Positive And Negative Integers, Rational Numbers
Positive And Negative Integers, Rational Numbers
Before the introduction of rational numbers it will help to recall that for two given integers a and b, their sum a + b, product a × b and the difference a - b are at all times integers.
However, it may not be possible at all time for a given integer to exactly divide another integer. This means that the result of division of an integer by a non- zero integers may or may not be an integer. For instance, when 9 is divided by 4, the result is not an integer because we are aware that 9/4 is a fraction.
Therefore, there is a requirement to extend the system of integers so that it may as well be possible to divide any given integer by any other given integer that differs from zero (due to the fact that it is not possible to be able to divide by zero).
Now, we ought to introduce the system of rational numbers, relationship between rational numbers, representation of rational numbers on the number line, a lot of operations on rational numbers and the properties of these operations on rational numbers.
A rational number is said to be negative if its numerator and denominator are of opposite signs in a manner that, one of them is positive integer and the other is a negative integer.
In other words, a rational number is negative, if its numerator and denominator are of the contrasting signs.
Every one of the rational numbers -1/6, 2/-7, -30/11, 13/-19, -15/23 are negative rational numbers, but -11/-18, 2/5, -3/-5, 1/3 are not negative rational numbers.
The meaning of rational numbers
The numbers of the form a/b, or a number which can be expressed in the form a/b, where ‘a’ and ‘b’ are integers and b ≠ 0, are called rational numbers. What this means is that a rational number is any number that can be expressed as the quotient of two integers with the condition that the dividend is not zero.
For instance; each of the numbers 2/3, 5/8, -3/14, -11/-5, 7/-9, 7/-15 and -6/-11 is a rational number.
Numerator and denominator:
If a/b is a rational number, then the integer a is known as its numerator and the integer b is referred to as the denominator.
Finding out if every negative integer a negative rational number
We know that -1 = -1/1, -2 = -2/1, -3 = -3/1, -4 = -4/1 etc …….
What this means is that any negative integer n can be written as n = n/1, here n is negative and 1 is positive.
Therefore, every negative integer is a negative (-ve) rational number.
Note: Although 0 is a rational number. It is neither positive nor negative.
Find out if the following rational numbers are negatives or not:
(i) 3/(-8)
3/(-8) is a negative rational numbers due to the fact that the numerator and denominator are of the opposite sign.
(ii) (-1)/(-5)
(-1)/(-5) is not a negative rational number and this is because both the numerator and denominator possess the same sign.
(iii) 11/29
11/29 is not a negative rational number and this is because both the numerator and denominator possess an equivalent sign.
(iv) 11/(-15)
11/(-15) is a negative rational number because both the numerator and denominator are of the different sign.
(v) (-71)/(-9)
(-71)/(-9) is not a negative rational number because both the numerator and denominator are of the same sign.
(vi) (-33)/7
(-33)/7 is a negative rational number because both the numerator and denominator are made up of the opposite sign.
(vii 21/22
21/22 is not a negative rational number because both the numerator and denominator have the same sign.
(viii) (-14)/39
(-14)/39 is a negative rational number because both the numerator and denominator are of the opposite sign.
From the ongoing, we can conclude that a rational number is negative, if its numerator and denominator are of the opposite sign and it is said to be positive if its numerator and denominator are either both positive integers or both negative integers.
This means that a rational number is positive, if its numerator and denominator are of the same sign.
Every one of the rational numbers 1/4, 2/9, -7/-11, -3/-13, 5/12 are positive rational numbers, but 2/-5, -3/10, -4/7, 11/-23 are not positive rational numbers.
Finding out if every natural number is a positive rational number
We are aware that, 1 = 1/1, 2 = 2/1, 3 = 3/1, 4 = 4/1 etc ……..
This means that any natural number n can be written as n = n/1, where n and 1 are positive integers.
Thus, every natural number is a positive (+ve) rational number.
Note: The rational number 0 is neither positive nor negative.
Find out if the following rational numbers are positives or not:
(i) (-11)/3
(-11)/3 is not a positive rational number due to the fact that both the numerator and denominator are of the opposite sign.
(ii) (-5)/(-7)
(-5)/(-7) is a positive rational number due to the fact that both the numerator and denominator are negative integers.
(iii) 13/19
13/19 is a positive rational number due to the fact that both the numerator and denominator are positive integers.
(iv) 21/(-17)
21/(-17) is not a positive rational due to the fact that both the numerator and denominator are of the opposite sign.
(v) (-105)/(-8)
(-105)/(-8) is a positive rational number due to the fact that both the numerator and denominator are negative integers.
(vi) (-3)/7
(-3)/7 is not a positive rational number due to the fact that both the numerator and denominator are of the opposite sign.
(vii) 27/31
27/31 is a positive rational number due to the fact that both the numerator and denominator are positive integers.
(viii) 25/(-27)
25/(-27) is not a positive rational number due to the fact that both the numerator and denominator are of the opposite sign.
Therefore, we can conclude from the above explanation that a rational number is positive, if its numerator and denominator are of the same sign.
Natural numbers are all numbers 1, 2, 3, 4… They are the numbers you normally count which continues to infinity.
Whole numbers are all natural numbers in addition to 0 For instance, 0, 1, 2, 3, 4…
Integers are made up of all whole numbers and their negative counterpart like…-4, -3, -2, -1, 0,1, 2, 3, 4,…
All integers are rational numbers. A rational number is a number
Where a and b are both integers.
Example:
The number 4 is an integer as well as a rational number. Since it can be written without a decimal constituent, it belongs to the integers. It is a rational number due to the fact that it can be written as:
4 ⁄ 2
or
8 ⁄ 4
or even
-8 ⁄ -2
while
1 ⁄ 5=0.2
is a rational number but not an integer.
A rational number written in a decimal form can either be ending as in:
1 ⁄ 5=0.2
Or repetitive like in
5 ⁄ 6=0.83333...
All rational numbers are real numbers.
When you examine a numeral line just like the one below,
You would observe that all integers, in addition to all rational numbers, are at a particular distance from 0. This distance between a number x and 0 is known as the number's absolute value. It is denoted with the symbol
If two numbers are at an equivalent distance from 0 like in the case of 10 and -10 they are known as opposites. Opposites are made up of equivalent absolute value due to the fact that they are both at the same distance from 0.
Equivalent rational numbers by multiplication:
If ab is a rational number and m is a non-zero integer then a×mb×m is a rational number equivalent to ab.
For instance, rational numbers 1215, 2025, −28−35, −48−60 are equivalent to the rational number 45.
We are aware that if we multiply the numerator and denominator of a fraction by the same positive integer, the value of the fraction does not alter.
For instance, the fractions 37 and 2149 are equivalent due to the fact that the numerator and the denominator of 2149 can be gotten by multiplying everyone of the numerator and denominator of 37 by 7.
Again, −34 = −3×(−1)4×(−1) = 3−4, −34 = −3×24×2 = −68, −34 = −3×(−2)4×(−2) = 6−8 etc …….
Thus, −34 = −3×(−1)4×(−1) = −3×24×2 = (−3)×(−2)4×(−2) etc …….
Note:
If the denominator of a rational number is a negative integer, then by making use of the property above, we can make it positive by multiplying its numerator and denominator by -1.
For instance, 5−7 = 5×(−1)(−7)×(−1) = −57
Equivalent rational numbers by division:
If ab is a rational number and m is a common divisor of a and b, then a÷ mb÷ m is a rational number equivalent to ab.
For instance, rational numbers −48−60, −28−35, 2025, 1215 are equivalent to the rational number 45.
We are aware that if we divide the numerator and denominator of a fraction by a common divisor, then the value of the fraction does not alter.
For instance, 4864 = 48÷1664÷16 = 34
In the same way, we have
−75100 = (−75)÷5100÷5 = −1520 = (−15)÷520÷5 = −34, and 42−56 = 42÷2(−56)÷2 = 21−28 = 21÷(−7)(−28)÷(−7) = −34
Calculated examples:
1. Find the two rational numbers that are equivalent to 37.
Solution:
37 = 3×47×4 = 1228 and
37 = 3×117×11 = 3377
Thus, the two rational numbers that are equivalent to 37 are 1228 and 3377
2. Determine the smallest equivalent rational number of 210462.
Solution:
210462 = 210÷2462÷2 = 105231 = 105÷3231÷3 = 3577 = 35÷777÷7 = 511 Thus, the least equivalent rational number of 210462 is 511
Write each of the following rational numbers with positive denominator:
3−7, 11−28, −19−13
Solution:
To express a rational number with positive denominator, we multiply its numerator and denominator by -1.
Thus,
3−7 = 3×(−1)(−7)×(−1) = −37,
11−28 = 11×(−1)(−28)×(−1) = −1128,
and −19−13 = (−19)×(−1)(−13)×(−1) = 1913
Express −37 as a rational number with numerator:
(i) -15; (ii) 21
Solution:
(i) In order to make -3 as a rational number with numerator -15, we first calculate a number which when multiplied by -3 gives -15.
Therefore, that number is (-15) ÷ (-3) = 5
Multiplying the numerator and denominator of −37 by 5, we obtain
−37 = (−3)×57×5 = −1535
Therefore, the required rational number is −1535.
(ii) In order to express −37 as a rational number with numerator 21, we at first calculate a number which when multiplied with -3 gives 21.
Evidently, that number is 21 ÷ (-3) = -7
Multiplying the numerator and denominator of −37 by (-7), we have
−37 = (−3)×(−7)7×(−7) = 21−49
Multiplication of a rational expression
To Multiply a rational expression:
1. Factor all numerators and denominators.
2. Cancel out all common factors.
3. Either multiply the denominators and numerators together or leave the solution in factored form.
Example 1
Multiply and then simplify the product
2x+4⁄x·3⁄6x+12
Multiply the following rational expressions:x2+6x+9⁄x2-9·3x-9⁄x2+2x-39=
Solution
1: Factor all numerators and denominators:
x2+6x+9⁄x2-9·3x-9⁄x2+2x-39=
2: Cancel out all common factors:
x2+6x+9⁄x2-9·3x-9⁄x2+2x-39=
3: Multiply the denominators and numerators:
x2+6x+9⁄x2-9·3x-9⁄x2+2x-39=
Addition and Subtraction of Rational (Fractional) Expressions
The Basic RULE for Adding and Subtracting Fractions: Get a Common Denominator
Fractions are fractions!
It doesn't matter if the fractions are made up of numbers, or of algebraic variables, the rules remain the same.
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