# 100

## Number

$100$ (**one hundred** or **a hundred**) is:

- $2^2 \times 5^2$

- The $2$nd power of $10$ after $(1)$, $10$:
- $100 = 10^2$

- The $4$th after $2$, $5$ and $10$ of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
**cent**

- The $8$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$:
- $100 = \dfrac {8 \paren {3 \times 8 + 1} } 2$

- The $8$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$:
- $\nexists m \in \Z_{>0}: m - \map \phi m = 100$

- where $\map \phi m$ denotes the Euler $\phi$ function

- The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $9$ different ways

- The $10$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$:
- $100 = 10 \times 10$

- The $14$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$

- The $16$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$:
- $100 = \dfrac {8 \paren {3 \times 8 + 1} } 2$

- The $20$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$, $91$, $94$, $97$:
- $100 \to 1^2 + 0^2 + 0^2 = 1$

- The $23$rd semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$, $84$, $88$, $90$, $96$:
- $100 = 5 + 20 + 25 + 50$

- The $49$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$ which cannot be expressed as the sum of distinct pentagonal numbers.

## Also see

*Previous ... Next*: Sequence of Powers of 10*Previous*: Letters of Names of Numbers in Alphabetical Order/French

*Previous ... Next*: Roman Numerals

*Previous ... Next*: Powerful Number*Previous ... Next*: Square Number

*Previous ... Next*: Noncototient

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous*: Semiperfect Number

*Previous ... Next*: Happy Number

## Historical Note

The number $100$ (one hundred) was at one time sometimes referred to in England as a **short hundred**.

This was to distinguish it from the term **long hundred** for the number $120$.

Both terms are now obsolete in England, although the term **great hundred** for $120$ is still used in Germany and Scandinavia.

The boiling point of water is defined as being $100$ degrees Celsius.

The number $100$ is expressed in Roman numerals as $\mathrm C$.

This originates from the first letter of the Latin word **centum**, meaning $100$.

The archetypal "big number", to small children, is $100$. This most likely stems from the fact that their first introduction to numbers is the exercise to count to $100$. As this is where the count stops, $100$ is the biggest number they know.

Once introduced to the concept of **one hundred and one**, however, their level of arithmetical sophistication is soon seen to increase.

## Linguistic Note

The Latin word for $100$ was **centum**.

Hence the prefix **centi-** was adopted in the metric system to mean **a hundredth part**.

For example, a centimetre is one hundredth part of a metre.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $100$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $100$