac{N_1}{D} < \frac{N_2}{D}$.
   • If $N_1 = N_2$, then $\frac{N_1}{D} = \frac{N_2}{D}$.

   If the common denominator $D$ is negative, the comparison of numerators is reversed, but for simplicity, we usually work with positive common denominators.

Answer:

Method 2: Cross-Multiplication

This method is particularly useful for quickly comparing just two fractions, say $\frac{a}{b}$ and $\frac{c}{d}$, where the denominators $b$ and $d$ are positive ($b > 0, d > 0$).

1. Calculate the two cross-products: $a \times d$ and $c \times b$.
2. Compare the two products:
   • If $a \times d > c \times b$, then $\frac{a}{b} > \frac{c}{d}$.
   • If $a \times d < c \times b$, then $\frac{a}{b} < \frac{c}{d}$.
   • If $a \times d = c \times b$, then $\frac{a}{b} = \frac{c}{d}$.

Derivation of the method: To compare $\frac{a}{b}$ and $\frac{c}{d}$, we can use the common denominator method. The common denominator is $bd$. The equivalent fractions are $\frac{ad}{bd}$ and $\frac{cb}{bd}$. Since $b>0$ and $d>0$, $bd > 0$. Comparing $\frac{ad}{bd}$ and $\frac{cb}{bd}$ is equivalent to comparing their numerators, $ad$ and $cb$. Thus, comparing $\frac{a}{b}$ and $\frac{c}{d}$ is equivalent to comparing $ad$ and $cb$.

Answer:

Special Cases for Comparing Fractions

• Fractions with the Same Denominator: If two fractions have the same positive denominator, the fraction with the larger numerator is the greater fraction. E.g., $\frac{3}{5} > \frac{2}{5}$ because $3 > 2$. If the denominator is negative, the comparison of numerators is reversed.
• Fractions with the Same Numerator: If two positive fractions have the same positive numerator, the fraction with the smaller positive denominator is the greater fraction. E.g., $\frac{1}{2} > \frac{1}{3}$ because dividing a whole into 2 parts results in larger parts than dividing it into 3 parts. For negative fractions with the same numerator (and positive denominators), the one with the smaller denominator is less (e.g., $-\frac{1}{2} < -\frac{1}{3}$).
• Comparing Negative Fractions: To compare negative fractions, you can compare their absolute values. The negative fraction with the smaller absolute value is greater (because it is closer to $0$ on the number line). E.g., compare $-\frac{3}{4}$ and $-\frac{5}{6}$. We know $\frac{3}{4} < \frac{5}{6}$ (since $3 \times 6 = 18$ and $5 \times 4 = 20$, and $18 < 20$). Since $|\!-\!\frac{3}{4}| < |\!-\!\frac{5}{6}|$, we have $-\frac{3}{4} > -\frac{5}{6}$. Alternatively, you can use the common denominator method or cross-multiplication directly with signed numerators.

Comparing Decimal Numbers

Decimal numbers are compared by examining their digits from left to right, starting from the highest place value.

1. Compare the whole number parts: Compare the integers to the left of the decimal point. The number with the larger whole number part is the greater number. (Handle negative numbers by comparing absolute values first, then reversing the comparison as done for integers).
2. If whole number parts are equal: Move to the decimal part and compare the digits from left to right, position by position (Tenths, Hundredths, Thousandths, and so on).
3. Identify the first differing digits: At the first place value where the digits of the two numbers are different, the number with the larger digit at that specific place is the greater number.
4. Adding trailing zeros: It can be helpful to add trailing zeros to the decimal with fewer decimal places so that both numbers have the same number of digits after the decimal point. This does not change the value of the number (e.g., $2.5 = 2.50 = 2.500$), but it aligns the place values and can make the digit-by-digit comparison clearer.

Answer:

Answer:

Comparing Different Types of Numbers

To compare numbers presented in different formats (e.g., a fraction and a decimal, a rational number and an irrational number), the most effective strategy is usually to convert them to a common format, typically decimal form.

1. Convert all numbers to decimal form:
   • If a number is a fraction $\frac{p}{q}$, perform the division $p \div q$ to get its decimal expansion.
   • If a number is already a decimal, keep it as is.
   • If a number is an irrational number (like $\sqrt{2}$ or $\pi$), use its known decimal approximation to a sufficient number of decimal places.
2. Compare the decimal expansions: Once all numbers are in decimal form, compare them using the digit-by-digit method described for decimal numbers. For non-terminating decimals (repeating or non-repeating), compare as many digits as necessary until a difference is found or until you have enough confidence for the comparison required.

Answer:

Answer:

Summary of Comparison Methods

Mastering these comparison methods is essential for solving inequalities and ordering various sets of numbers.

Absolute Value of Integers and Rational Numbers

Definition and Meaning of Absolute Value

The absolute value of a number is its distance from zero on the number line. Distance is always a non-negative quantity, so the absolute value of a number is always a non-negative number.

The absolute value of a number $x$ is denoted by placing the number between two vertical bars: $|x|$.

The formal definition of the absolute value of a real number $x$ is:

This definition can be interpreted as follows:

• If the number $x$ is positive or zero ($x \ge 0$), its absolute value is the number itself. For example, $|5| = 5$ and $|0| = 0$.
• If the number $x$ is negative ($x < 0$), its absolute value is the opposite of the number, which is positive. For example, $|-5| = -(-5) = 5$.

Visually, on the number line, the absolute value represents the magnitude of the number without considering its direction (positive or negative). Both a number and its additive inverse (opposite) are the same distance from zero.

(Note: The image shows a number line with the origin 0. Points are marked at -4 and 4. Arrows or labels indicate that the distance from 0 to -4 is 4 units, and the distance from 0 to 4 is also 4 units. Thus, $|-4|=4$ and $|4|=4$.)

Absolute Value of Integers ($\mathbb{Z}$)

The concept of absolute value applies directly to integers. For any integer $n$, its absolute value $|n|$ is its non-negative value, representing its distance from 0 on the number line.

• For a positive integer $n$, $|n| = n$. Example: $|10| = 10$, $|7| = 7$.
• For the integer zero, $|0| = 0$.
• For a negative integer $-n$ (where $n$ is a positive integer), $|-n| = -(-n) = n$. Example: $|-10| = -(-10) = 10$, $|-7| = -(-7) = 7$.

In simpler terms, to find the absolute value of a non-zero integer, just remove its sign.

Answer:

Absolute Value of Rational Numbers ($\mathbb{Q}$)

The definition and concept of absolute value apply equally to all rational numbers, which include integers, fractions, and terminating or repeating decimals. The absolute value of a rational number $q$ is its non-negative value, representing its distance from 0 on the number line.

For any rational number $q$:

• If $q \ge 0$, then $|q| = q$.
  • $|\frac{3}{4}| = \frac{3}{4}$
  • $|5.12| = 5.12$
  • $|0.\overline{7}| = 0.\overline{7}$
• If $q < 0$, then $|q| = -q$.
  • $|-\frac{3}{4}| = -(-\frac{3}{4}) = \frac{3}{4}$
  • $|-5.12| = -(-5.12) = 5.12$
  • $|-0.\overline{7}| = -(-0.\overline{7}) = 0.\overline{7}$

Answer:

Answer:

Key Properties of Absolute Value

The absolute value function has several important properties that are true for all real numbers, and therefore specifically for all integers and rational numbers. Let $x$ and $y$ be any real numbers.

• Non-negativity: The absolute value of any number is always greater than or equal to zero.

  $\quad |x| \ge 0$

  [Absolute value is non-negative]

  Also, the absolute value is zero if and only if the number itself is zero.

  $\quad |x| = 0 \iff x = 0$

• Symmetry: A number and its additive inverse (opposite) have the same absolute value.

  $\quad |-x| = |x|$

  [Symmetry property]

  Example: $|-15| = 15$ and $|15| = 15$. So, $|-15| = |15|$.

• Multiplicativity: The absolute value of a product of two numbers is equal to the product of their absolute values.

  $\quad |x \times y| = |x| \times |y|$

  [Multiplicativity property]

  Example: Let $x = -4$ and $y = 5$.

  $|x \times y| = |(-4) \times 5| = |-20| = 20$.

  $|x| \times |y| = |-4| \times |5| = 4 \times 5 = 20$.

  Since $20 = 20$, $|(-4) \times 5| = |-4| \times |5|$.

• Divisibility (for non-zero divisor): The absolute value of a quotient of two numbers is equal to the quotient of their absolute values, provided the divisor is not zero.

  $\quad |\frac{x}{y}| = \frac{|x|}{|y|}$, provided $y \neq 0$

  [Divisibility property]

  Example: Let $x = 12$ and $y = -3$.

  $|\frac{x}{y}| = |\frac{12}{-3}| = |-4| = 4$.

  $\frac{|x|}{|y|} = \frac{|12|}{|-3|} = \frac{12}{3} = 4$.

  Since $4 = 4$, $|\frac{12}{-3}| = \frac{|12|}{|-3|}$.

• Triangle Inequality: The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

  $\quad |x + y| \le |x| + |y|$

  [Triangle Inequality]

  Example: Let $x = 2$ and $y = 3$. $|2+3|=|5|=5$. $|2|+|3|=2+3=5$. $5 \le 5$.

  Let $x = -2$ and $y = 3$. $|-2+3|=|1|=1$. $|-2|+|3|=2+3=5$. $1 \le 5$.

  Let $x = -2$ and $y = -3$. $|-2+(-3)|=|-5|=5$. $|-2|+|-3|=2+3=5$. $5 \le 5$.

  This property is fundamental in various areas of mathematics, especially in defining distances and convergence.

• Reverse Triangle Inequality: A related property states that the absolute value of the difference of two absolute values is less than or equal to the absolute value of their difference.

  $\quad ||x| - |y|| \le |x - y|$

  [Reverse Triangle Inequality]

  Example: Let $x = 5$ and $y = 2$. $||5| - |2|| = |5 - 2| = |3| = 3$. $|5 - 2| = |3| = 3$. $3 \le 3$.

  Let $x = 2$ and $y = 5$. $||2| - |5|| = |2 - 5| = |-3| = 3$. $|2 - 5| = |-3| = 3$. $3 \le 3$.

  Let $x = -5$ and $y = 2$. $||-5| - |2|| = |5 - 2| = |3| = 3$. $|-5 - 2| = |-7| = 7$. $3 \le 7$.

Absolute value is used in various mathematical contexts, such as measuring errors (absolute error), defining distance between points on the number line ($|x-y|$ is the distance between $x$ and $y$), solving equations and inequalities involving absolute values, and in defining norms and metrics in higher mathematics.

Finding Numbers Between Two Given Numbers (Rational and Irrational)

The set of real numbers ($\mathbb{R}$) is continuous and dense, meaning there are no "gaps" on the number line. This density applies not only to the set of rational numbers ($\mathbb{Q}$) but also to the set of irrational numbers ($\mathbb{I}$). Furthermore, the real number line is "filled" in such a way that between any two distinct real numbers, no matter how close they are, there are infinitely many rational numbers and infinitely many irrational numbers.

Finding Rational Numbers Between Two Given Rational Numbers

Given two distinct rational numbers, say $r_1$ and $r_2$, with $r_1 < r_2$, we know that the set of rational numbers is dense. This means there are infinitely many rational numbers located between $r_1$ and $r_2$. Here are common methods to find such numbers:

Method 1: Using the Midpoint (Average)

The simplest rational number to find between two distinct rational numbers is their average (arithmetic mean). The average of $r_1$ and $r_2$ is $\frac{r_1 + r_2}{2}$.

As proven in the section on Rational Numbers, if $r_1$ and $r_2$ are rational, their sum $r_1+r_2$ is rational (due to closure under addition for rationals), and dividing a rational number by a non-zero rational number (2 in this case) results in a rational number (due to closure under division for rationals). Thus, $\frac{r_1+r_2}{2}$ is always rational.

Also, as proven in the section on Rational Numbers' density, if $r_1 < r_2$, then $r_1 < \frac{r_1 + r_2}{2} < r_2$. So, the average lies strictly between the two numbers.

Answer:

Method 2: Using Equivalent Fractions with a Larger Denominator

This method is useful for finding a specific number of rational numbers between two given rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ (assume $b > 0, d > 0$ and $\frac{a}{b} < \frac{c}{d}$).

1. Find a common denominator: Find the LCM of $b$ and $d$, let it be $D$. Convert $\frac{a}{b}$ to $\frac{a'}{D}$ and $\frac{c}{d}$ to $\frac{c'}{D}$.
2. Create space between numerators: If there are already enough integers between $a'$ and $c'$ (i.e., $c' - a' > n$, where $n$ is the number of rationals you want to find), you can use $\frac{a'+1}{D}, \frac{a'+2}{D}, ..., \frac{a'+n}{D}$ as rational numbers between them.
3. If there are not enough integers between $a'$ and $c'$ (i.e., $c' - a' \le n$), multiply the numerator and denominator of both equivalent fractions $\frac{a'}{D}$ and $\frac{c'}{D}$ by a suitable integer $k$. A simple choice for $k$ is $n+1$, or you can choose any integer large enough, e.g., 10, 100, etc., to ensure there are at least $n$ integers between the new numerators $a'k$ and $c'k$. The new equivalent fractions are $\frac{a'k}{Dk}$ and $\frac{c'k}{Dk}$.
4. Choose intermediate numerators: The rational numbers between $\frac{a'k}{Dk}$ and $\frac{c'k}{Dk}$ are fractions with denominator $Dk$ and numerators that are integers strictly between $a'k$ and $c'k$. You can pick any $n$ such integers, say $N_1, N_2, ..., N_n$, such that $a'k < N_1 < N_2 < ... < N_n < c'k$. The corresponding rational numbers are $\frac{N_1}{Dk}, \frac{N_2}{Dk}, ..., \frac{N_n}{Dk}$.

Answer:

Finding Irrational Numbers Between Two Given Numbers (Rational or Irrational)

The set of irrational numbers is also dense. Furthermore, between any two distinct real numbers (which could be two rationals, two irrationals, or one of each), there are infinitely many irrational numbers.

Between Two Rational Numbers ($r_1 < r_2$)

To find an irrational number between two rationals $r_1$ and $r_2$: One common strategy is to construct a non-terminating, non-repeating decimal whose value is strictly between $r_1$ and $r_2$. You can do this by starting with the decimal representation of $r_1$ and appending a non-repeating sequence of digits such that the resulting number remains less than $r_2$.

Answer:

Between Two Irrational Numbers ($i_1 < i_2$)

Between any two distinct irrational numbers, there are also infinitely many irrational numbers. Similar to the previous case, you can construct a non-terminating, non-repeating decimal that falls between their decimal expansions.

Answer:

Between Two Real Numbers (Including Rational and Irrational)

Between any two distinct real numbers, there are infinitely many rational numbers and infinitely many irrational numbers. The methods used above can be adapted.

Answer:

Answer:

The ability to find infinitely many rational and irrational numbers between any two distinct real numbers demonstrates the dense nature of both sets and the continuous nature of the real number line.

Comparing Very Large and Very Small Numbers

Very large and very small numbers are frequently encountered in various fields like science (e.g., distance to stars, size of atoms), engineering, economics (e.g., national debt, tiny tolerances in manufacturing), and technology. Writing these numbers in their full form (like $50,000,000,000$ or $0.0000000001$) is not only tedious but also makes it challenging to compare their magnitudes quickly and accurately.

Scientific notation (also known as standard form) is a widely accepted mathematical notation used to express such numbers concisely and to simplify calculations and comparisons.

Scientific Notation (Standard Form)

A number is written in scientific notation if it is expressed as the product of a number between $1$ and $10$ (including 1, but not 10) and an integer power of $10$.

The general form of a number in scientific notation is $a \times 10^n$, where:

• $a$ is the coefficient (or mantissa). It is a real number such that $1 \le |a| < 10$. This means the absolute value of $a$ is greater than or equal to 1 and strictly less than 10. The decimal point is placed after the first non-zero digit.
• $10^n$ is the power of 10.
• $n$ is the exponent. It is an integer (positive, negative, or zero) and indicates the order of magnitude of the number.

Converting a number to Scientific Notation:

To write a number in scientific notation, move the decimal point until there is only one non-zero digit to the left of the decimal point. The number of places the decimal point was moved determines the absolute value of the exponent $n$.

• If the original number is large (greater than 10) and the decimal point is moved to the left, the exponent $n$ is positive. The number of places moved is the value of $n$.

  Example: Speed of light $\approx 300,000,000 \text{ m/s}$. Move decimal 8 places left: $3.0 \times 10^8$. Here $a=3.0$ (which is $1 \le 3.0 < 10$) and $n=8$.

  Example: Population of India $\approx 1,400,000,000$. Move decimal 9 places left: $1.4 \times 10^9$. Here $a=1.4$ and $n=9$.

• If the original number is small (between 0 and 1) and the decimal point is moved to the right, the exponent $n$ is negative. The number of places moved is the absolute value of $n$.

  Example: Mass of an electron $\approx 0.0000000000000000000000000009109 \text{ kg}$. Move decimal 31 places right: $9.109 \times 10^{-31}$. Here $a=9.109$ and $n=-31$.

  Example: Diameter of a virus $\approx 0.000000025 \text{ m}$. Move decimal 8 places right: $2.5 \times 10^{-8}$. Here $a=2.5$ and $n=-8$.

• If the number is already between $1$ and $10$ (not including 10), or is $0$, the exponent $n$ is $0$.

  Example: $7.8 = 7.8 \times 10^0$.

  Example: $4.005 = 4.005 \times 10^0$.

  Note: The number 0 is sometimes written as $0 \times 10^n$ for any $n$, but it doesn't fit the $1 \le |a| < 10$ rule. For non-zero numbers, the coefficient $a$ must be non-zero.

Comparing Numbers in Scientific Notation

Comparing two numbers written in scientific notation, $a \times 10^n$ and $b \times 10^m$ (where $1 \le |a| < 10$ and $1 \le |b| < 10$), is straightforward. The comparison is primarily determined by the exponents of $10$, and secondarily by the coefficients if the exponents are the same.

1. Compare the Exponents ($n$ and $m$): This is the most important step. The number with the larger exponent is the greater number.
   • If $n > m$, then $a \times 10^n$ is greater than $b \times 10^m$.

     If $n > m$, then $a \times 10^n > b \times 10^m$.

     Example: $5 \times 10^8$ vs $9 \times 10^6$. Exponent 8 is greater than exponent 6. So $5 \times 10^8 > 9 \times 10^6$.

     Example: $2 \times 10^{-4}$ vs $8 \times 10^{-5}$. Exponent -4 is greater than exponent -5 (since -4 is to the right of -5 on the number line). So $2 \times 10^{-4} > 8 \times 10^{-5}$.

2. Compare the Coefficients ($a$ and $b$) if Exponents are Equal ($n = m$): If the exponents are the same, the numbers have the same order of magnitude. The comparison then depends on the coefficients.
   • If $n = m$, then compare the coefficients $a$ and $b$ using standard decimal comparison rules. The number with the larger coefficient is the greater number.

     If $n = m$ and $a > b$, then $a \times 10^n > b \times 10^m$.

     If $n = m$ and $a < b$, then $a \times 10^n < b \times 10^m$.

     If $n = m$ and $a = b$, then $a \times 10^n = b \times 10^m$.

     Example: $4.5 \times 10^7$ vs $3.2 \times 10^7$. Exponents are both 7. Compare 4.5 and 3.2. Since $4.5 > 3.2$, $4.5 \times 10^7 > 3.2 \times 10^7$.

     Example: $6.1 \times 10^{-9}$ vs $8.0 \times 10^{-9}$. Exponents are both -9. Compare 6.1 and 8.0. Since $6.1 < 8.0$, $6.1 \times 10^{-9} < 8.0 \times 10^{-9}$.

Examples of Comparing Numbers in Scientific Notation

Answer:

Answer:

Answer:

Scientific notation simplifies working with numbers across a vast range of scales and makes comparing their magnitudes intuitive and efficient.