IB Number and Algebra

Natural Numbers (N):

• The set of positive counting numbers.

Examples: 1, 2, 3, 4, 5, ...

• Used for discrete quantities.

Integers (Z):

• The set of all whole numbers, including negatives and zero

Examples: -2, -1, 0, 1, 2, ...

• Includes natural numbers and their negatives.

Rational Numbers (Q):

• The set of all numbers that can be expressed as fractions a/b.

Examples: ½, 0.75, -3.1

Irrational Numbers:

• The set of all numbers that are not expressible as fractions. Their decimal expansions are non-terminating and non-repeating.

Examples: sqrt(2), π, e

Real Numbers (R):

• The set of all rational and irrational numbers.

• Represents every point on the number line.

Rounding Up

• Numbers that are greater than 4 should be rounded up.

• Numbers that are less than or equal to 4 should be rounded down.

• Examples: 4.5 will be rounded up to 5.

      4.1 will be rounded down to 4. 

Significant Figures (s.f.) 

• Non-zero digits are always significant.

  • Example: 123.45 has 5 sig figs.

• Zeros between nonzero digits are significant.

  • Example: 1002 has 4 sig figs.

• Leading zeros (zeros before the first non-zero digit) are not significant.

  • Example: 0.0045 has 2 sig figs.

• Trailing zeros in a number with a decimal point are significant.

  • Example: 45.600 has 5 sig figs.

• Trailing zeros in a whole number without a decimal point are not significant unless specified by a bar or underline.

  • Example: 1500 has 2 sig figs (unless written as 1500̲, which indicates 4 sig figs).

• Default: 3 s.f.

Operations with s.f.:

• Addition/Subtraction:

  • The result should have the same number of decimal places as the number with the fewest decimal places.

  • Example: 12.11 + 0.3 = 12.4 (1 decimal place).

• Multiplication/Division:

  • The result should have the same number of sig figs as the number with the fewest sig figs.

  • Example: 4.56 × 1.4 = 6.4 (2 sig figs).

Scientific Notation/Standard Index Form:

a*10^k, 1 ≤ a < 10, k is integer

• Calculator format: 5.678112E14 = 5.67 * 10¹⁴ (3 s.f.)

• Ex: 167400 = 1.67 * 10⁵

Percentage Error:

Accuracy of an approximate value compared to the exact value, expressed as a percentage.

Formula: | (V_A-V_E)/V_E | * 100%

V_A - Approximate value

V_E - Exact value

Arithmetic Sequences:

A sequence with a constant difference (d) between consecutive terms.

Formula: U_n=U₁+d(n-1)

U₁: First term of sequence

d: common difference (U₂-U₁, U₃-U₂, etc)

U_n: n^th term of the sequence

• Sum formula: S_n=n*(U₁+U_n)/2

  • n is the position of the term you’re summing up to

Geometric Sequences:

A sequence with a constant ratio (r) between consecutive terms.

Formula: U_n=U₁*r^(n-1)

U₁: First term of sequence

r: The common ratio between the terms (U₂/U₁, U₃/U₂, etc.)

U_n: The n^th term

• Finite sum formula: S_n=U₁*(1-rⁿ)/(1-r)

  • n is the position of the term you’re summing up to

• Infinite sum formula: S=U₁/(1-r)

  • Converges if |r| < 1

Sigma Notation:

A mathematical way to represent the sum of a sequence of terms in a compact form using the Greek letter Σ (sigma).

Σ f(k), from k = a to b
where:

• Σ represents summation

• k is the index of summation (dummy variable)

• a is the lower limit (starting value of k)

• b is the upper limit (ending value of k)

• f(k) is the function being summed

Examples:

1. Sum of first 4 squares:
   Σ k², from k = 1 to 4
   = 1² + 2² + 3² + 4²
   = 1 + 4 + 9 + 16
   = 30

2. Sum of an arithmetic sequence:
   Find Σ (2k + 3), from k = 1 to 5
   2(1)+3=5, 2(2)+3=7,...
   5,7,...
   U₁=5, d=2
   S₅= 5*(U₁+[U₁+d(n-1)])/2
       = 5*(5+[5+2(5-1)])/2
       = 5*18/2
       = 45

Exponents and Logarithms:

Exponent Rules:

• a^m * aⁿ = a^m+n

• a^m/aⁿ = a^m-n

• (a^m)ⁿ = a^mn

• a⁰ = 1

• a^-n = 1/aⁿ

Logarithms:

• If a^b=c, then log_ac=b

• Rules:

  • log_a(mn) = log_am + log_an

  • log_a(m/n) = log_am - log_an

  • log_a(mⁿ) = nlog_am

  • Change of base: log_ab = (log_cb)/(log_ca)

• Natural log is when the base is e 

  • If e^a = b, then ln(b) = a
    
    

• Common log is when the base is 10

  • If 10^a=b, then log(b)=a

Binomial Theorem and Pascal's Triangle

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)ⁿ without direct multiplication.

Formula:

(a + b)ⁿ = Σ [ C(n, k) a⁽ⁿ⁻ᵏ⁾ bᵏ ], from k = 0 to n

where:

• n is the exponent

• k is the term index (starting from 0)

• a and b are the terms in the binomial

• C(n, k) (binomial coefficient) is given by:
  C(n, k) = n! / [k!(n - k)!], which represents the number of ways to choose k elements from n.

Example Expansion:

Expanding (x + y)³:

(x + y)³ = C(3,0) x³ + C(3,1) x²y + C(3,2) xy² + C(3,3) y³

= 1x³ + 3x²y + 3xy² + 1y³

= x³ + 3x²y + 3xy² + y³

Pascal’s Triangle

Pascal’s Triangle is a triangular array where each row contains the binomial coefficients for a given exponent n.

Construction Rules:

1. Start with 1 at the top.

2. Each row begins and ends with 1.

3. Each middle term is the sum of the two terms directly above it.

Pascal’s Triangle Structure:

      1

      1 1

     1 2 1

    1 3 3 1

   1 4 6 4 1

  1 5 10 10 5 1

• Row n gives coefficients for (a + b)ⁿ.

• The nᵗʰ row contains n + 1 elements.

Connection to Binomial Theorem

Each coefficient in the expansion (a + b)ⁿ corresponds to a number in Pascal's Triangle.

Example for (x + y)⁴:

• Pascal's Triangle row: 1 4 6 4 1

• Expansion:
  x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Compound Interest and Depreciation

Compound Interest

Compound interest is the interest calculated on both the initial principal and the accumulated interest from previous periods.

Formula:

A = P(1 + r/n)^(nt)

Where:

• A = Final amount after interest

• P = Initial principal (starting amount)

• r = Annual interest rate (decimal form)

• n = Number of times interest is compounded per year

• t = Number of years

Example:

If you invest $1,000 at an annual interest rate of 5%, compounded quarterly for 3 years, the final amount is:

A = 1000(1 + 0.05/4)^(4×3)
A = 1000(1.0125)^12
A ≈ 1157.63

After 3 years, the investment grows to $1,157.63.

Depreciation

Depreciation is the reduction in the value of an asset over time, often due to wear and tear or obsolescence.

Formula:

A = P(1 - r)ⁿ

Where:

• A = Final value after depreciation

• P = Initial value of the asset

• r = Depreciation rate (decimal form)

• n = Number of years

Example:

A car worth $20,000 depreciates by 15% per year. Its value after 3 years is:

A = 20000(1 - 0.15)³
A = 20000(0.85)³
A ≈ 12262.50

After 3 years, the car’s value drops to $12,262.50.

Loans and Annuities

Make/receive payments along the way

Loan - Borrow money to be paid back at regular intervals

Savings annuity - save money and make regular contributions

Gift annuity - receive a gift that is paid out at regular intervals

Use Finance app on calculator:

n - The number of payments made

I% - annual interest rate

PV - amt of money involved at the start

PMT - regular payment

FV - amt of money involved at the end

P/Y - frequency, per year, which payment is made

C/Y - frequency, per year, which interest is calculated