TOPIC 8: NUMBERS (II) ~ MATHEMATICS FORM 1
A Rational Number
Define a rational number
ARational Numberis a real number that can be written as a simple fraction (i.e. as aratio). Most numbers we use in everyday life are Rational Numbers.
| Number | As a Fraction | Rational? |
|---|---|---|
| 5 | 5/1 | Yes |
| 1.75 | 7/4 | Yes |
| .001 | 1/1000 | Yes |
| -0.1 | -1/10 | Yes |
| 0.111… | 1/9 | Yes |
| √2(square root of 2) | ? | NO ! |
The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they arenot rationalthey are calledIrrational.
The Basic Operations on Rational Numbers
Perform the basic operations on rational numbers
Addition of Rational Numbers:
To
add two or morerational numbers, the denominator of all the rational
numbers should be the same. If the denominators of all rational numbers
are same, then you can simply add all the numerators and the denominator
value will the same. If all the denominator values are not the same,
then you have to make the denominator value as same, by multiplying the
numerator and denominator value by a common factor.
add two or morerational numbers, the denominator of all the rational
numbers should be the same. If the denominators of all rational numbers
are same, then you can simply add all the numerators and the denominator
value will the same. If all the denominator values are not the same,
then you have to make the denominator value as same, by multiplying the
numerator and denominator value by a common factor.
Example 1
1⁄3+4⁄3=5⁄3
1⁄3 +1⁄5=5⁄15 +3⁄15 =8⁄15
Subtraction of Rational Numbers:
To
subtract two or more rational numbers, the denominator of all the
rational numbers should be the same. If the denominators of all rational
numbers are same, then you can simply subtract the numerators and the
denominator value will the same. If all the denominator values are not
the same, then you have to make the denominator value as same by
multiplying the numerator and denominator value by a common factor.
subtract two or more rational numbers, the denominator of all the
rational numbers should be the same. If the denominators of all rational
numbers are same, then you can simply subtract the numerators and the
denominator value will the same. If all the denominator values are not
the same, then you have to make the denominator value as same by
multiplying the numerator and denominator value by a common factor.
Example 2
4⁄3 -2⁄3 =2⁄3
1⁄3-1⁄5=5⁄15-3⁄15=2⁄15
Multiplication of Rational Numbers:
Multiplication
of rational numbers is very easy. You should simply multiply all the
numerators and it will be the resulting numerator and multiply all the
denominators and it will be the resulting denominator.
of rational numbers is very easy. You should simply multiply all the
numerators and it will be the resulting numerator and multiply all the
denominators and it will be the resulting denominator.
Example 3
4⁄3×2⁄3=8⁄9
Division of Rational Numbers:
Division
of rational numbers requires multiplication of rational numbers. If you
are dividing two rational numbers, then take the reciprocal of the
second rational number and multiply it with the first rational number.
of rational numbers requires multiplication of rational numbers. If you
are dividing two rational numbers, then take the reciprocal of the
second rational number and multiply it with the first rational number.
Example 4
4⁄3÷2⁄5=4⁄3×5⁄2=20⁄6=10⁄3
Real Numbers
Real Numbers
Define real numbers
he
type of number we normally use, such as 1, 15.82, −0.1, 3/4,
etc.Positive or negative, large or small, whole numbers or decimal
numbers are all Real Numbers.
type of number we normally use, such as 1, 15.82, −0.1, 3/4,
etc.Positive or negative, large or small, whole numbers or decimal
numbers are all Real Numbers.
They are called “Real Numbers” because they are not Imaginary Numbers.
Absolute Value of Real Numbers
Find absolute value of real numbers
The
absolute value of a number is the magnitude of the number without
regard to its sign. For example, the absolute value of 𝑥 𝑜𝑟 𝑥
written as 𝑥 . The sign before 𝑥 is ignored. This is because the
distance represented is the same whether positive or negative. For
example, a student walking 5 steps forward or 5 steps backwards will be
considered to have moved the same distance from where she originally
was, regardless of the direction.
absolute value of a number is the magnitude of the number without
regard to its sign. For example, the absolute value of 𝑥 𝑜𝑟 𝑥
written as 𝑥 . The sign before 𝑥 is ignored. This is because the
distance represented is the same whether positive or negative. For
example, a student walking 5 steps forward or 5 steps backwards will be
considered to have moved the same distance from where she originally
was, regardless of the direction.
The 5 steps forward (+5) and 5 steps backward (-5) have an absolute value of 5
Thus |𝑥| = 𝑥 when 𝑥 is positive (𝑥 ≥ 0), but |𝑥| = −𝑥 when 𝑥 is negative (𝑥 ≤ 0).
For example, |3| = 3 since 3 is positive (3 ≥ 0) And −3 = (−3) =3 since −3 is negative (3 ≤ 0)
Related Practical Problems
Solve related practical problems
Example 5
Solve for 𝑥 𝑖𝑓 |𝑥| = 5
Solution
For any number 𝑥, |𝑥| = 5, there are two possible values. Either 𝑥,= +5 𝑜𝑟 𝑥 = 5
Example 6
Solve for 𝑥, given that |𝑥 + 2| =4
Solution
