Open sentences
14 + □ = 17. What number in the box will make this true? You may have seen problems like this before. 14 + □ = 17 will be true if 3 goes in the box: 14 + 3 = 17 is true,
We say 14 + □ = 17 is an open sentence. Any value can go in the box, but usually, only one value can make an open sentence true.
Exercises
1. 3 + 2 = □
2. 8 – 7 = □
3. 7 + 7 = □
Letters for numbers
In mathematics we use letters of the alphabets to stand for numbers instead of boxes. We write 14 + x instead of 14 + □. Any letter can be used. For example, 14 + a would be just as good as 14 + x. Capital letters are not used; only small letters are used.
When using a letter instead of a number, the letter can stand for any number in general. Thus the value of 14 + x depends on the value of x.
For example,
If x stands for 2, 14 + x has the value 16;
If x stands for 12, 14 + x has the value 26;
If x stands for 5, 14 + x has the value 19.
When letters and numbers are used together in this way, the mathematics is called generalized arithmetic or algebra The word algebra comes to us from an important book written around AD830 by Mohammed Musa al Khowarizmi, a noted mathematician from Baghdad. The title of the book was Al-jabr wa’l Muqabalah.
The statement 14 + x = 17 is called an algebraic sentence. It means 14 plus a number x makes 17, or 14 plus x equals 17.
In the next exercise, each letter stands for a number. The number that the letter stands for sh0uld make the sentence true. For example, 14 + x = 17will be true if x stands for 3.
We write this:
If 14 + x = 17
Then x = 3
Exercise
Each sentence is true. Find the number that each letter stands for.
1. x = 2 + 7
2. x = 3 – 10
3. y = 9 + 5
4. z = 18 + 11
5. z = 15 – 8
What is the value of z + 6? The value of x + 6 depends on what x stands for.
If x = 3, then x + 6 = 3 + 6 = 9
If x = 8, then x + 8 + 6 = 14
