Exercise 1.1 - Step-by-Step Solutions

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Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where p and q are integers and q ≠ 0?

Step 1: Recall the definition A rational number can be written as $\frac{p}{q}$ where p, q are integers and q ≠ 0.

Step 2: Check if zero fits

• Zero = 0
• Can we write 0 as p/q? Yes!
• 0 = 0/1 (p = 0, q = 1)
• 0 = 0/2 (p = 0, q = 2)
• 0 = 0/(-5) (p = 0, q = -5)

Step 3: Verify conditions

• Is 0 an integer? ✅ Yes
• Is denominator ≠ 0? ✅ Yes (1 ≠ 0)

Answer: Yes, zero is a rational number. ✓

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Find six rational numbers between 3 and 4.

Method: Common Denominator

Step 1: Choose denominator = 7 (one more than numbers we need)

Step 2: Convert to equivalent fractions

• 3 = $\frac{21}{7}$ ( Just multiply and devide $\frac{3}{1}$ by 7)
• 4 = $\frac{28}{7}$ ( Just multiply and devide $\frac{4}{1}$ by 7)

Step 3: Find fractions between 21/7 and 28/7

• 22/7, 23/7, 24/7, 25/7, 26/7, 27/7

Answer: $\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}$

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Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ .

Step 1: Make denominators larger

Multiply and devide both by 6 (Since we need 5 numbers so we multiply devide by 1 more then 5):

• $\frac{3}{5}$ = $\frac{18}{30}$
• $\frac{4}{5}$ = $\frac{24}{30}$

Step 2: Find fractions between 18/30 and 24/30

• $\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}$

Step 3: Simplify if possible

• $\frac{20}{30}$ = $\frac{2}{3}$
• $\frac{21}{30}$ = $\frac{7}{10}$
• $\frac{22}{30}$ = $\frac{11}{15}$

Answer: $\frac{19}{30}, \frac{2}{3}, \frac{7}{10}, \frac{11}{15}, \frac{23}{30}$

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True or False with explanation:

(i) Every natural number is a whole number.

Step 1: Define both sets

• Natural numbers: {1, 2, 3, 4, ...}
• Whole numbers: {0, 1, 2, 3, 4, ...}

Step 2: Check containment Every element of N is also in W.

Answer: TRUE ✓

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(ii) Every integer is a whole number.

Step 1: Define integers

• Integers: {..., -2, -1, 0, 1, 2, ...}

Step 2: Find counterexample

• Is -1 a whole number? No!
• Is -2 a whole number? No!

Answer: FALSE ✗ (Negative integers are not whole numbers)

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(iii) Every rational number is a whole number.

Step 1: Find a counterexample

• Is $\frac{1}{2}$ a whole number? No!
• Is 0.75 a whole number? No!

Answer: FALSE ✗ (Many rationals like $\frac{1}{2}$ are not whole numbers)