Arithmetic Operations on Numbers

Fundamental Operations (Addition, Subtraction, Multiplication, Division) on Whole Numbers

Arithmetic operations are the core of numerical computations. They are the procedures used to combine numbers to produce a new value. The four fundamental arithmetic operations are Addition, Subtraction, Multiplication, and Division. We begin our study of these operations with the set of Whole Numbers ($\mathbb{W} = \{0, 1, 2, 3, ...\}$), which includes zero and the natural numbers.

Understanding how these operations behave within the set of whole numbers lays the groundwork for working with larger sets of numbers.

1. Addition (+) on Whole Numbers

Addition is the process of combining two or more numbers to find their total, called the sum. It can be visualized as bringing collections of objects together or moving to the right on the number line.

Properties of Addition on Whole Numbers: Let $a, b,$ and $c$ be any whole numbers.

• Closure Property: The sum of any two whole numbers is always a unique whole number.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a + b \in \mathbb{W}$.

  (Closure under Addition)

  Example: $5 \in \mathbb{W}, 8 \in \mathbb{W}$, then $5 + 8 = 13 \in \mathbb{W}$. $0 \in \mathbb{W}, 7 \in \mathbb{W}$, then $0 + 7 = 7 \in \mathbb{W}$.

• Commutative Property: The order in which two whole numbers are added does not affect the sum.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a + b = b + a$.

  (Commutativity of Addition)

  Example: $10 + 4 = 14$, and $4 + 10 = 14$. So, $10 + 4 = 4 + 10$.

• Associative Property: The grouping of three or more whole numbers when adding does not affect the sum.

  If $a, b, c \in \mathbb{W}$, then $(a + b) + c = a + (b + c)$.

  (Associativity of Addition)

  Example: $(2 + 3) + 5 = 5 + 5 = 10$. $2 + (3 + 5) = 2 + 8 = 10$. So, $(2 + 3) + 5 = 2 + (3 + 5)$.

• Additive Identity: Zero (0) is the additive identity for whole numbers. Adding 0 to any whole number results in the original number.

  For any $a \in \mathbb{W}$, $a + 0 = 0 + a = a$.

  (Additive Identity Property)

  Example: $17 + 0 = 17$, $0 + 42 = 42$.

Performing Addition: Addition of larger numbers is typically performed using column addition, aligning digits by their place value.

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2. Subtraction (-) on Whole Numbers

Subtraction is the inverse operation of addition. It is used to find the difference between two numbers or to determine how much remains when a quantity is taken away from another. For whole numbers, subtraction is only defined when the number being subtracted (the subtrahend) is less than or equal to the number from which it is being subtracted (the minuend).

Properties of Subtraction on Whole Numbers: Let $a$ and $b$ be whole numbers.

• Closure Property: Subtraction of whole numbers is not closed. The result of subtracting one whole number from another is not always a whole number.

  Example: $3 \in \mathbb{W}, 5 \in \mathbb{W}$, but $3 - 5 = -2$, which is an integer but not a whole number. $5 - 5 = 0$, which is a whole number, but not all differences are whole numbers.

• Commutative Property: Subtraction is not commutative for whole numbers. The order matters.

  Example: $5 - 3 = 2$, but $3 - 5 = -2$. Since $2 \neq -2$, $5 - 3 \neq 3 - 5$.

• Associative Property: Subtraction is not associative for whole numbers. The grouping matters.

  Example: Let $a=10, b=5, c=2$. $(10 - 5) - 2 = 5 - 2 = 3$. $10 - (5 - 2) = 10 - 3 = 7$. Since $3 \neq 7$, $(10 - 5) - 2 \neq 10 - (5 - 2)$.

• Subtraction from itself: Subtracting a whole number from itself results in 0.

  For any $a \in \mathbb{W}$, $a - a = 0$.

• Subtraction of zero: Subtracting 0 from a whole number results in the original number.

  For any $a \in \mathbb{W}$, $a - 0 = a$.

Performing Subtraction: Subtraction of larger numbers is typically performed using column subtraction, aligning digits by place value and using borrowing (or regrouping) when a digit is smaller than the digit below it.

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3. Multiplication ($\times$ or $\cdot$) on Whole Numbers

Multiplication is often introduced as repeated addition. For example, $3 \times 4$ means adding 3 to itself 4 times ($3 + 3 + 3 + 3 = 12$), or adding 4 to itself 3 times ($4 + 4 + 4 = 12$). The result of multiplication is called the product.

Properties of Multiplication on Whole Numbers: Let $a, b,$ and $c$ be any whole numbers.

• Closure Property: The product of any two whole numbers is always a unique whole number.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a \times b \in \mathbb{W}$.

  (Closure under Multiplication)

  Example: $6 \in \mathbb{W}, 7 \in \mathbb{W}$, then $6 \times 7 = 42 \in \mathbb{W}$. $0 \in \mathbb{W}, 15 \in \mathbb{W}$, then $0 \times 15 = 0 \in \mathbb{W}$.

• Commutative Property: The order in which two whole numbers are multiplied does not affect the product.

  If $a \in \mathbb{W}$ and $b \in \mathbb{W}$, then $a \times b = b \times a$.

  (Commutativity of Multiplication)

  Example: $5 \times 8 = 40$, and $8 \times 5 = 40$. So, $5 \times 8 = 8 \times 5$.

• Associative Property: The grouping of three or more whole numbers when multiplying does not affect the product.

  If $a, b, c \in \mathbb{W}$, then $(a \times b) \times c = a \times (b \times c)$.

  (Associativity of Multiplication)

  Example: $(3 \times 4) \times 2 = 12 \times 2 = 24$. $3 \times (4 \times 2) = 3 \times 8 = 24$. So, $(3 \times 4) \times 2 = 3 \times (4 \times 2)$.

• Multiplicative Identity: One (1) is the multiplicative identity for whole numbers. Multiplying any whole number by 1 results in the original number.

  For any $a \in \mathbb{W}$, $a \times 1 = 1 \times a = a$.

  (Multiplicative Identity Property)

  Example: $25 \times 1 = 25$, $1 \times 98 = 98$.

• Property of Zero: Multiplying any whole number by zero results in zero.

  For any $a \in \mathbb{W}$, $a \times 0 = 0 \times a = 0$.

  (Multiplication by Zero Property)

  Example: $50 \times 0 = 0$, $0 \times 123 = 0$.

• Distributive Property: Multiplication distributes over addition and subtraction.

  If $a, b, c \in \mathbb{W}$, then $a \times (b + c) = (a \times b) + (a \times c)$.

  (Distributivity over Addition)

  If $a, b, c \in \mathbb{W}$ and $b \ge c$, then $a \times (b - c) = (a \times b) - (a \times c)$.

  (Distributivity over Subtraction)

  Example: $5 \times (10 + 2) = 5 \times 12 = 60$. Also, $(5 \times 10) + (5 \times 2) = 50 + 10 = 60$.

  Example: $5 \times (10 - 2) = 5 \times 8 = 40$. Also, $(5 \times 10) - (5 \times 2) = 50 - 10 = 40$.

Performing Multiplication: For multiplying numbers with more than one digit, long multiplication based on place value is used.

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4. Division ($\div$ or /) on Whole Numbers

Division is the inverse operation of multiplication. It is used to share a quantity into equal parts (partitioning) or to find out how many times one quantity is contained within another (quotitioning). For example, $12 \div 4$ can mean sharing 12 into 4 equal groups (each has 3) or finding how many groups of 4 are in 12 (there are 3 groups).

Rules: Division of whole numbers is not closed, not commutative, and not associative. Division by zero is undefined.

• Closure Property: Division of whole numbers is not closed. The result of dividing one whole number by another (non-zero) whole number is not always a whole number.

  Example: $5 \in \mathbb{W}, 2 \in \mathbb{W}$, but $5 \div 2 = 2.5$, which is a rational number, not a whole number.

• Commutative Property: Division is not commutative for whole numbers.

  Example: $10 \div 2 = 5$, but $2 \div 10 = 0.2$. Since $5 \neq 0.2$, $10 \div 2 \neq 2 \div 10$.

• Associative Property: Division is not associative for whole numbers.

  Example: Let $a=24, b=6, c=2$. $(24 \div 6) \div 2 = 4 \div 2 = 2$. $24 \div (6 \div 2) = 24 \div 3 = 8$. Since $2 \neq 8$, $(24 \div 6) \div 2 \neq 24 \div (6 \div 2)$.

• Division by Zero: Division by zero is undefined. For any whole number $a$, the expression $a \div 0$ has no meaningful value. If we try to define $a \div 0 = q$, it would mean $a = q \times 0$. If $a \neq 0$, $a = 0$, which is false. If $a = 0$, $0 = q \times 0$, which is true for any value of $q$, meaning the quotient is not unique. Thus, division by zero is strictly avoided.
• Division by One: Dividing any whole number by 1 results in the number itself.

  For any $a \in \mathbb{W}$, $a \div 1 = a$.

• Zero Divided by a Non-Zero Number: Dividing zero by any non-zero whole number results in zero.

  For any $b \in \mathbb{W}$, $b \neq 0$, $0 \div b = 0$.

When a whole number $a$ is divided by a non-zero whole number $b$, we use the division algorithm, which states that there exist unique whole numbers, a quotient $q$ and a remainder $r$, such that:

If the remainder $r$ is $0$, the division is exact, and $b$ is a factor of $a$. The quotient $q = a \div b$.

Performing Division: For dividing larger numbers, long division is a standard method.

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Operations on Integers

The set of Integers ($\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$) includes all whole numbers and their negative counterparts. When performing arithmetic operations on integers, we need to consider the sign of each number, as the rules for operations involving positive and negative numbers differ from operations solely involving whole numbers.

1. Addition (+) of Integers

Adding integers depends on whether the numbers have the same sign or different signs.

• Adding Integers with the Same Sign:

  If two integers have the same sign (both positive or both negative), add their absolute values. The sum will have the same sign as the original integers.

  • (Positive) + (Positive) = Positive Sum. $|+a| + |+b| = a + b$. Result is $+(a+b)$.
    Example: $5 + 3 = 8$.
  • (Negative) + (Negative) = Negative Sum. $|-a| + |-b| = a + b$. Result is $-(a+b)$.
    Example: $(-5) + (-3)$. Absolute values are $|-5|=5$ and $|-3|=3$. Add them: $5+3=8$. Both numbers are negative, so the sum is negative. $(-5) + (-3) = -8$.

• Adding Integers with Different Signs:

  If two integers have different signs (one positive and one negative), subtract the smaller absolute value from the larger absolute value. The sum will have the sign of the integer with the larger absolute value.

  • Example: $7 + (-4)$. The integers are 7 (positive) and -4 (negative). Absolute values are $|7|=7$ and $|-4|=4$. The larger absolute value is 7 (from 7). Subtract the smaller from the larger absolute value: $7 - 4 = 3$. The sign of the result is the sign of the number with the larger absolute value (7 is positive), so the sum is positive. $7 + (-4) = 3$.
  • Example: $(-7) + 4$. The integers are -7 (negative) and 4 (positive). Absolute values are $|-7|=7$ and $|4|=4$. The larger absolute value is 7 (from -7). Subtract the smaller from the larger absolute value: $7 - 4 = 3$. The sign of the result is the sign of the number with the larger absolute value (-7 is negative), so the sum is negative. $(-7) + 4 = -3$.
  • Example: $4 + (-7)$. The integers are 4 (positive) and -7 (negative). Absolute values are $|4|=4$ and $|-7|=7$. The larger absolute value is 7 (from -7). Subtract the smaller from the larger absolute value: $7 - 4 = 3$. The sign of the result is the sign of the number with the larger absolute value (-7 is negative), so the sum is negative. $4 + (-7) = -3$.
• Adding Zero: Adding zero to any integer results in the integer itself.

  Example: $10 + 0 = 10$, $-15 + 0 = -15$, $0 + (-8) = -8$.

Properties of Addition on Integers: Addition of integers satisfies the same properties as for whole numbers, plus the existence of additive inverses:

• Closed: $a + b \in \mathbb{Z}$ for all $a, b \in \mathbb{Z}$.
• Commutative: $a + b = b + a$ for all $a, b \in \mathbb{Z}$.
• Associative: $(a + b) + c = a + (b + c)$ for all $a, b, c \in \mathbb{Z}$.
• Additive Identity: $a + 0 = a$ for any $a \in \mathbb{Z}$.
• Additive Inverse: For every integer $a$, there exists a unique integer $-a$ (its opposite) such that their sum is the additive identity (0).

  For any $a \in \mathbb{Z}$, there exists $-a \in \mathbb{Z}$ such that $a + (-a) = (-a) + a = 0$.

  [Additive Inverse Property]

  Example: The additive inverse of 5 is -5, because $5 + (-5) = 0$. The additive inverse of -12 is 12, because $-12 + 12 = 0$. The additive inverse of 0 is 0.

Visualizing Addition on Number Line: Starting from the position of the first number, move to the right for adding a positive number, and move to the left for adding a negative number. The distance moved is equal to the absolute value of the second number.

(Note: The image shows a number line. An arrow starts at 0 and goes to 3 (representing the first number). From 3, another arrow points left 5 units (representing adding -5). The second arrow ends at -2, showing that $3 + (-5) = -2$.)

2. Subtraction (-) of Integers

Subtraction of integers is defined as adding the additive inverse (opposite) of the number being subtracted.

To subtract an integer $b$ from an integer $a$, change the sign of $b$ to its opposite (its additive inverse, $-b$), and then add $a$ and $-b$ following the rules for integer addition.

Examples:

• $5 - 3$: This is $5 + (-3)$. Different signs, subtract absolute values $|5|-|3|=5-3=2$. Sign of larger absolute value (5 is positive) is positive. Result: $2$.
• $3 - 5$: This is $3 + (-5)$. Different signs, subtract absolute values $|-5|-|3|=5-3=2$. Sign of larger absolute value (-5 is negative) is negative. Result: $-2$.
• $5 - (-3)$: This is $5 + (-(-3))$. The opposite of -3 is 3. So, $5 + 3$. Same signs, add absolute values $|5|+|3|=5+3=8$. Sign is positive. Result: $8$. (Subtracting a negative is the same as adding a positive).
• $(-5) - 3$: This is $(-5) + (-3)$. Same signs, add absolute values $|-5|+|-3|=5+3=8$. Sign is negative. Result: $-8$.
• $(-5) - (-3)$: This is $(-5) + (-(-3))$. The opposite of -3 is 3. So, $(-5) + 3$. Different signs, subtract absolute values $|-5|-|3|=5-3=2$. Sign of larger absolute value (-5 is negative) is negative. Result: $-2$.

Properties of Subtraction on Integers:

• Closure Property: Subtraction of integers is closed. The difference between any two integers is always a unique integer.

  If $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, then $a - b \in \mathbb{Z}$.

  (Closure under Subtraction)

  Example: $4 \in \mathbb{Z}, -7 \in \mathbb{Z}$, then $4 - (-7) = 4 + 7 = 11 \in \mathbb{Z}$.

• Commutative Property: Subtraction is not commutative for integers.

  Example: $5 - 3 = 2$, but $3 - 5 = -2$.

• Associative Property: Subtraction is not associative for integers.

  Example: $(10 - 5) - 2 = 5 - 2 = 3$, but $10 - (5 - 2) = 10 - 3 = 7$.

Visualizing Subtraction on Number Line: Using the rule $a - b = a + (-b)$, subtraction is visualized as addition of the inverse. Starting from the position of the first number, move to the left for subtracting a positive number (adding a negative), and move to the right for subtracting a negative number (adding a positive). The distance moved is the absolute value of the number being subtracted.

(Note: The image shows a number line. An arrow starts at 0 and goes to 3. From 3, an arrow points right 5 units (representing subtracting -5, which is adding 5). The second arrow ends at 8, showing that $3 - (-5) = 3 + 5 = 8$).

3. Multiplication ($\times$ or $\cdot$) of Integers

Multiplying integers involves considering the signs of the numbers being multiplied. The rule for determining the sign of the product is critical.

Rules for Signs in Multiplication: For any positive integers $a$ and $b$:

• (Positive) $\times$ (Positive) = Positive. $(+a) \times (+b) = +(a \times b)$.
  Example: $3 \times 5 = 15$.
• (Positive) $\times$ (Negative) = Negative. $(+a) \times (-b) = -(a \times b)$.
  Example: $3 \times (-5) = -(3 \times 5) = -15$.
• (Negative) $\times$ (Positive) = Negative. $(-a) \times (+b) = -(a \times b)$.
  Example: $(-3) \times 5 = -(3 \times 5) = -15$.
• (Negative) $\times$ (Negative) = Positive. $(-a) \times (-b) = +(a \times b)$.
  Example: $(-3) \times (-5) = +(3 \times 5) = 15$.

Summary of Sign Rules:

• If the two integers have the same sign, their product is positive.
• If the two integers have different signs, their product is negative.

The magnitude of the product is found by multiplying the absolute values of the two integers: $|a \times b| = |a| \times |b|$.

Properties of Multiplication on Integers: Multiplication of integers shares the same properties as for whole numbers:

• Closed: $a \times b \in \mathbb{Z}$ for all $a, b \in \mathbb{Z}$.
• Commutative: $a \times b = b \times a$ for all $a, b \in \mathbb{Z}$.
• Associative: $(a \times b) \times c = a \times (b \times c)$ for all $a, b, c \in \mathbb{Z}$.
• Multiplicative Identity: $a \times 1 = a$ for any $a \in \mathbb{Z}$.
• Property of Zero: $a \times 0 = 0$ for any $a \in \mathbb{Z}$.
• Distributive over Addition: $a \times (b + c) = (a \times b) + (a \times c)$ for all $a, b, c \in \mathbb{Z}$.
• Distributive over Subtraction: $a \times (b - c) = (a \times b) - (a \times c)$ for all $a, b, c \in \mathbb{Z}$.

4. Division ($\div$ or /) of Integers

Division is the inverse operation of multiplication. For any integers $a$ and $b$ (with $b \neq 0$), $a \div b = q$ if and only if $a = b \times q$. The result of division is called the quotient.

The rules for the sign of the quotient are the same as the rules for the sign of the product (when the divisor is non-zero).

Rules for Signs in Division: For any integers $a$ and $b$, with $b \neq 0$:

• (Positive) $\div$ (Positive) = Positive. $(+a) \div (+b) = +(a \div b)$.
  Example: $15 \div 3 = 5$.
• (Positive) $\div$ (Negative) = Negative. $(+a) \div (-b) = -(a \div b)$.
  Example: $15 \div (-3) = -(15 \div 3) = -5$.
• (Negative) $\div$ (Positive) = Negative. $(-a) \div (+b) = -(a \div b)$.
  Example: $(-15) \div 3 = -(15 \div 3) = -5$.
• (Negative) $\div$ (Negative) = Positive. $(-a) \div (-b) = +(a \div b)$.
  Example: $(-15) \div (-3) = +(15 \div 3) = 5$.

Summary of Sign Rules:

• If the two integers have the same sign, their quotient is positive.
• If the two integers have different signs, their quotient is negative.

The magnitude of the quotient is found by dividing the absolute values: $|\frac{a}{b}| = \frac{|a|}{|b|}$.

Division by Zero: As with whole numbers, division by zero is undefined for integers.

Properties of Division on Integers:

• Closure Property: Division of integers is not closed. The result of dividing one integer by another (non-zero) integer is not always an integer.

  Example: $3 \div 2 = 1.5$, which is not an integer. $-7 \div 4 = -1.75$, not an integer. This lack of closure is why we extend the number system to rational numbers.

• Commutative Property: Division is not commutative for integers.
• Associative Property: Division is not associative for integers.

When integer division is not exact (i.e., the remainder is not 0), the result is not an integer. In such cases, the result is expressed as a rational number (a fraction) or a decimal.

Summary of Operations on Integers

Mastering these rules and properties is fundamental for working with rational and real numbers, as they build upon the operations defined for integers.

Operations on Fractions

Fractions are numbers that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is a non-zero integer ($q \neq 0$). Performing the fundamental arithmetic operations (addition, subtraction, multiplication, and division) with fractions requires specific rules that involve their numerators and denominators.

Let $\frac{a}{b}$ and $\frac{c}{d}$ be two fractions, where $a, b, c, d \in \mathbb{Z}$ and $b \neq 0, d \neq 0$.

1. Addition (+) of Fractions

Adding fractions depends on whether they have the same denominator or different denominators.

• Adding fractions with the same denominator:

  If two fractions have the same denominator, add their numerators and keep the common denominator. The common denominator should be non-zero.

  $\quad \frac{a}{d} + \frac{b}{d} = \frac{a+b}{d} \quad (\text{where } d \neq 0)$

  Example 1: $\frac{2}{5} + \frac{3}{5} = \frac{2+3}{5} = \frac{5}{5} = 1$.

  Example 2: $\frac{1}{7} + \frac{4}{7} = \frac{1+4}{7} = \frac{5}{7}$.

  Example 3: $\frac{-3}{8} + \frac{5}{8} = \frac{-3+5}{8} = \frac{2}{8}$. Simplify the result: $\frac{\cancel{2}^1}{\cancel{8}_4} = \frac{1}{4}$.

  Example 4: $\frac{-1}{6} + \frac{-4}{6} = \frac{-1+(-4)}{6} = \frac{-5}{6}$.

• Adding fractions with different denominators:

  If two fractions have different denominators, find a common denominator for both fractions. Convert each fraction into an equivalent fraction with this common denominator. Then, add the equivalent fractions as described above.

  The most efficient common denominator is the Least Common Multiple (LCM) of the denominators.

  $\quad \frac{a}{b} + \frac{c}{d} = \frac{a \times k_1}{b \times k_1} + \frac{c \times k_2}{d \times k_2}$, where $b k_1 = d k_2 = \text{LCM}(b,d)$

  $\quad = \frac{a k_1 + c k_2}{\text{LCM}(b,d)}$

  Example 1: $\frac{1}{2} + \frac{1}{3}$. The denominators are 2 and 3. LCM$(2, 3) = 6$.

  Convert $\frac{1}{2}$ to sixths: $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.

  Convert $\frac{1}{3}$ to sixths: $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.

  Now add the equivalent fractions: $\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}$.

  Example 2: $\frac{3}{4} + \frac{-5}{6}$. Denominators are 4 and 6. LCM$(4, 6) = 12$.

  Convert $\frac{3}{4}$ to twelfths: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

  Convert $\frac{-5}{6}$ to twelfths: $\frac{-5 \times 2}{6 \times 2} = \frac{-10}{12}$.

  Add the equivalent fractions: $\frac{9}{12} + \frac{-10}{12} = \frac{9+(-10)}{12} = \frac{9-10}{12} = \frac{-1}{12} = -\frac{1}{12}$.

Adding Mixed Numbers: To add mixed numbers, convert them into improper fractions first, and then add the improper fractions using the rules above. Alternatively, you can add the whole number parts and the fractional parts separately, regrouping if the sum of the fractional parts is an improper fraction.

Example: Add $2\frac{1}{2}$ and $1\frac{3}{4}$.

Method 1 (Improper Fractions): Convert to improper fractions: $2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$. $1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{7}{4}$.

Find LCM of denominators (2 and 4): LCM$(2, 4) = 4$. Convert $\frac{5}{2}$ to fourths: $\frac{5 \times 2}{2 \times 2} = \frac{10}{4}$.

Add: $\frac{10}{4} + \frac{7}{4} = \frac{10+7}{4} = \frac{17}{4}$. Convert back to mixed number: $\frac{17}{4} = 4$ with remainder 1, so $4\frac{1}{4}$.

Method 2 (Separate Parts): Add whole numbers: $2+1=3$. Add fractions: $\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}$. Convert improper fraction sum to mixed: $\frac{5}{4} = 1\frac{1}{4}$. Add the whole number sum and the mixed fraction sum: $3 + 1\frac{1}{4} = 4\frac{1}{4}$.

2. Subtraction (-) of Fractions

Subtracting fractions follows similar steps to adding fractions, especially regarding common denominators.

• Subtracting fractions with the same denominator:

  If two fractions have the same denominator, subtract the numerator of the second fraction from the numerator of the first fraction, and keep the common denominator (non-zero).

  $\quad \frac{a}{d} - \frac{b}{d} = \frac{a-b}{d} \quad (\text{where } d \neq 0)$

  Example 1: $\frac{4}{7} - \frac{2}{7} = \frac{4-2}{7} = \frac{2}{7}$.

  Example 2: $\frac{2}{5} - \frac{4}{5} = \frac{2-4}{5} = \frac{-2}{5} = -\frac{2}{5}$.

  Example 3: $\frac{1}{3} - \frac{-1}{3} = \frac{1 - (-1)}{3} = \frac{1+1}{3} = \frac{2}{3}$.

• Subtracting fractions with different denominators:

  Find a common denominator. Convert the fractions to equivalent fractions with the common denominator, then subtract the numerators.

  $\quad \frac{a}{b} - \frac{c}{d} = \frac{a \times k_1 - c \times k_2}{\text{LCM}(b,d)}$

  Example: $\frac{3}{4} - \frac{1}{6}$. Denominators are 4 and 6. LCM$(4, 6) = 12$.

  Convert $\frac{3}{4}$ to twelfths: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

  Convert $\frac{1}{6}$ to twelfths: $\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$.

  Subtract the equivalent fractions: $\frac{9}{12} - \frac{2}{12} = \frac{9-2}{12} = \frac{7}{12}$.

Subtracting Mixed Numbers: Convert mixed numbers to improper fractions first, then subtract. Or, subtract whole numbers and fractional parts separately, borrowing from the whole number part if the first fraction is smaller than the second fraction.

Example: Subtract $1\frac{3}{4}$ from $2\frac{1}{2}$.

Method 1 (Improper Fractions): Convert to improper fractions: $2\frac{1}{2} = \frac{5}{2}$, $1\frac{3}{4} = \frac{7}{4}$.

Find LCM of denominators (2 and 4): LCM$(2, 4) = 4$. Convert $\frac{5}{2}$ to fourths: $\frac{5 \times 2}{2 \times 2} = \frac{10}{4}$.

Subtract: $\frac{10}{4} - \frac{7}{4} = \frac{10-7}{4} = \frac{3}{4}$.

Method 2 (Separate Parts with Borrowing): $2\frac{1}{2} - 1\frac{3}{4}$. We can't subtract $\frac{3}{4}$ from $\frac{1}{2}$. Borrow 1 whole from 2: $2\frac{1}{2} = 1 + 1\frac{1}{2} = 1 + \frac{3}{2} = 1 + \frac{6}{4}$. So $2\frac{1}{2} = 1\frac{6}{4}$.

Now subtract: $1\frac{6}{4} - 1\frac{3}{4}$. Subtract whole numbers: $1-1=0$. Subtract fractions: $\frac{6}{4} - \frac{3}{4} = \frac{6-3}{4} = \frac{3}{4}$. Result: $0 + \frac{3}{4} = \frac{3}{4}$.

3. Multiplication ($\times$ or $\cdot$) of Fractions

Multiplying fractions is generally simpler than adding or subtracting them because it does not require finding a common denominator.

Rule: To multiply two fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.

It is often beneficial to simplify the fractions before multiplying by canceling out any common factors between any numerator and any denominator (known as cross-cancellation).

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Multiplying Mixed Numbers: Always convert mixed numbers to improper fractions before multiplying. Then, multiply the improper fractions using the rule.

Example: Multiply $1\frac{1}{2}$ by $2\frac{1}{3}$.

Convert to improper fractions: $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$. $2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}$.

Multiply the improper fractions: $\frac{3}{2} \times \frac{7}{3}$. Cancel the common factor 3:

The result $\frac{7}{2}$ can be left as an improper fraction or converted to a mixed number: $3\frac{1}{2}$.

4. Division ($\div$ or /) of Fractions

Dividing by a fraction is equivalent to multiplying the first fraction by the reciprocal of the second fraction (the divisor).

The reciprocal of a non-zero fraction $\frac{c}{d}$ is obtained by swapping its numerator and denominator, resulting in $\frac{d}{c}$. Note that the original fraction must be non-zero for its reciprocal to exist (i.e., $c \neq 0$).

Rule: To divide a fraction $\frac{a}{b}$ by a non-zero fraction $\frac{c}{d}$, multiply $\frac{a}{b}$ by the reciprocal of $\frac{c}{d}$.

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Dividing Mixed Numbers: Always convert mixed numbers to improper fractions before dividing. Then, divide the improper fractions using the rule (multiply by the reciprocal of the divisor).

Example: Divide $3\frac{1}{2}$ by $1\frac{1}{4}$.

Convert to improper fractions: $3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2}$. $1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{5}{4}$.

Divide the improper fractions: $\frac{7}{2} \div \frac{5}{4}$. The reciprocal of $\frac{5}{4}$ is $\frac{4}{5}$.

Multiply: $\frac{7}{2} \times \frac{4}{5}$. Cancel the common factor 2:

The result $\frac{14}{5}$ can be converted to a mixed number: $2\frac{4}{5}$.

Summary of Operations on Fractions

These rules allow us to perform all four basic arithmetic operations within the set of rational numbers (since integers and fractions cover all rational numbers), maintaining closure under addition, subtraction, and multiplication, and closure under division by any non-zero rational number.

Operations on Decimal Numbers

Decimal numbers are a convenient way to represent numbers, particularly rational numbers whose denominator is a power of 10. Operations on decimals are extensions of operations on whole numbers, requiring careful handling of the decimal point based on its place value significance.

1. Addition (+) of Decimal Numbers

To add decimal numbers, the key is to ensure that digits of the same place value are added together. This is achieved by aligning the decimal points vertically.

Steps for Adding Decimals:

1. Write the numbers vertically, aligning the decimal points.
2. Add trailing zeros to the end of any number(s) so that all numbers have the same number of decimal places. This helps align the place values visually, but is not strictly necessary mathematically.
3. Add the numbers column by column from right to left, just as you would with whole numbers. Carry over digits when a sum in a column is 10 or greater.
4. Place the decimal point in the sum directly below the aligned decimal points in the numbers being added.

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2. Subtraction (-) of Decimal Numbers

Subtracting decimal numbers also requires aligning the decimal points to ensure subtraction of digits from corresponding place values.

Steps for Subtracting Decimals:

1. Write the numbers vertically, aligning the decimal points. Place the number being subtracted (subtrahend) below the number it is being subtracted from (minuend).
2. Add trailing zeros to the minuend so that it has at least as many decimal places as the subtrahend. This is essential for correct borrowing.
3. Subtract the numbers column by column from right to left, just as you would with whole numbers. Use borrowing (regrouping) when a digit in the minuend is smaller than the corresponding digit in the subtrahend.
4. Place the decimal point in the difference directly below the aligned decimal points.

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3. Multiplication ($\times$ or $\cdot$) of Decimal Numbers

Multiplying decimal numbers involves two main steps: performing the multiplication as if they were whole numbers and then correctly placing the decimal point in the product.

Steps for Multiplying Decimals:

1. Multiply the numbers as if they were whole numbers, ignoring the decimal points for the moment.
2. Count the total number of decimal places in both of the original numbers being multiplied. This is the sum of the number of digits to the right of the decimal point in each factor.
3. Place the decimal point in the product obtained in Step 1 such that there are exactly that many decimal places (the total count from Step 2), counting from the rightmost digit of the product. Add leading zeros if necessary to place the decimal point correctly.

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4. Division ($\div$ or /) of Decimal Numbers

Dividing decimal numbers is often easiest when the divisor (the number you are dividing by) is a whole number. We can achieve this by multiplying both the divisor and the dividend (the number being divided) by the same power of 10. This does not change the value of the quotient.

Steps for Dividing Decimals:

1. If the divisor is a decimal, move the decimal point in the divisor to the right until it becomes a whole number. Count the number of places moved.
2. Move the decimal point in the dividend the same number of places to the right as you moved it in the divisor. Add trailing zeros to the dividend if necessary.
3. Perform the division as if you are dividing a whole number by a whole number.
4. Place the decimal point in the quotient directly above the new position of the decimal point in the dividend.
5. Continue the division process until you get a remainder of zero or reach the desired number of decimal places in the quotient.

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Operations on Rational Numbers

Rational numbers ($\mathbb{Q}$) encompass all numbers that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This includes integers, fractions, terminating decimals, and non-terminating repeating decimals. The arithmetic operations (addition, subtraction, multiplication, and division) can be performed on any pair of rational numbers by applying the rules for fractions or decimals.

The set of rational numbers, along with addition and multiplication, forms a mathematical structure known as a field, due to satisfying a specific set of properties.

Properties of Rational Numbers under Operations

Let $a, b,$ and $c$ be any three rational numbers.

• Closure Property: The set of rational numbers is closed under addition, subtraction, and multiplication. The result of performing these operations on any two rational numbers is always a unique rational number. It is also closed under division by any non-zero rational number.

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$, then $a+b \in \mathbb{Q}$, $a-b \in \mathbb{Q}$, and $a \times b \in \mathbb{Q}$.

  (Closure for Addition, Subtraction, Multiplication)

  If $a \in \mathbb{Q}$ and $b \in \mathbb{Q}$ with $b \neq 0$, then $a \div b \in \mathbb{Q}$.

  (Closure for Division by non-zero rational)

  Example: $\frac{1}{2} + \frac{1}{3} = \frac{5}{6} \in \mathbb{Q}$. $\frac{3}{4} - \frac{1}{5} = \frac{11}{20} \in \mathbb{Q}$. $-\frac{2}{3} \times \frac{1}{4} = -\frac{2}{12} = -\frac{1}{6} \in \mathbb{Q}$. $\frac{3}{5} \div \frac{2}{7} = \frac{21}{10} \in \mathbb{Q}$.

• Commutative Property: Addition and multiplication are commutative for rational numbers. The order of the operands does not affect the result.

  $\quad a + b = b + a$

  (Commutativity of Addition)

  $\quad a \times b = b \times a$

  (Commutativity of Multiplication)

  Example: $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$, and $\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$. $(-2) \times \frac{3}{4} = -\frac{6}{4} = -\frac{3}{2}$, and $\frac{3}{4} \times (-2) = -\frac{6}{4} = -\frac{3}{2}$.

• Associative Property: Addition and multiplication are associative for rational numbers. The grouping of operands does not affect the result.

  $\quad (a + b) + c = a + (b + c)$

  (Associativity of Addition)

  $\quad (a \times b) \times c = a \times (b \times c)$

  (Associativity of Multiplication)

  Example: $(\frac{1}{2} + \frac{1}{3}) + \frac{1}{4} = \frac{5}{6} + \frac{1}{4} = \frac{10+3}{12} = \frac{13}{12}$. $\frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) = \frac{1}{2} + \frac{7}{12} = \frac{6+7}{12} = \frac{13}{12}$.

  Example: $(\frac{1}{2} \times \frac{1}{3}) \times \frac{1}{4} = \frac{1}{6} \times \frac{1}{4} = \frac{1}{24}$. $\frac{1}{2} \times (\frac{1}{3} \times \frac{1}{4}) = \frac{1}{2} \times \frac{1}{12} = \frac{1}{24}$.

• Identity Property (Existence of Identity Element):
  • Additive Identity: Zero (0) is the additive identity for rational numbers. Adding 0 to any rational number results in the original rational number.

    For any $a \in \mathbb{Q}$, $a + 0 = 0 + a = a$.

    (Additive Identity Property)

    Since $0 = \frac{0}{1}$, $0$ is a rational number.

  • Multiplicative Identity: One (1) is the multiplicative identity for rational numbers. Multiplying any rational number by 1 results in the original rational number.

    For any $a \in \mathbb{Q}$, $a \times 1 = 1 \times a = a$.

    (Multiplicative Identity Property)

    Since $1 = \frac{1}{1}$, $1$ is a rational number.

• Inverse Property (Existence of Inverse Element):
  • Additive Inverse: For every rational number $a$, there exists a unique rational number $-a$ (its opposite) such that their sum is the additive identity (0).

    For any $a \in \mathbb{Q}$, there exists $-a \in \mathbb{Q}$ such that $a + (-a) = (-a) + a = 0$.

    (Additive Inverse Property)

    If $a = \frac{p}{q}$, then $-a = -\frac{p}{q} = \frac{-p}{q}$, which is also rational. Example: Additive inverse of $\frac{2}{3}$ is $-\frac{2}{3}$. Additive inverse of $-5$ is $5$.

  • Multiplicative Inverse (Reciprocal): For every non-zero rational number $a$, there exists a unique rational number $\frac{1}{a}$ (its reciprocal) such that their product is the multiplicative identity (1).

    For any $a \in \mathbb{Q}, a \neq 0$, there exists $\frac{1}{a} \in \mathbb{Q}$ such that $a \times \frac{1}{a} = \frac{1}{a} \times a = 1$.

    (Multiplicative Inverse Property)

    If $a = \frac{p}{q}$ and $a \neq 0$ (meaning $p \neq 0$), then its reciprocal is $\frac{q}{p}$, which is also rational (since $q \in \mathbb{Z}$ and $p \in \mathbb{Z}, p \neq 0$). Example: Multiplicative inverse of $\frac{2}{3}$ is $\frac{3}{2}$. Multiplicative inverse of $-5 = \frac{-5}{1}$ is $\frac{1}{-5} = -\frac{1}{5}$. The number 0 has no multiplicative inverse.

• Distributive Property: Multiplication of rational numbers distributes over both addition and subtraction.

  If $a, b, c \in \mathbb{Q}$, then $a \times (b + c) = (a \times b) + (a \times c)$.

  (Distributivity over Addition)

  If $a, b, c \in \mathbb{Q}$, then $a \times (b - c) = (a \times b) - (a \times c)$.

  (Distributivity over Subtraction)

  Example: $\frac{1}{2} \times (\frac{1}{3} + \frac{1}{4}) = \frac{1}{2} \times \frac{7}{12} = \frac{7}{24}$. Also $(\frac{1}{2} \times \frac{1}{3}) + (\frac{1}{2} \times \frac{1}{4}) = \frac{1}{6} + \frac{1}{8} = \frac{4+3}{24} = \frac{7}{24}$.

These properties confirm that rational numbers behave consistently under the basic operations, forming a solid algebraic structure.

Examples of Operations on Rational Numbers

To perform operations on rational numbers given in different forms (fractions, decimals), you can either convert all numbers to fractions and apply fraction rules, or convert all numbers to decimals and apply decimal rules.

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Operations on Real Numbers

The set of Real Numbers ($\mathbb{R}$) comprises both rational numbers ($\mathbb{Q}$) and irrational numbers ($\mathbb{I}$). Operations on real numbers build upon the rules for rational numbers and include methods for handling expressions involving irrational components. The set of real numbers, under the operations of addition and multiplication, forms a field, meaning it satisfies a comprehensive set of algebraic properties.

Properties of Real Numbers under Operations

For any real numbers $a, b,$ and $c$, the fundamental arithmetic operations satisfy the following properties:

• Closure Property: The sum, difference, and product of any two real numbers are always unique real numbers. The quotient of a real number divided by a non-zero real number is also a unique real number.

  If $a, b \in \mathbb{R}$, then $a+b \in \mathbb{R}$, $a-b \in \mathbb{R}$, and $a \times b \in \mathbb{R}$.

  (Closure for Addition, Subtraction, Multiplication)

  If $a, b \in \mathbb{R}$ with $b \neq 0$, then $a \div b \in \mathbb{R}$.

  (Closure for Division by non-zero real)

  Example: $\sqrt{2} + \sqrt{3} \in \mathbb{R}$. $\pi - 5 \in \mathbb{R}$. $\sqrt{2} \times \sqrt{2} = 2 \in \mathbb{R}$. $\pi \div 2 \in \mathbb{R}$.

• Commutative Property: Addition and multiplication are commutative for real numbers.

  $\quad a + b = b + a$

  (Commutativity of Addition)

  $\quad a \times b = b \times a$

  (Commutativity of Multiplication)

  Example: $\sqrt{2} + 5 = 5 + \sqrt{2}$. $\sqrt{3} \times \pi = \pi \times \sqrt{3}$.

• Associative Property: Addition and multiplication are associative for real numbers.

  $\quad (a + b) + c = a + (b + c)$

  (Associativity of Addition)

  $\quad (a \times b) \times c = a \times (b \times c)$

  (Associativity of Multiplication)

  Example: $(2 + \sqrt{3}) + 5 = 2 + (\sqrt{3} + 5)$. $(2 \times \sqrt{3}) \times 5 = 2 \times (\sqrt{3} \times 5)$.

• Identity Property:
  • Additive Identity: $0$ is the additive identity ($a+0=a$). $0 \in \mathbb{R}$.
  • Multiplicative Identity: $1$ is the multiplicative identity ($a \times 1 = a$). $1 \in \mathbb{R}$.
• Inverse Property:
  • Additive Inverse: For every $a \in \mathbb{R}$, there exists $-a \in \mathbb{R}$ such that $a + (-a) = 0$.
  • Multiplicative Inverse: For every $a \in \mathbb{R}, a \neq 0$, there exists $\frac{1}{a} \in \mathbb{R}$ such that $a \times \frac{1}{a} = 1$.

  Example: Additive inverse of $\sqrt{5}$ is $-\sqrt{5}$. Multiplicative inverse of $\sqrt{5}$ is $\frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$. Multiplicative inverse of $\pi$ is $\frac{1}{\pi}$.

• Distributive Property: Multiplication distributes over addition and subtraction.

  $\quad a \times (b + c) = (a \times b) + (a \times c)$

  (Distributivity over Addition)

  $\quad a \times (b - c) = (a \times b) - (a \times c)$

  (Distributivity over Subtraction)

  Example: $3 \times (\sqrt{2} + 5) = 3\sqrt{2} + 15$. $\sqrt{7} \times (10 - \sqrt{3}) = 10\sqrt{7} - \sqrt{21}$.

These properties are the foundation for algebraic manipulations involving real numbers.

Operations Involving Irrational Numbers

While the properties hold for all real numbers, the result of operations specifically involving irrational numbers might be either rational or irrational. This was highlighted in the "Broad Classification of Numbers" section.

• Sum/Difference: Rational $\pm$ Irrational is usually Irrational (unless adding additive inverses). Irrational $\pm$ Irrational can be Rational or Irrational.
• Product/Quotient: Non-zero Rational $\times$/$\div$ Irrational is always Irrational. Irrational $\times$/$\div$ Irrational can be Rational or Irrational (non-zero divisor).

When performing operations, treat irrational parts (especially square roots) somewhat like variables in algebra. Combine 'like' irrational terms (those with the same radical part). Use properties like $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ (for $a, b \ge 0$), $(\sqrt{a})^2 = a$, and the distributive property.

Examples of Operations on Real Numbers

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Rationalizing the Denominator

Rationalizing the denominator is the process of converting a fraction with an irrational number in the denominator into an equivalent fraction with a rational number in the denominator. This is often done to simplify expressions or to make calculations easier.

• If the denominator is a single term square root ($\sqrt{a}$): Multiply the numerator and denominator by $\sqrt{a}$.
• If the denominator is a binomial involving a square root ($a \pm \sqrt{b}$ or $\sqrt{a} \pm \sqrt{b}$): Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $x+y$ is $x-y$, and vice-versa. Multiplying a binomial by its conjugate uses the difference of squares formula: $(x+y)(x-y) = x^2 - y^2$, which removes the square root if $x$ or $y$ is a square root term.

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Order of Operations (BODMAS/PEMDAS)

When a mathematical expression involves more than one arithmetic operation, the sequence in which these operations are performed is crucial for obtaining a unique and correct result. To standardize calculations, a fixed order of operations has been established and is universally followed. This order is commonly memorized using acronyms like BODMAS or PEMDAS.

The BODMAS Rule

In India and many other parts of the world, the acronym BODMAS is used to remember the standard order of operations. Each letter stands for a type of operation:

• B - Brackets (or Parentheses): Operations inside grouping symbols must be performed first.
• O - Orders (or Exponents/Indices): Powers and roots (square roots, cube roots, etc.) are evaluated next.
• D - Division: Division operations.
• M - Multiplication: Multiplication operations.
• A - Addition: Addition operations.
• S - Subtraction: Subtraction operations.

The sequence of performing operations is as follows:

1. Evaluate any expressions contained within Brackets (Parentheses, Curly Braces $\{\}$, Square Brackets []). If there are brackets nested within other brackets, perform the operations in the innermost brackets first and work outwards.
2. Evaluate all Orders, which include exponents (powers like $2^3$) and roots (like $\sqrt{9}$).
3. Perform all Division and Multiplication operations. These two operations have the same level of priority. When both division and multiplication appear in an expression, perform them in the order they appear from left to right.
4. Perform all Addition and Subtraction operations. These two operations also have the same level of priority. When both addition and subtraction appear in an expression, perform them in the order they appear from left to right.

It's important to remember that Division and Multiplication are on the same level (left to right) and Addition and Subtraction are on the same level (left to right).

The PEMDAS Rule

Another widely used acronym for the order of operations is PEMDAS, which stands for:

• P - Parentheses
• E - Exponents
• M - Multiplication
• D - Division
• A - Addition
• S - Subtraction

PEMDAS is effectively the same rule as BODMAS, just using slightly different terminology (Parentheses instead of Brackets, Exponents instead of Orders/Indices) and listing Multiplication before Division (M/D) and Addition before Subtraction (A/S). The priority levels and left-to-right evaluation for M/D and A/S remain the same. In the Indian context, BODMAS is generally the preferred acronym.

Examples Applying BODMAS

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Consistently applying the order of operations (BODMAS/PEMDAS) is critical for accurately evaluating complex arithmetic expressions.

Simplification of Expressions involving various Number Types

Simplifying mathematical expressions that involve different types of numbers (such as integers, fractions, decimals, and irrational numbers like square roots) and multiple arithmetic operations requires a systematic approach. The key is to apply the rules for each specific operation correctly and strictly adhere to the established order of operations (BODMAS/PEMDAS).

While it's often helpful to convert all numbers within an expression to a single format (either fractions or decimals) before performing operations, sometimes the expression's structure or the nature of the numbers involved might make it simpler to work with mixed formats or keep irrational numbers in their symbolic form.

Steps for Simplifying Expressions

Here's a general approach to simplifying expressions with various number types:

1. Understand the Expression: Identify all the numbers and the operations involved. Note the grouping symbols (brackets).
2. Convert to a Common Format (Optional but Often Helpful): Decide whether to work with fractions, decimals, or a mix.
   • If the expression primarily contains fractions, converting decimals to fractions is usually a good idea.
   • If the expression primarily contains decimals or involves non-terminating repeating decimals where conversion to fraction is complex, working with decimals might be easier (you may need to round irrational numbers if an exact answer isn't possible).
   • Irrational numbers like $\sqrt{2}$ or $\pi$ can often be kept in their symbolic form until the final step, unless operations within brackets or orders require calculation involving their approximate values.
3. Apply the Order of Operations (BODMAS/PEMDAS): Proceed through the operations in the specified order:
   • B - Brackets (Parentheses): Simplify expressions within all grouping symbols, starting from the innermost ones.
   • O - Orders (Exponents/Roots): Evaluate powers, roots, and other operations indicated by superscripts or radical signs.
   • D & M - Division and Multiplication: Perform all division and multiplication operations in the order they appear from left to right.
   • A & S - Addition and Subtraction: Perform all addition and subtraction operations in the order they appear from left to right.
4. Perform Operations and Simplify: Execute the calculations according to the rules for the specific number types you are working with (fraction addition, decimal multiplication, integer subtraction, etc.). Simplify results at each stage where possible (e.g., reduce fractions to lowest terms).

Examples of Simplification

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Practicing with expressions involving different number types and operations is crucial for building fluency and accuracy in numerical applications.