Surds, indices, logarithms (as required) - Additional Mathematics IGCSE Study Notes

Let's break down these fancy words into simple ideas:

• Indices (or Powers): Think of indices like a special 'count how many times' button on a calculator. When you see something like 2³ (read as '2 to the power of 3' or '2 cubed'), it just means you multiply the number by itself that many times. So, 2³ is 2 × 2 × 2 = 8. The '3' is the index (or power), and the '2' is the base (the number being multiplied).

  • Analogy: Imagine you have a magic duplicating machine. If you put one toy car in and set the 'power' to 2, it makes 2 × 2 = 4 cars. If you set it to 3, it makes 2 × 2 × 2 = 8 cars. The power tells you how many times the 'duplication' process happens.

• Surds: These are numbers that involve a square root (like √ ) or a cube root (like ³√ ) that can't be simplified into a neat whole number or a simple fraction. For example, √4 is 2 (a whole number), so it's NOT a surd. But √2 is approximately 1.41421356... and it goes on forever without repeating. That's a surd! We keep it as √2 to be perfectly accurate.

  • Analogy: Think of surds as a perfectly wrapped present. You don't open it and try to guess what's inside (which would be like rounding √2). You keep it wrapped (as √2) until you absolutely need to use its approximate value.

• Logarithms (Logs): These are like asking a question: "What power do I need to raise this base number to, to get this other number?" For example, if you see log₂8, it's asking: "What power do I raise 2 to, to get 8?" Since 2³ = 8, then log₂8 = 3.

  • Analogy: Imagine you have a secret code. The base is your secret key, the answer you want is the locked message, and the logarithm tells you the number of times you need to 'turn' the key to open it.

Let's look at how indices help us with something super common: compound interest (how money grows in a bank).

Imagine your rich aunt gives you $100 for your birthday, and you put it in a special bank account that gives you 5% interest every year. This means your money grows by 5% each year.

• Year 1: You start with $100. After one year, you have $100 + (5% of $100) = $100 + $5 = $105. Or, more simply, $100 × 1.05.
• Year 2: Now you have $105. After another year, you get 5% of $105. So, $105 × 1.05 = $110.25. Notice this is the same as $100 × 1.05 × 1.05.
• Year 3: You'd have $110.25 × 1.05 = $115.76. This is the same as $100 × 1.05 × 1.05 × 1.05.

See the pattern? If you want to know how much money you'll have after 10 years, you don't have to do it year by year! You can use an index:

Amount after 10 years = $100 × (1.05)¹⁰

The '¹⁰' is the index, telling you to multiply 1.05 by itself 10 times. This is a much faster way to calculate growth! This is why indices are so powerful for financial calculations, population growth, and even how quickly a virus spreads.