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Are There Numbers Whose Squares Are Smaller Than The Numbers Themselves?

1. Yes, there are numbers whose squares are smaller than the numbers themselves. Any number between 0 and 1, excluding the endpoints, will have a square smaller than the original number. For example, the square of 1/2 is 1/4, which is smaller than 1/2. 2. No, there is no real number whose square is -1. The square of any real number will always be positive. However, mathematicians use an "imaginary number" i to represent the square root of -1. Complex numbers combine real and imaginary parts in the form a + bi, where a is real and bi is imaginary.

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411 views2 pages

Are There Numbers Whose Squares Are Smaller Than The Numbers Themselves?

1. Yes, there are numbers whose squares are smaller than the numbers themselves. Any number between 0 and 1, excluding the endpoints, will have a square smaller than the original number. For example, the square of 1/2 is 1/4, which is smaller than 1/2. 2. No, there is no real number whose square is -1. The square of any real number will always be positive. However, mathematicians use an "imaginary number" i to represent the square root of -1. Complex numbers combine real and imaginary parts in the form a + bi, where a is real and bi is imaginary.

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1. Are there numbers whose squares are smaller than the numbers themselves?

- Yes, there are. The square of any number between 0 and 1


(excluding the extremes) is smaller than the number itself.

For example, 1/2 squared is equal to 1/4; which is smaller than 1/2. Squaring a Fraction

Yes, and they are called proper fractions and their decimal forms.

Step-by-step explanation:

Squaring is when we multiple a number by itself, represented by an exponent of two.


For example, we can write 5 x 5 as , and we will still get the same answer, which is
25.

Now, when we square a natural number (except for 1, which will always give us 1 as the
product), normally, the squares are larger than the original number. However, when we
square proper fractions (fractions wherein the numerator has a smaller number than the
denominator), the square is always smaller than the original number.

For example:

 x   or  is equal to ¼


0.25 x 0.25 = 0.0625

Or None.

Step-by-step explanation:

A square of a number is always greater than the number itself.

Example:

The number is -2

(-2)² = 4

-2 < 4

The number is 5.

(5)² = 25

5 < 25
The number is always smaller than its square or the square of a number is always greater than
the number.

Numbers greater than zero but less than one have squares that are smaller. Consider the
number one-half (1/2); half of a half is a quarter (1/4 < 1/2)! Of course, the square of a negative
number is greater (since a negative squared is positive). Only the numbers zero and one are
equal to their squares. another explanation: Yes . In fact there is an infinite number of numbers
that have that condition. For example (1/2)² = 1/4. In fact any positive number ’n’ such that
0<n<1 is a number which has a smaller number when squared. 

2. Is there a real number whose square is −1?

- No. the square of any real number, either rational or irrational, whether a whole number
or a fraction, is always going to be positive number. There are no exceptions to this rule.
It is no. However mathematicians have a ‘number’ they use whose value they interpret
as square root of negative one. This number is often written as ‘i’. It is referred to as an
IMAGINARY NUMBER.

This imaginary number can also be combined with a real number to form a COMPLEX
NUMBER. A complex number comes in the form a+bi where a is the real part and bi is the
imaginary part (b is a real number and i stands for the square root of negative one). Examples
of complex numbers: 1+5i, 6–7i, 4i and so on.
However mathematicians have a ‘number’ they use whose value they interpret as square root of
negative one. This number is often written as ‘i’. It is referred to as an IMAGINARY NUMBER.

This imaginary number can also be combined with a real number to form a COMPLEX
NUMBER. A complex number comes in the form a+bi where a is the real part and bi is the
imaginary part (b is a real number and i stands for the square root of negative one). Examples
of complex numbers: 1+5i, 6–7i, 4i and so on.

CHECK YOUR PROGRESS


For all real numbers X, if x is greater than 2 then X² is greater than 4
a. If a real number is greater than 2, then its square is ___ greater than 4

b. For all real numbers greater than 2, _____then X² is greater than 4


c. If X _>2__, then ____>4

d. The square of any real number greater than 2 is _____ greater than 4
e. All real numbers greater than 2 have ____ X² is greater than 4

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