A Story of Numbers · Lesson 5 of 8
Advantages of a Base-n System
“See why regular base systems simplify regrouping, arithmetic and abacus representation.”
• Explain why regrouping is simple in a base-n system. • Perform addition using regrouping of landmark numbers. • Explain why products of landmark numbers remain landmark numbers. • Interpret a decimal abacus as powers-of-10 rows. • Identify the main shortcoming of the Egyptian symbol system.
Advantages of a Base-n System
The real strength of a base system appears during calculation. Because every landmark is formed from a fixed number of copies of the previous landmark, regrouping follows one consistent rule.
In base 10, ten 1s regroup as one 10, ten 10s regroup as one 100, and so on. In base 5, five 1s regroup as one 5, five 5s regroup as one 25, and so on.
Problem
Suppose an addition produces 15 ones. How should they be regrouped?
- 1.Ten ones make one 10.
- 2.15 ones = 10 ones + 5 ones.
- 3.Replace ten of the ones with one 10 landmark.
- 4.The result has one extra 10 and five remaining ones.
Problem
Suppose a base-5 addition produces 8 copies of the 1-landmark. Regroup them.
- 1.Five 1s make one 5-landmark.
- 2.8 = 5 + 3.
- 3.So regroup as one 5 and three 1s.
Multiplication also becomes regular. Each landmark is a power of the base, so multiplying landmark numbers adds exponents. The result is another landmark number.
Problem
Multiply the landmarks 10² and 10³.
- 1.10² represents 100 and 10³ represents 1000.
- 2.10² × 10³ = 10⁵.
- 3.10⁵ is another landmark number in the base-10 system.
This regularity is what Roman numerals lack. Their landmarks 1, 5, 10, 50, 100, ... do not all arise by repeatedly multiplying by one fixed number.
Abacus that Makes Use of the Decimal System
A decimal abacus can organise counters by powers of 10. One row represents ones, the next tens, then hundreds, then thousands. The number of counters in each row tells how many copies of that landmark are present.
Problem
An abacus has 3 counters on 1000, 4 on 100, 2 on 10 and 6 on 1. What number is shown?
- 1.3×1000 = 3000.
- 2.4×100 = 400.
- 3.2×10 = 20.
- 4.6×1 = 6.
- 5.Total = 3426.
The same regrouping idea explains carrying in addition. If 7 ones and 3 ones come together, they make 10 ones, which can be replaced by one counter in the tens row.
III. Shortcomings of the Egyptian System
The Egyptian landmark structure makes grouping and arithmetic efficient, but the written system still needs a different symbol for every new power of 10. If numbers keep getting larger, new symbols must keep being invented.
So the original problem returns in a new form: the landmark values are elegant, but the notation is not yet finite. The next breakthrough is to reuse the same symbols by letting position tell us which landmark value they represent.
Quiz
In base 10, what happens to 10 ones during regrouping?
Why is multiplication of landmark numbers simple in a base system?
What does the 100-row on a decimal abacus represent?
What is the main written-notation weakness of the Egyptian system described?
What idea solves the need for endlessly many landmark symbols?
Practice Problems
- Regroup 27 ones in base 10 into tens and ones.
- Regroup 19 units in a base-5 system into 5s and 1s.
- Find 10³×10⁴ and explain why the result is another landmark number.
- Represent 2907 on a decimal abacus by stating the counters needed on each power-of-10 row.
- Explain why a base structure can simplify arithmetic even if the written symbol system is still inconvenient.
Key Takeaways
• Base systems use one consistent regrouping rule. • In base 10, ten copies of one landmark become one copy of the next. • In base 5, five copies of one landmark become one copy of the next. • Products of landmark powers remain landmark powers. • A decimal abacus organises counters by successive powers of 10. • The Egyptian system still required new symbols for higher and higher landmarks. • Reusing symbols through position is the next major improvement.