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Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 1

Real Numbers, Exponents and Logarithms

What to remember

  • Every real number is either rational or irrational. A rational number can be written as p/q (q ≠ 0); its decimal form either ends or repeats. An irrational number has a decimal form that never ends and never repeats.
  • A fraction p/q in lowest terms has a terminating decimal only if q has no prime factor except 2 and 5. Any other prime factor in q gives a non-terminating repeating decimal.
  • Laws of exponents and logarithms are two views of one idea. a^x = N is the same as log_a N = x. Product becomes sum, quotient becomes difference, power becomes multiplier.

The number family

Natural numbers N = 1, 2, 3, …. Whole numbers W = 0, 1, 2, …. Integers Z = …, −2, −1, 0, 1, 2, …. Rational numbers Q are all numbers p/q with p, q integers and q ≠ 0. Irrational numbers are real numbers that are not rational. Real numbers R = rational numbers together with irrational numbers. Each set sits inside the next: N ⊂ W ⊂ Z ⊂ Q ⊂ R.

TypeMeaningExamples
NaturalCounting numbers1, 2, 3
WholeNatural numbers and zero0, 1, 2
IntegerWhole numbers and their negatives−3, 0, 5
Rationalp/q, q ≠ 03/4, −7, 0.25, 0.333…
IrrationalNot p/q√2, √3, π
RealRational and irrational togetherevery point on the number line

Between any two real numbers there are infinitely many rational numbers and infinitely many irrational numbers. Every real number matches exactly one point on the number line, and every point matches one real number.

Euclid's division lemma and the fundamental theorem

For positive integers a and b there are unique integers q and r such that a = bq + r, with 0 ≤ r < b. Repeating this step, with the divisor becoming the new dividend, gives the HCF. The last non-zero divisor is the HCF (Euclid's division algorithm).

Worked example: HCF of 96 and 404. 404 = 96 × 4 + 20. 96 = 20 × 4 + 16. 20 = 16 × 1 + 4. 16 = 4 × 4 + 0. The HCF is 4.

The fundamental theorem of arithmetic says every composite number can be written as a product of primes, and this product is unique apart from the order of the factors. Use it for HCF and LCM: HCF takes the smallest power of each common prime; LCM takes the greatest power of every prime that appears. For two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. This rule works for two numbers only, not for three.

Irrational numbers

A number is irrational if it cannot be written as p/q. The square root of any prime is irrational, so √2, √3, √5, √7 are irrational. The square root of a perfect square, such as √16 = 4, is rational. The proof that √2 is irrational uses contradiction: if √2 = p/q in lowest terms, then p² = 2q², so 2 divides p, then 2 divides q as well, which contradicts "lowest terms".

Rules to remember:

  • Rational + irrational is irrational. Non-zero rational × irrational is irrational. So 3 + √2 and 5√3 are irrational.
  • The sum, difference, product or quotient of two irrationals can be rational: √2 × √2 = 2 and (3 + √2)(3 − √2) = 7.
  • So "the sum of two irrationals is irrational" is false in general.
  • The number π is irrational. The fraction 22/7 is only a close rational approximation.

Rationalising: 1/(√a + √b) = (√a − √b)/(a − b). The conjugate of √a + √b is √a − √b.

Classroom angle: show √2 on the number line by drawing a right triangle with both legs 1 unit; the hypotenuse is √2. Then use a compass to mark it on the line. Students often believe 22/7 equals π, so address this directly.

Decimal expansions

A rational number p/q (lowest terms) has a terminating decimal expansion exactly when q = 2^m × 5^n for whole numbers m and n. Otherwise the expansion is non-terminating and repeating (recurring). A non-terminating, non-repeating expansion means the number is irrational.

NumberDecimal formKind
7/80.875Terminating (q = 2³)
3/200.15Terminating (q = 2² × 5)
1/30.333…Repeating
1/70.142857142857…Repeating, block of 6 digits
√21.41421356…Never ends, never repeats
0.1010010001…pattern with growing gapsIrrational

Converting a repeating decimal to p/q: let x = 0.777…; then 10x = 7.777…; subtract to get 9x = 7, so x = 7/9. For two repeating digits multiply by 100. Note that 0.999… = 1 exactly, since 9x = 9 gives x = 1.

Laws of exponents

For a > 0 and real or rational m, n:

LawStatement
Producta^m × a^n = a^(m+n)
Quotienta^m ÷ a^n = a^(m−n)
Power of a power(a^m)^n = a^(mn)
Power of a product(ab)^n = a^n × b^n
Power of a quotient(a/b)^n = a^n / b^n
Zero exponenta^0 = 1 (a ≠ 0)
Negative exponenta^(−n) = 1/a^n
Fractional exponenta^(m/n) = (nth root of a)^m

Examples: 27^(2/3) = (3)² = 9. 16^(3/4) = 2³ = 8. 8^(−1/3) = 1/2. 2^5 × 2^3 = 2^8 = 256. Always change to the same base before comparing: 4^x = 32 means 2^(2x) = 2^5, so x = 5/2. Note that a^m + a^n cannot be simplified by a law; and (a + b)^n is not a^n + b^n.

Logarithms

Definition: if a^x = N (a > 0, a ≠ 1, N > 0), then log_a N = x. The base cannot be 1 and the number must be positive. Log base 10 is the common logarithm; base e is the natural logarithm.

LawStatement
Productlog_a(mn) = log_a m + log_a n
Quotientlog_a(m/n) = log_a m − log_a n
Powerlog_a(m^n) = n log_a m
Base changelog_a b = log_c b / log_c a
Reciprocallog_a b × log_b a = 1
Special valueslog_a 1 = 0 and log_a a = 1

Examples: log_2 32 = 5, because 2^5 = 32. log_3 81 = 4. log_10 0.001 = −3. log_5 (1/25) = −2. log 2 + log 5 = log 10 = 1 (base 10). If log_x 64 = 3, then x³ = 64 and x = 4. Also a^(log_a N) = N.

More worked examples: simplify log 8 + log 125 (base 10) with the product law: log (8 × 125) = log 1000 = 3. Solve 2^(x+1) = 32: write 32 = 2^5, so x + 1 = 5 and x = 4. Evaluate (3 + √2)(3 − √2) = 9 − 2 = 7. Find the HCF and LCM of 12 and 18: 12 = 2² × 3 and 18 = 2 × 3², so HCF = 2 × 3 = 6 and LCM = 2² × 3² = 36; check 6 × 36 = 216 = 12 × 18.

Classroom angle: introduce logarithms as the "inverse" of exponents, in the same way that subtraction undoes addition. A table that pairs powers of 2 with their exponents helps students see the laws.

Exam traps

  • 22/7 and π: 22/7 is rational; π is irrational.
  • √16 and √2: √16 = 4 is rational, √2 is irrational. Only roots of non-perfect squares are irrational.
  • Sum of irrationals: can be rational (√2 + (−√2) = 0), so no general rule.
  • 0.333… and 0.101001…: the first repeats and is rational; the second has a growing pattern and is irrational.
  • 1/12 versus 1/20: 12 has the factor 3, so 1/12 does not terminate; 20 = 2² × 5, so 1/20 = 0.05 terminates. Check the lowest form first.
  • HCF × LCM rule: valid only for two numbers.
  • (a + b)^n: is not a^n + b^n. Likewise log(m + n) is not log m + log n.
  • log(m^n): equals n log m, not (log m)^n and not log m + n.

One-liners

  • 1. Every integer is a rational number, since n = n/1.
  • 2. Zero is a whole number and an integer, but not a natural number.
  • 3. Euclid's lemma: a = bq + r with 0 ≤ r < b.
  • 4. The last non-zero remainder-divisor in Euclid's algorithm is the HCF.
  • 5. HCF × LCM = product of the two numbers.
  • 6. The square root of a prime number is irrational.
  • 7. A terminating decimal has denominator 2^m 5^n in lowest terms.
  • 8. 0.999… equals 1.
  • 9. a^0 = 1 for a ≠ 0, and a^(−n) = 1/a^n.
  • 10. log_a 1 = 0 and log_a a = 1.
  • 11. log_a(mn) = log_a m + log_a n; log_a(m/n) = log_a m − log_a n.
  • 12. log_a b × log_b a = 1.

Practice questions

  1. The decimal expansion of a rational number is

    1. always terminating
    2. always non-terminating and non-repeating
    3. either terminating or non-terminating repeating
    4. terminating and repeating at once
    Answer

    C. either terminating or non-terminating repeating

    A rational number always ends or repeats in decimal form.

  2. Which of the following numbers is irrational?

    1. 0.333…
    2. 7/8
    3. √25
    4. √2
    Answer

    D. √2

    √2 is the root of a non-square prime, so it is irrational; the others are rational.

  3. In Euclid's division lemma a = bq + r, the remainder r satisfies

    1. 0 < r ≤ b
    2. 0 ≤ r < b
    3. b < r < 2b
    4. 0 ≤ r ≤ b
    Answer

    B. 0 ≤ r < b

    The lemma states 0 ≤ r < b.

  4. log_a(mn) is equal to

    1. log_a m + log_a n
    2. log_a m × log_a n
    3. log_a m ÷ log_a n
    4. log_a m − log_a n
    Answer

    A. log_a m + log_a n

    Product law of logarithms.

  5. The value of 7^0 is

    1. 0
    2. 1
    3. 7
    4. Not defined
    Answer

    B. 1

    Any non-zero number raised to the power 0 is 1.

  6. The HCF of two numbers is 6 and their LCM is 84. If one number is 12, the other is

    1. 72
    2. 84
    3. 42
    4. 36
    Answer

    C. 42

    Other number = (6 × 84)/12 = 42.

  7. The value of log_2 32 is

    1. 4
    2. 6
    3. 16
    4. 5
    Answer

    D. 5

    2^5 = 32, so log_2 32 = 5.

  8. The value of log_10 0.001 is

    1. 3
    2. −2
    3. −4
    4. −3
    Answer

    D. −3

    0.001 = 10^(−3).

  9. The product of 2 and √3 is

    1. an irrational number
    2. a whole number
    3. an integer
    4. a rational number
    Answer

    A. an irrational number

    A non-zero rational times an irrational is irrational.

  10. The decimal expansion of 7/8 is

    1. 0.785
    2. 0.78
    3. 0.875
    4. 0.875875…
    Answer

    C. 0.875

    7 ÷ 8 = 0.875, which terminates because 8 = 2³.

  11. The decimal expansion of 1/7 is

    1. non-terminating and repeating
    2. non-terminating and non-repeating
    3. terminating
    4. terminating after 6 digits
    Answer

    A. non-terminating and repeating

    7 is not of the form 2^m 5^n, so the decimal repeats.

  12. Which fraction has a terminating decimal expansion?

    1. 17/6
    2. 13/3125
    3. 29/343
    4. 11/30
    Answer

    B. 13/3125

    3125 = 5^5, so 13/3125 terminates. The others have factors 3 or 7 in the denominator.

  13. The value of 0.999… is

    1. 1
    2. 0.9
    3. just less than 1
    4. 10
    Answer

    A. 1

    Let x = 0.999…; 9x = 9, so x = 1.

  14. The value of 2^5 × 2^3 is

    1. 512
    2. 256
    3. 128
    4. 64
    Answer

    B. 256

    2^(5+3) = 2^8 = 256.

  15. The value of (2^3)^2 is

    1. 12
    2. 128
    3. 32
    4. 64
    Answer

    D. 64

    (2^3)^2 = 2^6 = 64.

  16. The value of 27^(2/3) is

    1. 18
    2. 3
    3. 81
    4. 9
    Answer

    D. 9

    27^(1/3) = 3, then 3² = 9.

  17. The value of 16^(3/4) is

    1. 8
    2. 12
    3. 4
    4. 64
    Answer

    A. 8

    16^(1/4) = 2 and 2³ = 8.

  18. The value of 8^(−1/3) is

    1. −2
    2. 1/2
    3. 2
    4. −1/2
    Answer

    B. 1/2

    8^(1/3) = 2, so 8^(−1/3) = 1/2.

  19. The value of log_3 81 is

    1. 3
    2. 9
    3. 4
    4. 27
    Answer

    C. 4

    3^4 = 81.

  20. The value of log 2 + log 5 (base 10) is

    1. 7
    2. 10
    3. 0
    4. 1
    Answer

    D. 1

    log 2 + log 5 = log 10 = 1.

  21. If log_x 64 = 3, then x is

    1. 4
    2. 8
    3. 16
    4. 2
    Answer

    A. 4

    x³ = 64, so x = 4.

  22. Using Euclid's algorithm, the HCF of 96 and 404 is

    1. 2
    2. 4
    3. 8
    4. 12
    Answer

    B. 4

    404 = 96×4 + 20; 96 = 20×4 + 16; 20 = 16 + 4; 16 = 4×4. HCF = 4.

  23. The fundamental theorem of arithmetic states that every composite number can be written as a product of primes in a way that is

    1. unique including the order
    2. possible in more than one way
    3. unique apart from the order of factors
    4. possible only for even numbers
    Answer

    C. unique apart from the order of factors

    Prime factorisation is unique up to the order.

  24. The value of (3 + √2)(3 − √2) is

    1. 11
    2. 9 − √2
    3. 3
    4. 7
    Answer

    D. 7

    9 − 2 = 7.

  25. On rationalising 1/(√3 + √2) the result is

    1. √3 − √2
    2. √3 + √2
    3. 1
    4. √6
    Answer

    A. √3 − √2

    Multiply by (√3 − √2)/(√3 − √2); the denominator becomes 3 − 2 = 1.

  26. If 2^(x+1) = 32, then x is

    1. 5
    2. 4
    3. 3
    4. 6
    Answer

    B. 4

    32 = 2^5, so x + 1 = 5 and x = 4.

  27. The value of log_5 (1/25) is

    1. 2
    2. −5
    3. −2
    4. 1/2
    Answer

    C. −2

    1/25 = 5^(−2).

  28. To show √2 on the number line, a teacher draws a right triangle whose legs are each 1 unit. The hypotenuse is √2 because of

    1. the remainder theorem
    2. the factor theorem
    3. the Pythagoras theorem
    4. Euclid's division lemma
    Answer

    C. the Pythagoras theorem

    1² + 1² = 2, so the hypotenuse is √2 by Pythagoras.

  29. The LCM of 12 and 18 is

    1. 216
    2. 6
    3. 36
    4. 72
    Answer

    C. 36

    12 = 2²×3, 18 = 2×3²; LCM = 2²×3² = 36.

  30. The value of log_2 8 × log_8 2 is

    1. 3
    2. 1
    3. 1/3
    4. 8
    Answer

    B. 1

    log_a b × log_b a = 1.

  31. The number of rational numbers between 1 and 2 is

    1. infinite
    2. ten
    3. one
    4. zero
    Answer

    A. infinite

    Between any two real numbers lie infinitely many rationals.

  32. A fraction p/q in lowest terms has a terminating decimal form. Which could q be?

    1. 21
    2. 30
    3. 66
    4. 160
    Answer

    D. 160

    160 = 2^5 × 5. The others contain 3 or 7 or 11.

  33. The simplest form of log 8 + log 125 (base 10) is

    1. 3
    2. 2
    3. 1000
    4. 13
    Answer

    A. 3

    log (8×125) = log 1000 = 3.

  34. The value of log_9 3 is

    1. 2
    2. 1/2
    3. 3
    4. 1/3
    Answer

    B. 1/2

    9^(1/2) = 3, so log_9 3 = 1/2.

  35. The number 0.1010010001… (gaps between the 1s keep growing) is

    1. an integer
    2. a recurring decimal
    3. rational
    4. irrational
    Answer

    D. irrational

    Its expansion never ends and never repeats.

  36. If x = 0.777…, then x as a fraction is

    1. 77/100
    2. 7/99
    3. 7/9
    4. 7/10
    Answer

    C. 7/9

    10x − x = 7, so x = 7/9.

  37. Consider the statements: 1. Every integer is a rational number. 2. Every rational number is an integer.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Any integer n = n/1, but 1/2 is rational and not an integer.

  38. Consider the statements: 1. π is an irrational number. 2. 22/7 is exactly equal to π.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    22/7 is only an approximation of π, which is irrational.

  39. Consider the statements: 1. The sum of two irrational numbers is always irrational. 2. The product of a non-zero rational number and an irrational number is irrational.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    √2 + (−√2) = 0 shows statement 1 is false; statement 2 is a standard result.

  40. Consider the statements: 1. log_a(m/n) = log_a m − log_a n. 2. log_a(m^n) = log_a m + n.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    The power law is log_a(m^n) = n log_a m, so statement 2 is wrong.

  41. Consider the statements: 1. log_a 1 = 0 for a valid base a. 2. log_a a = 1 for a valid base a.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Since a^0 = 1 and a^1 = a, both are correct.

  42. Consider the statements: 1. √9 is an irrational number. 2. The decimal 0.1212… is a rational number.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    √9 = 3 is rational; 0.1212… = 12/99 = 4/33 is rational.

  43. Consider the statements: 1. The base of a logarithm can be 1. 2. log_a m is defined for every real number m.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    The base must be positive and not 1, and m must be positive.

  44. Consider the statements: 1. 2√3 is irrational. 2. √2 × √8 is rational.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    2√3 is rational times irrational; √2 × √8 = √16 = 4.

  45. Consider the statements: 1. 1/20 has a terminating decimal expansion. 2. 1/12 has a terminating decimal expansion.

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    20 = 2²×5 terminates; 12 = 2²×3 has a factor 3, so it repeats.

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