LEARN WITH ALICIA · SENIOR 1 · OFFLINE COMPANION

Number Bases

One number can be written in different number languages. This guide helps you understand what a base means before you begin converting.

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THE SITUATION

A scoreboard with a different counting rule

A school club uses a scoreboard that can show only the digits 0, 1, 2, 3 and 4. After 4, it must regroup. The screen shows 1325 points. The teacher asks: How many points is that in our usual base-ten system?

Your job: Use place value to explain what 1325 means, then convert it to base ten.

1. What does “base” mean?

A base tells us two things:

Base ten

Uses digits 0–9. The places are powers of 10:

… 1000, 100, 10, 1

Base five

Uses digits 0–4. The places are powers of 5:

… 125, 25, 5, 1

Quick check: A base-five number can never contain the digit 5, 6, 7, 8 or 9. So 2055 is not a valid base-five number.

2. Convert another base to base ten

Write the place values, multiply each digit by its place value, then add.

Base-five number5² = 255¹ = 55⁰ = 1
13251 × 253 × 52 × 1

1325 = 25 + 15 + 2 = 4210

Answer to the situation: the scoreboard shows 42 points in base ten.

Worked example: base two

Convert 10112 to base ten.

(1×8) + (0×4) + (1×2) + (1×1)
= 8 + 0 + 2 + 1
= 1110

Worked example: base eight

Convert 378 to base ten.

(3×8) + (7×1)
= 24 + 7
= 3110

3. Convert base ten to another base

Use repeated division. Divide by the new base, keep each remainder, then read the remainders from bottom to top.

Worked example: convert 1310 to base two

  1. 13 ÷ 2 = 6 remainder 1
  2. 6 ÷ 2 = 3 remainder 0
  3. 3 ÷ 2 = 1 remainder 1
  4. 1 ÷ 2 = 0 remainder 1

Read upwards: 1310 = 11012

Check: 8 + 4 + 0 + 1 = 13.

4. Your turn

  1. List all the digits allowed in base three.
  2. Is 1434 a valid base-four number? Explain.
  3. Convert 101012 to base ten.
  4. Convert 2435 to base ten.
  5. Convert 528 to base ten.
  6. Convert 1910 to base two.
  7. Convert 3810 to base five.
  8. A store labels a box 1245. How many items are in the box in base ten? Show the place values.

ANSWER CHECK

Check after you have tried

  1. 0, 1 and 2.
  2. No. Base four allows only 0, 1, 2 and 3.
  3. 2110.
  4. 7310.
  5. 4210.
  6. 100112.
  7. 1235.
  8. (1×25) + (2×5) + (4×1) = 39 items.

TEACHER OR STUDY PARTNER

Questions that reveal understanding