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Rational Numbers

Properties of Rational Numbers

What is Additive Inverse?

The additive inverse of a rational number \(a\) is the number that, when added to \(a\), gives zero (the additive identity). For any rational number \(\frac{p}{q}\), its additive inverse is \(\frac{-p}{q}\).

  • Formula: \(a + (-a) = 0\)
  • Example: Additive inverse of \(\frac{3}{5}\) is \(\frac{-3}{5}\) because \(\frac{3}{5} + \frac{-3}{5} = 0\)

What is Multiplicative Inverse (Reciprocal)?

The multiplicative inverse of a non-zero rational number \(a\) is the number that, when multiplied by \(a\), gives 1 (the multiplicative identity). For \(\frac{p}{q} \neq 0\), its multiplicative inverse is \(\frac{q}{p}\).

  • Formula: \(a \times \frac{1}{a} = 1\) (where \(a \neq 0\))
  • Example: Multiplicative inverse of \(\frac{2}{3}\) is \(\frac{3}{2}\) because \(\frac{2}{3} \times \frac{3}{2} = 1\)

Real-life Connection: Think of additive inverse like walking forward 5 steps and then backward 5 steps to return to start. Multiplicative inverse is like doubling a recipe then halving it to get back to original!

Representation on Number Line

Rational numbers can be plotted on a number line - a straight line with numbers placed at equal intervals. Zero is at the center, positive numbers to the right, negative numbers to the left.

Steps to represent a rational number on number line:

  • Draw a horizontal line and mark 0 in the middle
  • Mark equal divisions (based on the denominator)
  • Locate the number based on numerator (positive = right, negative = left)
Operations on Rational NumbersAddition1/3 + 1/62/6 + 1/6= 1/2Subtraction3/4 − 1/42/4= 1/2Multiplication2/3 × 3/46/12= 1/2Division2/3 ÷ 4/52/3 × 5/4= 10/12Key: For + and −, find LCM of denominators first. For ÷, multiply by the reciprocal.
Example 1: What is the additive identity for rational numbers?
0, because adding 0 changes nothing.
Example 2: What is the multiplicative inverse of ⅔?
3/2, since ⅔ × 3/2 = 1.
Example 3: Example 1: Find the additive inverse and multiplicative inverse of \(\frac{-5}{7}\).
- Step 1: Additive inverse of \(\frac{p}{q}\) is \(\frac{-p}{q}\) - Step 2: Here \(p = -5\), so \(-p = -(-5) = 5\) - Step 3: Additive inverse = \(\frac{5}{7}\) - Step 4: Check: \(\frac{-5}{7} + \frac{5}{7} = 0\) ✓ - Step 5: Multiplicative inverse of \(\frac{p}{q}\) is \(\frac{q}{p}\) (where \(p \neq 0\)) - Step 6: Here \(\frac{p}{q} = \frac{-5}{7}\), so multiplicative inverse = \(\frac{7}{-5} = \frac{-7}{5}\) - Step 7: Check: \(\frac{-5}{7} \times \frac{-7}{5} = \frac{35}{35} = 1\) ✓
Example 4: Example 2: Represent \(\frac{5}{3}\) and \(\frac{-2}{3}\) on a number line.
- Step 1: Draw a horizontal line and mark 0 at the center - Step 2: \(\frac{5}{3}\) is positive, so it lies to the right of 0 - Step 3: \(\frac{5}{3} = 1\frac{2}{3}\), so it lies between 1 and 2 - Step 4: Divide the segment from 0 to 1 and 1 to 2 into 3 equal parts - Step 5: Count 5 parts from 0 to reach \(\frac{5}{3}\) - Step 6: For \(\frac{-2}{3}\), it is negative, so it lies to the left of 0 - Step 7: Divide the segment from 0 to -1 into 3 equal parts - Step 8: Count 2 parts left from 0 to reach \(\frac{-2}{3}\)
Example 5: Example 3: The sum of a rational number and its additive inverse is zero. If \(\frac{3}{4} + x = 0\), find \(x\). Also, if \(\frac{3}{4} \times y = 1\), find \(y\). Then plot \(x\), \(\frac{3}{4}\), and \(y\) on a number line.
- Step 1: For \(\frac{3}{4} + x = 0\), \(x\) is the additive inverse of \(\frac{3}{4}\) - Step 2: \(x = -\frac{3}{4}\) - Step 3: For \(\frac{3}{4} \times y = 1\), \(y\) is the multiplicative inverse of \(\frac{3}{4}\) - Step 4: \(y = \frac{4}{3} = 1\frac{1}{3}\) - Step 5: Now plot \(-\frac{3}{4}\), \(\frac{3}{4}\), and \(\frac{4}{3}\) on number line - Step 6: \(-\frac{3}{4}\) is negative, lies left of 0 between -1 and 0 - Step 7: \(\frac{3}{4}\) is positive, lies right of 0 between 0 and 1 - Step 8: \(\frac{4}{3}\) is positive, lies between 1 and 2
Quick recap
  • Commutative, associative for + and ×; distributive of × over +.
  • Additive inverse of a is −a; reciprocal of a/b is b/a.
  • Additive inverse of \(\frac{p}{q}\) is \(\frac{-p}{q}\); their sum is always 0
  • Multiplicative inverse of \(\frac{p}{q}\) (\(p \neq 0\)) is \(\frac{q}{p}\); their product is always 1
  • Zero has no multiplicative inverse (division by zero is undefined)
  • The additive identity is 0, and the multiplicative identity is 1
  • On a number line: positive numbers are to the right of 0, negatives to the left
  • To plot \(\frac{p}{q}\), divide unit length into \(q\) equal parts and count \(p\) parts from 0
  • Mixed fractions should be converted to improper fractions before plotting
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✓ Quick check
What is the multiplicative inverse of 5?
5 × 1/5 = 1, so the reciprocal is 1/5.
What is the additive inverse of −3/7?
−3/7 + 3/7 = 0, so the additive inverse is 3/7.

Rational Numbers on the Number Line

What are Rational Numbers?

A rational number is any number that can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). The word "rational" comes from the word "ratio".

Examples of rational numbers: \(\frac{1}{2}, \frac{-3}{4}, \frac{5}{1}, 0, -2, \frac{7}{-8}\)

What are Properties of Rational Numbers?

Properties are rules that always hold true when we perform operations (like addition, subtraction, multiplication, division) on rational numbers.

The Four Main Properties:

PropertyMeaningAddition ExampleMultiplication Example
Closure PropertyWhen you add/multiply two rational numbers, the result is also a rational number\(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) (rational)\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\) (rational)
Commutative PropertyChanging the order does not change the result\(\frac{1}{4} + \frac{3}{4} = \frac{3}{4} + \frac{1}{4}\)\(\frac{2}{5} \times \frac{1}{3} = \frac{1}{3} \times \frac{2}{5}\)
Associative PropertyChanging the grouping does not change the result\((\frac{1}{2}+\frac{1}{3})+\frac{1}{6} = \frac{1}{2}+(\frac{1}{3}+\frac{1}{6})\)\((\frac{1}{2}\times\frac{1}{3})\times\frac{1}{4} = \frac{1}{2}\times(\frac{1}{3}\times\frac{1}{4})\)
Distributive PropertyMultiplication distributes over addition\(a \times (b + c) = a \times b + a \times c\)-

Real-life Example: When sharing pizza among friends, if you have \(\frac{1}{2}\) pizza and get another \(\frac{1}{3}\) pizza, the total \(\frac{5}{6}\) pizza is also a rational number (closure property).

Important Notes:

  • Subtraction and division are NOT commutative ( \(5 - 3 \neq 3 - 5\) )
  • Subtraction and division are NOT associative ( \((8-4)-2 \neq 8-(4-2)\) )
  • The distributive property connects multiplication and addition beautifully!
The Number System — Where Rational Numbers FitReal Numbers ℝRational Numbers ℚIntegers ℤNon-Integers p/qNatural ℕWhole W-3/407/22-511/3← Examples of Rational Numbers (any p/q where q ≠ 0) →
Example 1: Between which whole numbers does −½ lie?
Between −1 and 0.
Example 2: Between which whole numbers does ¾ lie?
Between 0 and 1.
Example 3: Example 1: Check if \(\frac{2}{3} + \frac{1}{6}\) is a rational number. Also verify the commutative property for addition.
- Step 1: Add \(\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}\) - Step 2: \(\frac{5}{6}\) is in the form \(\frac{p}{q}\) with \(q \neq 0\), so it is a rational number - Step 3: For commutative property, check \(\frac{1}{6} + \frac{2}{3} = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}\) - Step 4: Both orders give \(\frac{5}{6}\), so commutative property holds
Example 4: Example 2: Verify the associative property of multiplication for the rational numbers \(\frac{1}{2}, \frac{2}{3},\) and \(\frac{3}{4}\).
- Step 1: Left side: \((\frac{1}{2} \times \frac{2}{3}) \times \frac{3}{4}\) - Step 2: First multiply \(\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}\) - Step 3: Then \(\frac{1}{3} \times \frac{3}{4} = \frac{3}{12} = \frac{1}{4}\) - Step 4: Right side: \(\frac{1}{2} \times (\frac{2}{3} \times \frac{3}{4})\) - Step 5: First multiply \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) - Step 6: Then \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\) - Step 7: Both sides equal \(\frac{1}{4}\)
Example 5: Example 3: Using the distributive property, simplify: \(\frac{2}{3} \times (\frac{1}{4} + \frac{1}{2})\)
- Step 1: Apply distributive law: \(a \times (b + c) = a \times b + a \times c\) - Step 2: Here \(a = \frac{2}{3}, b = \frac{1}{4}, c = \frac{1}{2}\) - Step 3: \((\frac{2}{3} \times \frac{1}{4}) + (\frac{2}{3} \times \frac{1}{2})\) - Step 4: First term: \(\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}\) - Step 5: Second term: \(\frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}\) - Step 6: Add: \(\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\) - Step 7: Check by solving inside brackets first: \(\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}\); then \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) ✓
Quick recap
  • Negatives lie to the left of 0.
  • A proper fraction lies between two consecutive integers.
  • Closure Property: Sum or product of two rational numbers is always a rational number
  • Commutative Property: \(a + b = b + a\) and \(a \times b = b \times a\) for rational numbers
  • Associative Property: \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\)
  • Distributive Property: \(a \times (b + c) = a \times b + a \times c\)
  • Subtraction and division do NOT follow commutative or associative properties
  • These properties help us simplify complex calculations and solve equations easily
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✓ Quick check
Which number lies between 0 and 1?
½ = 0.5 lies between 0 and 1.
Between which integers does −5/4 lie?
−5/4 = −1.25 lies between −2 and −1.

Rationals Between Two Rationals (Mean Method)

What is Standard Form of a Rational Number?

A rational number \(\frac{p}{q}\) is in standard form (or simplest form) when:

  • \(q > 0\) (denominator is positive)
  • \(p\) and \(q\) have no common factor other than 1 (they are co-prime)

Steps to convert to standard form:

  • Make denominator positive (multiply numerator and denominator by -1 if needed)
  • Find the HCF (GCD) of numerator and denominator
  • Divide both numerator and denominator by the HCF

How to Compare Rational Numbers?

Method 1 (Same denominator): Compare numerators directly

  • \(\frac{3}{7} > \frac{2}{7}\) because \(3 > 2\)

Method 2 (Different denominators): Use cross-multiplication

  • For \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a \times d\) and \(c \times b\)
  • If \(a \times d > c \times b\), then \(\frac{a}{b} > \frac{c}{d}\)

Method 3 (Decimal conversion): Convert to decimals and compare

Ordering of Rational Numbers

Ascending order (smallest to largest): Arrange from left to right on number line Descending order (largest to smallest): Arrange from right to left on number line

Operations on Rational Numbers

OperationMethodExample
AdditionFind LCM of denominators, convert, add numerators\(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)
SubtractionSame as addition, then subtract numerators\(\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}\)
MultiplicationMultiply numerators, multiply denominators\(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)
DivisionMultiply by reciprocal of divisor\(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\)
Comparing & Ordering Rational Numbers-2-3/2-1-1/201/213/22Ordering: −2 < −3/2 < −1 < −1/2 < 0 < 1/2 < 1 < 3/2 < 2Standard Form: p/q where q > 0 and gcd(p,q) = 1Example: −6/−4 = 6/4 = 3/2 (positive denominator, fully reduced)
Example 1: Find a rational number between ½ and ¾.
(½ + ¾) ÷ 2 = (5/4) ÷ 2 = 5/8.
Example 2: Find a rational number between 1 and 2.
(1 + 2) ÷ 2 = 3/2.
Example 3: Example 1: Express \(\frac{-24}{36}\) in standard form and compare it with \(\frac{-2}{3}\).
- Step 1: Denominator is positive (36 > 0) ✓ - Step 2: Find HCF of 24 and 36 - Step 3: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Step 4: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 - Step 5: HCF = 12 - Step 6: Divide numerator and denominator by 12: \(\frac{-24 \div 12}{36 \div 12} = \frac{-2}{3}\) - Step 7: To compare, they are equal! \(\frac{-24}{36} = \frac{-2}{3}\)
Example 4: Example 2: Arrange the rational numbers \(\frac{3}{5}, \frac{-1}{2}, \frac{7}{10}, \frac{-3}{4}\) in ascending order.
- Step 1: Convert to decimals for easy comparison (or use cross multiplication) - Step 2: \(\frac{3}{5} = 0.6\) - Step 3: \(\frac{-1}{2} = -0.5\) - Step 4: \(\frac{7}{10} = 0.7\) - Step 5: \(\frac{-3}{4} = -0.75\) - Step 6: Arrange from smallest to largest: \(-0.75, -0.5, 0.6, 0.7\) - Step 7: Write in original form: \(\frac{-3}{4}, \frac{-1}{2}, \frac{3}{5}, \frac{7}{10}\) - Step 8: Verify using number line: \(-3/4\) is leftmost, then \(-1/2\), then \(3/5\), then \(7/10\)
Example 5: Example 3 (Word Problem): Riya ate \(\frac{2}{5}\) of a pizza. Her brother ate \(\frac{1}{3}\) of the remaining pizza. What fraction of the whole pizza is left?
- Step 1: Whole pizza = 1 - Step 2: Riya ate \(\frac{2}{5}\), so remaining after Riya = \(1 - \frac{2}{5}\) - Step 3: \(1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}\) - Step 4: Brother ate \(\frac{1}{3}\) of remaining = \(\frac{1}{3} \times \frac{3}{5} = \frac{3}{15} = \frac{1}{5}\) - Step 5: Total eaten = Riya's share + Brother's share = \(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\) - Step 6: Leftover = \(1 - \frac{3}{5} = \frac{2}{5}\)
Example 6: Example 4 (Word Problem - Multi-step): A car travels \(\frac{2}{3}\) of its journey on the first day and \(\frac{1}{4}\) of the remaining journey on the second day. If the total journey is 240 km, how much distance is left to cover on the third day?
- Step 1: Total journey = 240 km - Step 2: Distance covered on first day = \(\frac{2}{3} \times 240 = 160\) km - Step 3: Remaining after day 1 = \(240 - 160 = 80\) km - Step 4: Distance covered on second day = \(\frac{1}{4} \times 80 = 20\) km - Step 5: Total covered = \(160 + 20 = 180\) km - Step 6: Distance left = \(240 - 180 = 60\) km
Quick recap
  • There are infinitely many rationals between any two.
  • Mean method: (a + b) ÷ 2 lies halfway between a and b.
  • Standard form: Denominator positive and numerator & denominator have HCF = 1
  • Comparing fractions: Use cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) if \(a \times d > c \times b\)
  • Ascending order: Smallest to largest; Descending order: Largest to smallest
  • Addition/Subtraction: Find LCM of denominators first
  • Multiplication: Multiply numerators and denominators directly
  • Division: Multiply by the reciprocal of the divisor
  • Always simplify your final answer to standard form
  • For word problems, identify what is given and what needs to be found, then break into steps
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✓ Quick check
Find a rational number between ¼ and ½.
(¼ + ½) ÷ 2 = (3/4) ÷ 2 = 3/8.
What is the mean of ⅓ and ⅔?
(⅓ + ⅔) ÷ 2 = 1 ÷ 2 = ½.
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