Number Bases

Converting to and from Base 10

Primary Seven Mathematics — Term I, Topic 2

What You Will Learn Today

By the end of this lesson you will be able to:

1. Explain what a number base is and read subscript notation
2. Convert base 5, 8, or 2 to base 10 using expand-and-add
3. Convert base 10 to base 5, 8, or 2 by repeated division

Prior Knowledge: Place Value as Powers

Each column value is a power of the base (10).
This structure works in any base.

What Is a Number Base?

Base 10: 10 fingers — digits 0 through 9

Base 5: one hand — digits 0 through 4

Base 2 (binary): computers use ON/OFF — digits 0 and 1

Every phone and server runs on base 2.

Column Values as Powers of the Base

In base , columns are (right to left).

Valid digits in base b: 0 through b−1 only.

Base 5: Reading the Column Values

Valid digits in base 5: {0, 1, 2, 3, 4} only

Base 2 (Binary): Column Values

Powers of 2 — each column doubles the previous:

Valid digits: {0, 1} — called bits. 8 bits = 1 byte.

Reading Subscript Notation for Bases

Always write the subscript — else a number is assumed base 10.

Converting to Base 10: Expand-and-Add

Method — three steps:

1. Label each column with its power value (right to left: )
2. Multiply each digit by its column value
3. Add all products

Write every digit — including zeros — in the expansion.

Example 1: Base 10

Example 2: Base 10

Base 8 column values: …64, 8, 1

Example 3: Base 10

Column labels:

The zero term still holds its column.

Why We Must Include Zero Terms

Always write every term. Drop zeros from the sum, never the expansion.

Practise Now: Convert to Base 10

Convert to base 10 — show labels, expansion, sum:

a)   b)   c)

Answers: ;   ;  

Converting from Base 10: Repeated Division

Method — keep dividing until quotient = 0:

Divide by the target base. Record the remainder.
Divide the quotient by the base. Record the remainder.
Repeat until quotient = 0.
Read remainders from bottom to top.

The last remainder is the highest-place digit.

Converting 82 to Base 5

Read remainders bottom to top: 3, 1, 2 →

Verify, Then Convert 37 to Base 8

Verify 312₅: ✓

Convert 37 to base 8:

Bottom to top → . Verify: ✓

Converting 25 to Base 2

Result: Reading 25 as

Read remainders bottom to top: 1, 1, 0, 0, 1 →

Verify using column values 16, 8, 4, 2, 1:

Practise Now: Repeated Division Conversions

Convert using repeated division, then verify:

a) base 5
b) base 2

Answers: (verify );   (verify )

Misconception 1: Using an Invalid Digit

Rule: In base , digits are 0 through .

Base 2 → {0, 1}   Base 5 → {0–4}   Base 8 → {0–7}

Error: — digit 8 is invalid in base 5.
Fix: A remainder ≥ the base means a division error.

Misconceptions 2 and 3: Order and Zeros

Error 2 — Reading top-to-bottom:

remainders 2, 1, 3.
Wrong:   Correct: ← bottom to top

Error 3 — Skipping zero terms:

Summary: The Two Conversion Directions

Always verify by converting back.

Next: Binary Arithmetic (whole-numbers-04)