Zero as a placeholder
An empty slot needs a symbol, or numbers get confused.
A gap on the tablet
A Mesopotamian scribe wrote two digits with a blank gap between them: 2 ▁ 5. Gaps were not always the same width, so readers had to guess how many slots the gap held.
Could the same writing be a different number if the gap holds a different number of empty slots?
What this lesson covers
The idea
Blank spaces for missing powers make place value numerals ambiguous, so a placeholder symbol for nothingness, zero, is indispensable for writing numbers unambiguously in a place value system.
A gap on the tablet
A Mesopotamian scribe wrote two digits with a blank gap between them: 2 ▁ 5. Gaps were not always the same width, so readers had to guess how many slots the gap held.
Could the same writing be a different number if the gap holds a different number of empty slots?
- Yes, it could
- No, it is always the same number
- Only for even numbers
Read the gap
Choose how many empty slots the gap holds, and read the number. Then write a placeholder 0 in each empty slot.
Zero holds the empty places
Blank spaces for missing powers make place value numerals ambiguous, so a placeholder symbol for nothingness, zero, is indispensable for writing numbers unambiguously in a place value system.
With zeros: 205, 2005 and 20005 look different. With blanks, they all look like 2 ▁ 5.
Notes
Blank spaces for missing powers make place value numerals ambiguous, so a placeholder symbol for nothingness, zero, is indispensable for writing numbers unambiguously in a place value system.
Check yourself
A numeral has the digit 3 in the hundreds slot, 7 in the ones slot, and the tens slot is empty. Writing a 0 in the tens slot, what number do we write?
Answer: 307
3 hundreds + 0 tens + 7 ones = 307. The 0 keeps the 7 in the ones place.
Write 4009 with a placeholder. How many zeros does it need?
Answer: 2
4009 = 4 × 1000 + 0 × 100 + 0 × 10 + 9 × 1. The two empty slots, hundreds and tens, each hold a zero.
Why do we need the symbol zero in a place value system?
Without a placeholder, which numbers could all be written as 3 ▁ 8?
- It marks an empty place so that numbers are written without confusion — correct. Yes! Zero is a placeholder for nothingness.
- It makes numbers bigger. Zero does not make numbers bigger. It holds an empty place.
- It is only used at the end of a number. Zero is also needed in the middle, like in 205. It holds any empty place.
- 308, 3008 and 30008 — correct. Yes! The blank could hold one, two or three empty slots.
- 38 only. There is a gap, so the digits are not side by side. The gap could hold one or more empty slots.
- 83 and 38. The digits stay in the same order, 3 first and 8 last. Only the number of empty slots between them is unknown.