NUMBER SYSTEMS - Q&A

1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

Yes, zero is a rational number. It can be written in the form p/q where p and q are integers and q ≠ 0.
For example: 0/1, 0/2, 0/10, etc.
Here, p = 0 and q can be any non-zero integer.

2. Find six rational numbers between 3 and 4.

To find 6 rational numbers, we can multiply and divide the numbers by 6 + 1 = 7.
3 = (3 × 7) / 7 = 21/7
4 = (4 × 7) / 7 = 28/7
Now, we can choose numbers between 21/7 and 28/7.
The six rational numbers are: 22/7, 23/7, 24/7, 25/7, 26/7, 27/7.

3. Find five rational numbers between 3/5 and 4/5.

To find 5 rational numbers, we multiply and divide the fractions by 5 + 1 = 6.
3/5 = (3 × 6) / (5 × 6) = 18/30
4/5 = (4 × 6) / (5 × 6) = 24/30
Now, we choose numbers between 18/30 and 24/30.
The five rational numbers are: 19/30, 20/30, 21/30, 22/30, 23/30.

4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

True. The collection of whole numbers contains all natural numbers (1, 2, 3...) along with 0.

(ii) Every integer is a whole number.

False. Negative integers (like -2, -3) are not whole numbers.

(iii) Every rational number is a whole number.

False. Rational numbers like 1/2 or 3/5 are not whole numbers.

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

True. The collection of real numbers is made up of all rational and irrational numbers.

(ii) Every point on the number line is of the form √m, where m is a natural number.

False. Negative numbers on the number line cannot be the square root of a natural number (e.g., no natural number 'm' exists such that √m = -2).

(iii) Every real number is an irrational number.

False. Real numbers include rational numbers as well. For example, 2 is a real number but it is not irrational.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

No, the square roots of all positive integers are not irrational.
Example: √4 = 2, which is a rational number.
Example: √9 = 3, which is a rational number.

3. Show how √5 can be represented on the number line.

To represent √5 on the number line:
1. We know that 5 = 2² + 1².
2. Draw a number line and mark point O at 0 and A at 2.
3. Construct a perpendicular AB of length 1 unit at A.
4. Join O and B. By Pythagoras theorem, OB = √(OA² + AB²) = √(2² + 1²) = √(4+1) = √5.
5. Using a compass with center O and radius OB, draw an arc intersecting the number line at point P.
Point P represents √5 on the number line.

1. Write the following in decimal form and say what kind of decimal expansion each has:

(i) 36/100

36/100 = 0.36
Type: Terminating.

(ii) 1/11

1/11 = 0.090909... = 0.̅09
Type: Non-terminating recurring (repeating).

(iii) 4 1/8

4 1/8 = 33/8 = 4.125
Type: Terminating.

(iv) 3/13

3/13 = 0.230769230769... = 0.̅230769
Type: Non-terminating recurring.

(v) 2/11

2/11 = 0.181818... = 0.̅18
Type: Non-terminating recurring.

(vi) 329/400

329/400 = 0.8225
Type: Terminating.

2. You know that 1/7 = 0.̅142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?

Yes, we can predict them by multiplying the value of 1/7 by 2, 3, 4, 5, and 6 respectively.
2/7 = 2 × 0.̅142857 = 0.̅285714
3/7 = 3 × 0.̅142857 = 0.̅428571
4/7 = 4 × 0.̅142857 = 0.̅571428
5/7 = 5 × 0.̅142857 = 0.̅714285
6/7 = 6 × 0.̅142857 = 0.̅857142

3. Express the following in the form p/q, where p and q are integers and q ≠ 0.

(i) 0.̅6

Let x = 0.6666...
Multiply by 10: 10x = 6.6666...
Subtract x from 10x: 9x = 6
x = 6/9 = 2/3.

(ii) 0.4̅7

Let x = 0.4777...
Multiply by 10: 10x = 4.777...
Multiply by 100: 100x = 47.777...
Subtract 10x from 100x: 90x = 43
x = 43/90.

(iii) 0.̅001

Let x = 0.001001...
Multiply by 1000: 1000x = 1.001001...
Subtract x from 1000x: 999x = 1
x = 1/999.

4. Express 0.99999... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Let x = 0.99999...
10x = 9.99999...
Subtracting x from 10x:
9x = 9
x = 9/9 = 1.
The answer is 1. It makes sense because the difference between 1 and 0.999... is infinitesimally small, so they are mathematically equal.

5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.

The maximum number of digits in the repeating block is 16 (since divisor is 17, max remainder is 16).
Performing division: 1/17 = 0.0588235294117647...
The repeating block is 0588235294117647, which has 16 digits.

6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

The prime factorization of q must contain only powers of 2 or powers of 5 or both.

7. Write three numbers whose decimal expansions are non-terminating non-recurring.

Examples of such irrational numbers are:
1. 0.101001000100001...
2. 0.202002000200002...
3. π (3.14159...)

8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.

5/7 = 0.714285...
9/11 = 0.818181...
We need irrational numbers between 0.71... and 0.81...
1. 0.720720072000...
2. 0.750750075000...
3. 0.808008000...

9. Classify the following numbers as rational or irrational:

(i) √23

Irrational (23 is not a perfect square).

(ii) √225

Rational (√225 = 15).

(iii) 0.3796

Rational (Terminating decimal).

(iv) 7.478478...

Rational (Non-terminating recurring).

(v) 1.101001000100001...

Irrational (Non-terminating non-recurring).

1. Classify the following numbers as rational or irrational:

(i) 2 - √5

Irrational (Difference of rational and irrational is irrational).

(ii) (3 + √23) - √23

Rational (Simplifies to 3).

(iii) 2√7 / 7√7

Rational (Simplifies to 2/7).

(iv) 1 / √2

Irrational (Quotient of rational and irrational).

(v) 2π

Irrational (Product of rational and irrational).

2. Simplify each of the following expressions:

(i) (3 + √3)(2 + √2)

= 3(2) + 3(√2) + √3(2) + √3(√2)
= 6 + 3√2 + 2√3 + √6.

(ii) (3 + √3)(3 - √3)

Using (a+b)(a-b) = a² - b²
= 3² - (√3)²
= 9 - 3 = 6.

(iii) (√5 + √2)²

Using (a+b)² = a² + 2ab + b²
= (√5)² + 2(√5)(√2) + (√2)²
= 5 + 2√10 + 2
= 7 + 2√10.

(iv) (√5 - √2)(√5 + √2)

Using (a-b)(a+b) = a² - b²
= (√5)² - (√2)²
= 5 - 2 = 3.

3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π = c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

There is no contradiction. When we measure a length with a scale or any other device, we only get an approximate rational value. Therefore, we may not realize that either c or d is irrational. The exact ratio c/d remains irrational.

4. Represent √9.3 on the number line.

Steps:
1. Draw a line segment AB = 9.3 units.
2. Extend AB to C such that BC = 1 unit.
3. Find the midpoint O of AC.
4. Draw a semicircle with center O and radius OA (or OC).
5. Draw a perpendicular to AC at B, intersecting the semicircle at D.
6. The length BD is √9.3. Taking B as center and BD as radius, draw an arc to cut the number line (extended AC) at E. Point E represents √9.3.

5. Rationalise the denominators of the following:

(i) 1 / √7

Multiply numerator and denominator by √7:
= (1 × √7) / (√7 × √7)
= √7 / 7.

(ii) 1 / (√7 - √6)

Multiply by conjugate (√7 + √6):
= (√7 + √6) / [(√7 - √6)(√7 + √6)]
= (√7 + √6) / (7 - 6)
= √7 + √6.

(iii) 1 / (√5 + √2)

Multiply by conjugate (√5 - √2):
= (√5 - √2) / [(√5 + √2)(√5 - √2)]
= (√5 - √2) / (5 - 2)
= (√5 - √2) / 3.

(iv) 1 / (√7 - 2)

Multiply by conjugate (√7 + 2):
= (√7 + 2) / [(√7 - 2)(√7 + 2)]
= (√7 + 2) / (7 - 4)
= (√7 + 2) / 3.

1. Find:

(i) 64^1/2

64^1/2 = (8²)^1/2 = 8^2×1/2 = 8¹ = 8.

(ii) 32^1/5

32^1/5 = (2⁵)^1/5 = 2^5×1/5 = 2¹ = 2.

(iii) 125^1/3

125^1/3 = (5³)^1/3 = 5^3×1/3 = 5¹ = 5.

2. Find:

(i) 9^3/2

9^3/2 = (3²)^3/2 = 3³ = 27.

(ii) 32^2/5

32^2/5 = (2⁵)^2/5 = 2² = 4.

(iii) 16^3/4

16^3/4 = (2⁴)^3/4 = 2³ = 8.

(iv) 125^-1/3

125^-1/3 = (5³)^-1/3 = 5^-1 = 1/5.

3. Simplify:

(i) 2^2/3 · 2^1/5

Using a^m · aⁿ = a^m+n
= 2^{(2/3 + 1/5)}
= 2^{(10/15 + 3/15)}
= 2^13/15.

(ii) (1 / 3³)⁷

Using (a^m)ⁿ = a^mn
= (3^-3)⁷
= 3^-21 or 1/3²¹.

(iii) 11^1/2 / 11^1/4

Using a^m / aⁿ = a^m-n
= 11^{(1/2 - 1/4)}
= 11^{(2/4 - 1/4)}
= 11^1/4.

(iv) 7^1/2 · 8^1/2

Using a^m · b^m = (ab)^m
= (7 × 8)^1/2
= 56^1/2.