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Primary Mathematics: Number Bases, Place-Value Codes and Numeral Systems | Worked Learning Guide

Our usual decimal notation is powerful, but it is not inevitable. Changing the base reveals what place value is actually doing.

This guide is Primary Mathematics enrichment. It uses alternative number bases and coding systems to deepen understanding of place value, regrouping, representation and arithmetic. Learners should already be comfortable with ordinary base-ten place value. Formal computer science is not required.

Wait, What? “10” Does Not Always Mean Ten

In ordinary decimal notation, 10 means one ten and zero ones because the base is ten. In base five, the written numeral 10 means one group of five and zero ones, so its decimal value is five. The digits are interpreted by their place-value system.

This is not wordplay. It exposes the core idea of positional notation: a digit’s value depends on both the digit and its position.

1. Base Ten Revisited

In 4,372:

• 4 is in the thousands place: 4×1000
• 3 is in the hundreds place: 3×100
• 7 is in the tens place: 7×10
• 2 is in the ones place: 2×1

Place values are powers of ten: 1, 10, 100, 1000,…

2. Base Five

Base five uses place values 1,5,25,125,… and digits 0,1,2,3,4. A digit 5 is not allowed because five ones regroup as one five.

Worked Example 1: Read 243₅

243₅ = 2×25 + 4×5 + 3 = 50+20+3 = 73 in base ten.

Worked Example 2: Read 101₅

101₅ = 1×25 + 0×5 + 1 = 26.

3. Convert Base Ten to Base Five by Grouping

Worked Example 3: Convert 18 to Base Five

18 contains 3 groups of 5 with 3 ones remaining.

Therefore 18 = 33₅.

Worked Example 4: Convert 73 to Base Five

Largest place not exceeding 73 is 25.

73 = 2×25 + 23.

23 = 4×5 + 3.

So 73 = 243₅.

4. Base Two

Base two uses only digits 0 and 1. Place values are 1,2,4,8,16,32,…

Worked Example 5: Read 1011₂

1011₂ = 1×8 + 0×4 + 1×2 + 1 = 11.

Worked Example 6: Convert 13 to Base Two

13 = 8 + 4 + 1.

So the 8,4,2,1 places are 1,1,0,1.

13 = 1101₂.

5. Why Binary Is a Useful Place-Value Laboratory

Every position is either used or not used. This makes decomposition very visible. 101101₂ means 32+8+4+1=45.

A learner who understands this is using place value, powers of two and additive decomposition—not memorising a code.

6. Regrouping Depends on the Base

In base ten, ten ones regroup as one ten. In base five, five ones regroup as one five. In base two, two ones regroup as one two.

Worked Example 7: Add in Base Five

23₅ + 14₅.

Ones: 3+4=7 decimal = 1 group of five and 2 ones. Write 2, carry 1.

Fives: 2+1+1=4.

23₅ + 14₅ = 42₅.

Check in decimal: 13+9=22; 42₅=4×5+2=22.

Worked Example 8: Add in Base Two

101₂ + 11₂.

Decimal check: 5+3=8.

8 in binary is 1000₂.

7. Subtraction in Another Base

Worked Example 9: Base Five Subtraction

42₅ − 13₅.

Decimal values: 22−8=14, which is 24₅.

Direct regrouping reaches the same answer: borrow one five as five ones.

42₅−13₅=24₅.

8. Invalid Numerals

Each base permits digits from 0 up to one less than the base.

• Base 2: only 0,1.
• Base 5: 0,1,2,3,4.
• Base 8: 0 through 7.

Worked Example 10: Is 352₅ Valid?

No. Digit 5 is not allowed in base five.

Worked Example 11: Is 1201₂ Valid?

No. Digit 2 is not allowed in base two.

9. Base Eight

Base eight uses place values 1,8,64,512,… and digits 0–7.

Worked Example 12: Read 157₈

1×64 + 5×8 + 7 = 64+40+7 = 111.

Worked Example 13: Convert 100 Decimal to Base Eight

100 = 1×64 + 36.

36 = 4×8 + 4.

So 100 = 144₈.

10. Compare Numerals Across Bases

Worked Example 14

Compare 100₅ and 100₂.

100₅ = 25 decimal.

100₂ = 4 decimal.

Identical written digits can represent very different values because the place-value bases differ.

11. Base Conversion Through Decimal as a Bridge

For Primary enrichment, the safest route between two unfamiliar bases is often:

1. Convert the original numeral to decimal value.
2. Regroup that decimal value into the target base.

Worked Example 15: 34₅ to Base Two

34₅ = 3×5+4 = 19 decimal.

19 = 16+2+1 = 10011₂.

12. Place-Value Codes

A code can assign symbols to place values. For example, suppose ▲=25, ■=5 and ●=1. Then ▲▲■■●●● represents 2×25+2×5+3=63.

This is not positional notation if symbols themselves carry fixed values, but it still teaches decomposition and regrouping.

Worked Example 16: Decode a Symbol System

Two ▲, three ■ and four ● represent 2×25+3×5+4 = 69.

13. Why Positional Systems Are Efficient

In a positional system, the same small set of digits can represent arbitrarily large numbers by position. Roman-style additive symbols require more explicit symbols or subtraction conventions. Place value compresses repeated grouping.

Worked Example 17: Compare Representations

Decimal 64 can be written 1000000₂, 224₅ or 100₈. The numeral lengths differ because each base packages quantities into different group sizes.

14. Last Digit and Remainders

In base ten, the last digit gives the remainder when dividing by 10. In base five, the last digit gives the remainder when dividing by 5.

Worked Example 18

243₅ ends in 3, so its decimal value leaves remainder 3 when divided by 5. Indeed 73 ÷ 5 = 14 remainder 3.

15. Trailing Zeros

In base ten, multiplying a whole number by 10 shifts it one place left and appends a zero. In base five, multiplying by five appends a base-five zero. In base two, multiplying by two appends a binary zero.

Worked Example 19

23₅ × 5(decimal) = 230₅.

13 decimal ×5=65 decimal; 230₅=2×25+3×5=65.

16. Fractions of a Base Unit

Alternative bases can also extend to the right of a point, but this is advanced enrichment. In base two, 0.1₂ means one-half because the first fractional place is 1/2. 0.01₂ means one-quarter.

Worked Example 20

0.11₂ = 1/2 + 1/4 = 3/4.

This reveals that decimal tenths and hundredths are specifically base-ten fractional place values.

17. Arithmetic Patterns Depend on Representation

In decimal, 9+1=10. In base five, 4₅+1₅=10₅. In binary, 1₂+1₂=10₂. The written “rollover” occurs when the digit reaches the base.

Worked Example 21

44₅ + 1₅ = 100₅, because 44₅ is 24 decimal and one more is 25 decimal.

18. Common Errors

• Assuming “10” always means ten.
• Using a digit that is not allowed in the base.
• Using decimal place values when reading another base.
• Regrouping after ten units instead of after the stated base.
• Comparing numerals from different bases without converting value.
• Forgetting that trailing-zero scaling depends on the base.
• Treating a symbol-value code as though it were positional.
• Using advanced fractional-base notation without understanding ordinary fractions first.

19. Error Repair

Error A: 100₂ = 100

Binary place values are 4,2,1. Therefore 100₂=4.

Error B: 25₅ Is a Valid Base-Five Numeral

Digit 5 is forbidden in base five. Decimal 15 would be written 30₅, not 25₅.

Error C: 44₅ + 1 = 45₅

Base five cannot use digit 5. Five ones regroup to one five, and five fives regroup to one twenty-five: 100₅.

20. Practice: 24 Original Questions

Questions 1–8: Read and Validate

1. Convert 243₅ to decimal.
2. Convert 101₅ to decimal.
3. Convert 1011₂ to decimal.
4. Convert 157₈ to decimal.
5. Is 352₅ valid? Explain.
6. Is 1201₂ valid? Explain.
7. Convert 100₅ to decimal.
8. Convert 100₂ to decimal.

Questions 9–16: Convert and Calculate

9. Convert decimal 18 to base five.
10. Convert decimal 73 to base five.
11. Convert decimal 13 to binary.
12. Convert decimal 100 to base eight.
13. Calculate 23₅+14₅.
14. Calculate 42₅−13₅.
15. Calculate 101₂+11₂.
16. Convert 34₅ to binary.

Questions 17–24: Structure and Transfer

17. What remainder does 243₅ leave when divided by 5?
18. Write decimal 64 in base two.
19. Write decimal 64 in base eight.
20. Multiply 23₅ by 5 decimal and write the result in base five.
21. What decimal fraction is 0.1₂?
22. What decimal/fractional value is 0.11₂?
23. Explain why 4₅+1₅=10₅.
24. Compare 100₅ and 1111₂. Which represents the larger decimal value?

21. Worked Answers

1. 73. 2. 26. 3. 11. 4. 111. 5. No; digit 5 is invalid. 6. No; digit 2 is invalid. 7. 25. 8. 4.

9. 33₅. 10. 243₅. 11. 1101₂. 12. 144₈. 13. 42₅. 14. 24₅. 15. 1000₂. 16. 10011₂.

17. 3. 18. 1000000₂. 19. 100₈. 20. 230₅. 21. 1/2. 22. 3/4. 23. Five ones form one group of five, so the representation rolls over to 10₅. 24. 100₅=25; 1111₂=15, so 100₅ is larger.

22. Checking Ladder

• What base is being used?
• Which digits are allowed?
• What are the place values?
• Does regrouping happen after the correct number of units?
• Can I convert to decimal to check?
• Does the result’s size make sense?
• Am I comparing values or merely written digit strings?

23. Transfer Test

Design a base-four place-value system. List the first four place values, write decimal values 1–20 in base four, explain why digit 4 is forbidden, and calculate 23₄+12₄. Then create a symbol-value code using values 16,4 and1 and compare its strengths and weaknesses with positional base four.

24. Delayed Return

Three days later, read one base-five numeral, one binary numeral and one base-eight numeral, convert one decimal number into each of those bases, and explain the regrouping rule without looking at the examples.

25. Parent and Tutor Guide

Use physical bundling first. For base five, bundle five counters into one larger unit; bundle five of those into a twenty-five unit. Let the child experience the regrouping before writing 10₅ or 100₅. The purpose is to deepen ordinary place value, not to turn notation into a memory test.

26. Mastery Receipt

• I understand that numeral value depends on base and position.
• I identify allowed digits in a base.
• I convert simple numerals between base and decimal.
• I regroup according to the stated base.
• I add and subtract simple alternative-base numerals.
• I use decimal conversion as an independent check.
• I connect trailing zeros to multiplication by the base.
• I see base ten as one positional system rather than the only possible one.

Official Reference Route

Singapore Ministry of Education — Primary Mathematics Syllabus P1–P6, updated October 2025

This guide is enrichment. Alternative-base notation is used to illuminate place value and is not presented as a universal Primary syllabus requirement.

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The Quiet Return

Alternative bases make the hidden machinery of decimal notation visible: grouping, place value, allowed digits and regrouping. Once that machinery is understood, ordinary arithmetic becomes less mysterious.