Extending the number system

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1. Rational exponents in exponential functions

Warm-up

Three quick questions from earlier lessons.

Discover

A bacteria population doubles every day. By what factor does it grow in half a day?

Hint: whatever the factor ff is, growing for two half-days must double the population, so f⋅f=2f \cdot f = 2.

Key idea

Exponents that are fractions are roots.

b1n=bnb^{\frac{1}{n}} = \sqrt[n]{b} and bmn=(bn)mb^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m

So the half-day factor is 212=2≈1.412^{\frac12} = \sqrt{2} \approx 1.41. A negative exponent means a reciprocal: b−r=1brb^{-r} = \dfrac{1}{b^{r}}.

Worked example

Evaluate 8238^{\frac{2}{3}}.

823=(83)28^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2
The denominator 3 means a cube root; the numerator 2 means a square.
=22= 2^2
The cube root of 8 is 2.
=4= 4
Square it.

Your turn, guided

Your turn: evaluate 272327^{\frac{2}{3}}. Fill in the missing line.

2723=(273)2=3227^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2
Cube root, then square.
=9= 9
3² = 9.

Practice ★

  1. Evaluate 365236^{ \frac{ 5 }{2} }.
  2. Evaluate 813281^{ \frac{ 3 }{2} }.
  3. Evaluate 815281^{ \frac{ 5 }{2} }.
  4. Evaluate 4324^{ \frac{ 3 }{2} }.
  5. Evaluate 495249^{ \frac{ 5 }{2} }.

Watch out!

Practice ★★

  1. Evaluate 125−23125^{ -\frac{ 2 }{3} }.
  2. Evaluate 125−13125^{ -\frac{ 1 }{3} }.
  3. Evaluate 64−1364^{ -\frac{ 1 }{3} }.
  4. Evaluate 8−138^{ -\frac{ 1 }{3} }.
  5. Evaluate 27−1327^{ -\frac{ 1 }{3} }.

Exit ticket

Three questions to check you have got it.

  1. Evaluate 125−23125^{ -\frac{ 2 }{3} }.
  2. Evaluate 125−13125^{ -\frac{ 1 }{3} }.
  3. Evaluate 64−1364^{ -\frac{ 1 }{3} }.

Summary card

Rational exponents

  • b1n=bnb^{\frac{1}{n}} = \sqrt[n]{b}; bmn=(bn)mb^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m.
  • b−r=1brb^{-r} = \dfrac{1}{b^{r}}: the value stays positive.
  • A factor bb per unit means a factor b12b^{\frac12} per half-unit.

2. Rational and irrational numbers; repeating decimals

Warm-up

Three quick questions from earlier lessons.

Discover

13=0.333…\tfrac{1}{3} = 0.333\ldots goes on forever. Is it rational?

  1. Yes
  2. No: it never ends

Key idea

Rational numbers are fractions; their decimals end or repeat.

A rational number can be written ab\tfrac{a}{b} with integers aa, b≠0b \neq 0; its decimal expansion terminates or repeats. An irrational number cannot: its decimal never ends and never repeats. n\sqrt{n} is irrational unless nn is a perfect square; π\pi is irrational.

Worked example

Write 0.27‾0.\overline{27} as a fraction.

100x=27.2727…100x = 27.2727\ldots
Let x = 0.2727…
99x=2799x = 27
Subtract x.
x=2799=311x = \tfrac{27}{99} = \tfrac{3}{11}
Simplify.

Your turn, guided

Your turn: write 0.5‾0.\overline{5} as a fraction.

59\tfrac{5}{9}
10x − x = 5.

Practice ★

  1. Which number is irrational?
    1. 2\sqrt{ 2 }
    2. 4\sqrt{ 4 }
    3. 0.1‾0.\overline{ 1 }
  2. Which number is irrational?
    1. 10\sqrt{ 10 }
    2. 16\sqrt{ 16 }
    3. 0.3‾0.\overline{ 3 }
  1. Write 0.1‾0.\overline{ 1 } (the digit 1 repeating forever) as a fraction.

Watch out!

Practice ★★

  1. Write 0.48‾0.\overline{ 48 } (the digits 48 repeating forever) as a fraction.
  2. Write 0.70‾0.\overline{ 70 } (the digits 70 repeating forever) as a fraction.

Exit ticket

Three questions to check you have got it.

  1. Write 0.1‾0.\overline{ 1 } (the digit 1 repeating forever) as a fraction.
  2. Write 0.2‾0.\overline{ 2 } (the digit 2 repeating forever) as a fraction.
  3. Write 0.8‾0.\overline{ 8 } (the digit 8 repeating forever) as a fraction.

Summary card

Rational and irrational

Ends or repeats: rational. Neither: irrational.

Watch out: a repeating decimal is a fraction over 9, 99, …

3. The number i and complex numbers

Warm-up

Three quick questions from earlier lessons.

Discover

No real number squares to −1-1. Mathematicians define ii with i2=−1i^2 = -1. What is (3i)2(3i)^2?

  1. −9-9
  2. 99

Key idea

A new number that makes negative square roots possible.

i2=−1i^2 = -1. For a>0a > 0: −a=ia\sqrt{-a} = i\sqrt{a}. A complex number is a+bia + bi with real part aa and imaginary part bb. Powers of ii cycle: i,−1,−i,1,…i, -1, -i, 1, \dots (period 4).

Worked example

Simplify −49\sqrt{-49} and i10i^{10}.

−49=7i\sqrt{-49} = 7i
√49 · √−1.
i10=i2i^{10} = i^2
10 = 4·2 + 2.
i10=−1i^{10} = -1
i² = −1.

Your turn, guided

Your turn: i7i^7.

−i-i
7 = 4 + 3: i³ = −i.

Practice ★

  1. Write −121\sqrt{-121} in terms of ii.
  2. Write −4\sqrt{-4} in terms of ii.
  1. Which number is real?
    1. (9i)2(9i)^2
    2. −9\sqrt{-9}
    3. 9+i9 + i

Watch out!

Practice ★★

  1. Simplify i10i^{ 10 }.
  2. Simplify i13i^{ 13 }.
  3. Simplify i5i^{ 5 }.

Exit ticket

Three questions to check you have got it.

  1. Write −121\sqrt{-121} in terms of ii.
  2. Write −4\sqrt{-4} in terms of ii.
  3. Write −25\sqrt{-25} in terms of ii.

Summary card

The number i

i2=−1i^2 = -1; −a=ia\sqrt{-a} = i\sqrt{a}; powers of ii repeat every 4.

Watch out: −16\sqrt{-16} is 4i4i, not −4-4.

4. Adding, subtracting and multiplying complex numbers

Warm-up

Three quick questions from earlier lessons.

Discover

(2+3i)+(4+5i)(2 + 3i) + (4 + 5i). Treat ii like a variable. What do you get?

  1. 6+8i6 + 8i
  2. 1414

Key idea

Calculate as with polynomials in ii, then use i2=−1i^2 = -1.

(a+bi)±(c+di)=(a±c)+(b±d)i(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i. (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i, since bd i2=−bdbd\,i^2 = -bd. The conjugate of a+bia + bi is a−bia - bi, and (a+bi)(a−bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2.

Worked example

(2+3i)(1−4i)(2 + 3i)(1 - 4i).

2−8i+3i−12i22 - 8i + 3i - 12i^2
Distribute.
2−5i+122 - 5i + 12
i² = −1.
14−5i14 - 5i
Combine.

Your turn, guided

Your turn: (3+2i)(3−2i)(3 + 2i)(3 - 2i).

1313
9 + 4.

Practice ★

  1. Add: (9+8i)+(2+1i)(9 + 8i) + (2 + 1i).
  2. Add: (3+4i)+(9+2i)(3 + 4i) + (9 + 2i).
  1. Multiply: (2+8i)(2−8i)(2 + 8i)(2 - 8i).
  2. Multiply: (7+2i)(7−2i)(7 + 2i)(7 - 2i).

Watch out!

Practice ★★

  1. Expand (3+6i)2(3 + 6i)^2.
  2. Expand (1+6i)2(1 + 6i)^2.

Exit ticket

Three questions to check you have got it.

  1. Multiply: (2+8i)(2−8i)(2 + 8i)(2 - 8i).
  2. Multiply: (7+2i)(7−2i)(7 + 2i)(7 - 2i).
  3. Multiply: (8+4i)(8−4i)(8 + 4i)(8 - 4i).

Summary card

Complex arithmetic

Add parts separately; multiply out and use i2=−1i^2 = -1; conjugates give a real product.

Watch out: i2i^2 is −1-1, not 11.

5. Adding, subtracting and multiplying polynomials

Warm-up

Three quick questions from earlier lessons.

Discover

A box made from a 20 cm by 30 cm sheet, with squares of side xx cut from the corners, has volume x(20−2x)(30−2x)x(20 - 2x)(30 - 2x). Multiplied out, what is the highest power of xx?

  1. x3x^3
  2. x2x^2

Key idea

Polynomials combine like numbers.

To add or subtract, combine like terms (same power of xx). To multiply, multiply every term of one by every term of the other, then combine like terms. The degree of a product is the sum of the degrees.

Worked example

(x+3)(x2−2x+5)(x + 3)(x^2 - 2x + 5).

x3−2x2+5xx^3 - 2x^2 + 5x
x times each term.
3x2−6x+153x^2 - 6x + 15
3 times each term.
x3+x2−x+15x^3 + x^2 - x + 15
Combine like terms.

Your turn, guided

Your turn: the coefficient of xx in (x+4)(x+5)(x + 4)(x + 5).

99
5x + 4x.

Practice ★

  1. Expand (x+6)(x+9)(x + 6)(x + 9). What is the coefficient of xx?
  2. Expand (x+1)(x+5)(x + 1)(x + 5). What is the coefficient of xx?
  3. Expand (x+7)(x+3)(x + 7)(x + 3). What is the coefficient of xx?

Watch out!

Practice ★★

  1. What is the constant term of (x+9)(x2+1x+6)(x + 9)(x^2 + 1x + 6)?
  2. What is the constant term of (x+7)(x2+7x+3)(x + 7)(x^2 + 7x + 3)?
  1. A polynomial of degree 5 is multiplied by a polynomial of degree 5. What is the degree of the product?

Exit ticket

Three questions to check you have got it.

  1. Expand (x+6)(x+9)(x + 6)(x + 9). What is the coefficient of xx?
  2. Expand (x+1)(x+5)(x + 1)(x + 5). What is the coefficient of xx?
  3. Expand (x+7)(x+3)(x + 7)(x + 3). What is the coefficient of xx?

Summary card

Polynomial arithmetic

  • Add: combine like terms.
  • Multiply: every term by every term; degrees add.

Watch out: xx-terms add; constants multiply.

Answers: Extending the number system

1. Rational exponents in exponential functions

A

  1. 77767776
  2. 729729
  3. 5904959049
  4. 88
  5. 1680716807

B

  1. 125\frac{1}{25}
  2. 15\frac{1}{5}
  3. 14\frac{1}{4}
  4. 12\frac{1}{2}
  5. 13\frac{1}{3}

C

  1. 125\frac{1}{25}
  2. 15\frac{1}{5}
  3. 14\frac{1}{4}

2. Rational and irrational numbers; repeating decimals

A

  1. 2\sqrt{ 2 }
  2. 10\sqrt{ 10 }

B

  1. 19\frac{1}{9}

C

  1. 1633\frac{16}{33}
  2. 7099\frac{70}{99}

D

  1. 19\frac{1}{9}
  2. 29\frac{2}{9}
  3. 89\frac{8}{9}

3. The number i and complex numbers

A

  1. 11i11 i
  2. 2i2 i

B

  1. (9i)2(9i)^2

C

  1. −1-1
  2. ii
  3. ii

D

  1. 11i11 i
  2. 2i2 i
  3. 5i5 i

4. Adding, subtracting and multiplying complex numbers

A

  1. 11+9i11 + 9 i
  2. 12+6i12 + 6 i

B

  1. 6868
  2. 5353

C

  1. −27+36i-27 + 36 i
  2. −35+12i-35 + 12 i

D

  1. 6868
  2. 5353
  3. 8080

5. Adding, subtracting and multiplying polynomials

A

  1. 1515
  2. 66
  3. 1010

B

  1. 5454
  2. 2121

C

  1. 1010

D

  1. 1515
  2. 66
  3. 1010