Letters that stand for numbers
A letter can stand for a number.
Shabnam and Aftab
Shabnam is 3 years older than Aftab. When Aftab is 10, Shabnam is 13.
Aftab keeps getting older. Can one short sentence give Shabnam’s age for any age of Aftab?
What this lesson covers
The idea
A letter used to stand for a number is called a letter-number.
Shabnam and Aftab
Shabnam is 3 years older than Aftab. When Aftab is 10, Shabnam is 13.
Aftab keeps getting older. Can one short sentence give Shabnam’s age for any age of Aftab?
- No. We must know Aftab’s age first.
- Yes: Shabnam’s age is always Aftab’s age plus 3.
- Yes: Shabnam’s age is always 13.
Give the age a name
Aftab’s age keeps changing, so give it a short name: the letter a. Call Shabnam’s age s. Set *a* to any age you like.
A letter stands for a number
*a* can be 10 today, 23 another time, 40 another. The one sentence s = a + 3 works for every age.
A letter used to stand for a number is called a letter-number. In s = a + 3, both *a* and *s* are letter-numbers.
- Aftab’s age (a) | Shabnam’s age (s = a + 3)
- 10 | 13
- 23 | 26
- 40 | 43
Notes
A letter used to stand for a number is a letter-number. One letter-number can stand for many different numbers: in s = a + 3, *a* is Aftab’s age whatever it is.
Check yourself
Aftab is 23 years old. Shabnam’s age is s = a + 3. How old is Shabnam?
Answer: 26 years
s = 23 + 3 = 26 years.
In s = a + 3, which are the letter-numbers?
Aftab is 10 now and will be 11 next year. Can the letter-number *a* stand for both 10 and 11?
Any letter will do. Let *x* stand for Aftab’s age, so Shabnam’s age is x + 3. If *x* is 18, what is Shabnam’s age?
Answer: 21 years
x + 3 = 18 + 3 = 21 years.
- a and 3. 3 is an ordinary number. Only *a* is a letter here.
- a and s — correct. Yes! Each stands for a number: *a* for Aftab’s age and *s* for Shabnam’s age.
- s and 3. 3 is an ordinary number, not a letter.
- only s. *a* is a letter-number too: it stands for Aftab’s age.
- No. A letter-number is fixed at 10 for ever.. It is the opposite: a letter can stand for any number we choose.
- No. A letter cannot stand for a number at all.. A letter-number is exactly a letter that stands for a number.
- Yes. A letter-number can stand for different numbers at different times. — correct. Yes! That is why Aftab’s age needs a letter: it does not stay the same.