Square roots
Here is how we write a square root... It's made up of a radical sign and something inside called the radicand.
The square root of a number (the radicand) is a number that produces the radicand when it is squared.
Example:
HERE ARE THE FIRST TEN square numbers and their roots:
Square numbers 1 4 9 16 25 36 49 64 81 100
Square roots 1 2 3 4 5 6 7 8 9 10
We write, for example,
= 5.
"The square root of 25 is 5."
This mark is called the radical sign (after the Latin radix = root). The number under the radical sign is called the radicand. In the example, 25 is the radicand.
Problem 1. Evaluate the following.
a) = 8
b = 12
c) = 20
d) = 17
e) = 1
f)
= 7
9
Example 1. Evaluate .
Solution. = 13.
For, 13• 13 is a square number. And the square root of 13• 13 is 13
If a is any positive number, then a• a is obviously a square number, and
Problem 2. Evaluate the following.
a) = 28.
b) = 135.
c) = 2• 3• 5 = 30.
We can state the following theorem:
A square number times a square number is itself a square number.
For example,
36• 81 = 6• 6• 9• 9 = 6• 9• 6• 9 = 54• 54
Problem 3. Without multiplying the given square numbers, each product of square numbers is equal to what square number?
a) 25• 64 = 5• 8• 5• 8 = 40• 40
b) 16• 49 = 4• 7• 4• 7 = 28• 28
c) 4• 9• 25 = 2• 3• 5• 2• 3• 5 = 30• 30
Rational and irrational numbers
The rational numbers are the numbers of arithmetic: the whole numbers, fractions, mixed numbers, and decimals; together with their negative images.
That is what a rational number is. As for what it looks
like, it will take the form a
b , where a and b are
integers (b ≠ 0).
Problem 4. Which of the following numbers are rational?
1 −6 3½ 4
5 − 13
5 0 7.38609
All of them!
At this point, the student might wonder, What is a number that is not rational?
An example of such a number is ("Square root of 2"). is not a number of arithmetic. There is no whole number, no fraction, and no
decimal whose square is 2. 7
5 is close, because
7
5 • 7
5 = 49
25
-- which is almost 2.
But to prove that there is no rational number whose square is 2, then
suppose there were. Then we could express it as a fraction m
n in lowest
terms. That is, suppose
m
n • m
n = m• m
n• n = 2.
But that is impossible. Because since m
n is in lowest terms, then
m and n have no common divisors except 1. Therefore, m• m and n• n also have no common divisors -- they are relatively prime -- and it will be impossible to divide n• n into m• m and get 2
There is no rational number whose square is 2. Therefore we call an irrational number.
Question. Which square roots are rational?
Answer. Only the square roots of square numbers.
= 1 Rational
Irrational
Irrational
= 2 Rational
, , , Irrational
= 3 Rational
And so on.
Only the square roots of square numbers are rational.
The existence of these irrationals was first realized by Pythagoras in the 6th century B.C. He called them "unnamable" or "speechless" numbers. For, if we ask, "How much is ? -- we cannot say. We can only call it, "Square root of 2."
Problem 5. Say the name of each number.
a) Square root of 3
b) Square root of 8
c) 3
d)
2
5 e) Square root of 10
As for the decimal representation of both irrational and rational numbers, see Topic 2 of Precocious.
An equation x² = a, and the principal square root
Example 2. Solve this equation:
x² = 25.
Solution. x = 5 or −5, because (−5)² = 25, also.
In other words,
x = or − .
We say however that the positive value 5 is the principal square root. That is, we say that "the square root of 25" is 5.
= 5.
As for −5, it is "the negative of the square root of 25."
− = −5.
Thus the symbol refers to one non-negative number.
Example 3. Solve this equation:
x² = 10.
Solution. x = or − .
Problem 6. Solve for x.
a) x² = 9 implies x = ±3 b) x² = 144 implies x = ±12
c) x² = 5 implies x = ± d) x² = 3 implies x = ±
e) x² = a − b implies x = ±
