IP Maths Notes (Lower Sec, Year 1-2): 01) Number Systems & Indices

Number sense underpins every algebraic manipulation you will meet from Year 2 onward. This guide cleans up conversion fluency, index laws, surds, and estimation so you never lose calculator-free marks.

Learning targets

• Convert between fractions, decimals, and percentages without a calculator.
• Apply the five index laws, including negative and fractional indices.
• Standardise answers to the correct number of significant figures.
• Present surds in rationalised form.

1. Number systems refresher

1.1 Fraction-decimal-percentage triad

• Fractions express parts of a whole. Simplify by dividing numerator and denominator by their highest common factor.
• Decimals express fractions in base-ten. For terminating decimals, use place value; for recurring decimals, note the repeating block.
• Percentages compare against 100. Use the identity \( \text{value} = \text{base} \times \frac{\text{percentage}}{100} \).

Example: Convert \( 0.375 \) into fraction and percentage.

\[ 0.375 = \frac{375}{1000} = \frac{3}{8}, \quad 0.375 = 37.5\%. \]

1.2 Ordering rational and irrational numbers

Place numbers on a single number line by converting to decimals. Surds such as \( \sqrt{5} \approx 2.236 \) give context for comparisons against rationals like \( \frac{9}{4} = 2.25 \).

2. Index laws

The five core laws hold for any non-zero base \( a \) and integers \( m, n \):

1. Product: \( a^m \times a^n = a^{m+n} \)
2. Quotient: \( \frac{a^m}{a^n} = a^{m-n} \)
3. Power of a power: \( (a^m)^n = a^{mn} \)
4. Zero index: \( a^0 = 1 \)
5. Negative index: \( a^{-n} = \frac{1}{a^n} \)

Extend to fractional indices via roots: \( a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} \).

Worked example 1 — Mixed indices

Simplify \( \frac{27^{-2} b^3}{9^{-1} b^{-2}} \).

\[ \begin{aligned} \frac{27^{-2} b^3}{9^{-1} b^{-2}} &= 27^{-2} \times 9^{1} \times b^{3 - (-2)} \\ &= (3^3)^{-2} \times (3^2)^{1} \times b^{5} \\ &= 3^{-6} \times 3^{2} \times b^{5} \\ &= 3^{-4} b^{5} = \frac{b^{5}}{3^{4}} = \frac{b^{5}}{81}. \end{aligned} \]

Worked example 2 — Fractional indices & surds

Simplify \( 16^{\frac{3}{4}} \times 4^{-\frac{1}{2}} \).

\[ 16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{3} = 8, \quad 4^{-\frac{1}{2}} = (2^2)^{-\frac{1}{2}} = 2^{-1} = \frac{1}{2}. \]

\[ 16^{\frac{3}{4}} \times 4^{-\frac{1}{2}} = 8 \times \frac{1}{2} = 4. \]

3. Standard form & significant figures

Scientific notation compresses large/small numbers: \( N = a \times 10^{k} \) where \( 1 \leq a < 10 \) and \( k \in \mathbb{Z} \).

• To convert \( 45,600 \) into standard form: \( 4.56 \times 10^{4} \).
• To convert \( 0.0032 \): \( 3.2 \times 10^{-3} \).

Significant figures checklist

• First non-zero digit counts as the first significant figure.
• Zeros between non-zero digits count, trailing zeros only count if the number has a decimal point.
• When rounding, look at the next digit to decide whether to bump up.

Example: Round \( 0.007864 \) to 2 significant figures → \( 0.0079 \).

4. Rationalising simple surds

Express denominators without surds to streamline later algebra.

Example: Rationalise \( \frac{5}{\sqrt{3}} \).

\[ \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}. \]

For binomial surds, multiply by the conjugate: \( \frac{1}{\sqrt{3} - 2} \times \frac{\sqrt{3} + 2}{\sqrt{3} + 2} \).

Try it yourself

1. Evaluate \( 125^{\frac{2}{3}} \times 5^{-1} \).
2. Convert \( \frac{7}{24} \) into recurring decimal form.
3. Simplify \( \frac{3a^{-2} b^4}{9a b^{-3}} \).
4. Rationalise \( \frac{4}{2 - \sqrt{5}} \) and simplify.

Check your answers against the worked solutions at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-01-Number-Systems-and-Indices after attempting each question.

Exam watchpoints

• Always state the number of significant figures used when summarising data.
• Separate the sign from the power of ten: write \( 3.2 \times 10^{-4} \), not \( 3.2 \times -10^{-4} \).
• When rationalising, ensure the denominator multiplies out exactly to an integer.

Next: proceed to https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-02-Algebraic-Expressions-and-Equations for algebra workflows.