Q: What does IP Maths Notes (Lower Sec, Year 1-2): 01) Number Systems & Indices cover? A: Master integer, fractional, and decimal conversions, index laws, and significant figure rounding for Year 1 IP maths.
The core idea is simple: Number sense prevents algebra mistakes later.
Use it as a working check: Convert fractions, decimals, and percentages fluently, then apply index laws step by step. Keep bases consistent before combining powers.
Then go one layer deeper: Work through the examples to practise conversion, ordering, negative indices, fractional indices, surds, and rounding without relying on calculator prompts.
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.
These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).
Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.
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.
Choosing the first move
Mixed number questions become easier when you identify the structure before calculating. Use this map to choose the first tool, then show one clean line of working for that choice.
Question cue
First move
What to check
Fraction, decimal, or percentage conversion
Move everything into one form first.
Simplify fractions and keep recurring decimals clearly marked.
Rewrite each base as a power of the same prime number.
Apply index laws only after the bases match.
Negative index
Move the factor across the fraction line.
The base changes position; the sign of the value does not automatically change.
Fractional index
Treat the denominator of the index as a root.
Use brackets so 1643 is handled as a fourth-root power, not a decimal estimate.
Standard form
Place the decimal after the first non-zero digit.
Count the number of places moved and keep the sign of the power consistent.
Significant figures
Find the first non-zero digit, then count from there.
Look at the next digit before rounding.
Surd in the denominator
Multiply by the matching surd or conjugate.
The final denominator should no longer contain a surd.
Common trap: A negative index means reciprocal, not a negative answer. For example, 5−2=251, not −25.
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 value=base×100percentage.
Example: Convert 0.375 into fraction and percentage.
0.375=1000375=83,0.375=37.5%.
1.2 Ordering rational and irrational numbers
Place numbers on a single number line by converting to decimals. Surds such as 5≈2.236 give context for comparisons against rationals like 49=2.25.
2 Index laws
The five core laws hold for any non-zero base a and integers m,n:
Product: am×an=am+n
Quotient: anam=am−n
Power of a power: (am)n=amn
Zero index: a0=1
Negative index: a−n=an1
Extend to fractional indices via roots: anm=(am)n1.
Same-base index checkpoint
Index laws work only after the bases match. Before adding, subtracting, or multiplying powers, pause and rewrite each base as a power of the same prime or simple base.
Expression type
First rewrite
Law to use after rewriting
Common trap
82×43
(23)2×(22)3
Power of a power, then product law.
Adding exponents before making both bases 2.
93272
(32)3(33)2
1643
(24)43
5−2
521
Negative index means reciprocal.
Making the answer negative.
Worked check:82×43=(23)2(22)3=26×26=212. The bases had to become 2 before the product law could be used.
Scientific notation compresses large/small numbers: N=a×10k where 1≤a<10 and k∈Z.
To convert 45,600 into standard form: 4.56×104.
To convert 0.0032: 3.2×10−3.
Standard-form direction checkpoint
When converting to standard form, track whether the original number is large or small before deciding the sign of the power of ten.
Original number
Decimal move to make 1≤a<10
Power of ten
Common trap
45600
Move decimal 4 places left to get 4.56.
104
Writing 10−4 because the decimal moved left.
0.0032
Move decimal 3 places right to get 3.2.
10−3
Writing 103
7.8
Decimal is already after the first non-zero digit.
100
Forcing a power of ten when standard form is already reached.
Worked check: 0.00054 becomes 5.4 after moving the decimal 4 places right. Because the original number is smaller than 1, the standard form is 5.4×10−4.
Misconception check: the direction you move the decimal is not copied directly into the sign. The power of ten must make the new number equal to the original number.
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.
Rationalising denominator checkpoint
Choose the multiplier from the whole denominator, not from the first surd you see.
Denominator type
Multiplier to use
Why it works
Common trap
Single surd, such as 3
Multiply top and bottom by the same surd.
3×3=3.
Multiplying only the denominator and changing the value of the fraction.
Whole-number multiple, such as 25
Multiply top and bottom by 5
Binomial surd, such as 2−5
Multiply top and bottom by the conjugate 2+5
Worked check: to rationalise 2−54, multiply by the conjugate:
2−54×2+52+5=4−54(2+5)=−8−45.
Misconception check: the conjugate belongs to a two-term denominator. For a single 3 denominator, there is no plus or minus sign to flip.
Example: Rationalise 35.
35×33=353.
For binomial surds, multiply by the conjugate: 3−21×3+23+2.
Practice Quiz
Test your conversion fluency, index-law manipulations, and surd rationalisation with the interactive drill below.
