What are rational numbers

In this publication, we will consider what rational numbers are, how to compare them with each other, and also what arithmetic operations can be performed with them (addition, subtraction, multiplication, division and exponentiation). We will accompany the theoretical material with practical examples for a better understanding.

Definition of a rational number

Rational is a number that can be represented as . The set of rational numbers has a special notation – Q.

Rules for comparing rational numbers:

1. Any positive rational number is greater than zero. Indicated by “greater than” special sign “>“.

   For example: 5>0, 12>0, 144>0, 2098>0, etc.

2. Any negative rational number is less than zero. Indicated by the “less than” symbol “<“.

   For example: -3<0, -22<0, -164<0, -3042<0 etc.

3. Of two positive rational numbers, the one with the larger absolute value is greater.

   For example: 10>4, 132>26, 1216<1516 и т.д.

4. Of two negative rational numbers, the larger one is the one with the smaller absolute value.

   For example: -3>-20, -14>-202, -54<-10 and т.д.

Arithmetic operations with rational numbers

Addition

1. To find the sum of rational numbers with the same signs, simply add them up, then put their sign in front of the resulting result.

For example:

• 5 + 2 = + (5 + 2) = +7 = 7
• 13 + 8 + 4 = + (13 + 8 + 4) = +25 = 25
• -9 + (-11) = – (9 + 11) =-20
• -14 + (-53) + (-3) = – (14 + 53 + 3) =-70

Note: If there is no sign before the number, it means “+“, i.e. it is positive. Also in the result “a plus” can be lowered.

2. In order to find the sum of rational numbers with different signs, we add to a number with a large modulus those whose sign coincides with it, and subtract numbers with opposite signs (we take absolute values). Then, before the result, we put the sign of the number from which we subtracted everything.

For example:

• -6 + 4 = – (6 – 4) =-2
• 15 + (-11) = + (15 – 11) = +4 = 4
• -21 + 15 + 2 + (-4) = – (21 + 4 – 15 – 2) =-8
• 17 + (-6) + 10 + (-2) = + (17 + 10 – 6 – 2) = 19

Subtraction

To find the difference between two rational numbers, we add the opposite number to the one being subtracted.

For example:

• 9 – 4 = 9 + (-4) = 5
• 3 – 7 = 3 + (-7) = – (7 – 3) =-4

If there are several subtrahends, then first add up all positive numbers, then all negative ones (including the reduced one). Thus, we get two rational numbers, the difference of which we find using the algorithm above.

For example:

• 12 – 5 – 3 = 12 – (5 + 3) = 4
• 22 – 16 – 9 = 22 – (16 + 9) = 22 – 25 = – (25 – 22) =-3

Multiplication

To find the product of two rational numbers, simply multiply their modules, then put before the resulting result:

• sign “+“if both factors have the same sign;
• sign “–“if the factors have different signs.

For example:

• 3 7 = 21
• -15 4 = -60

When there are more than two factors, then:

1. If all numbers are positive, then the result will be signed. “a plus”.
2. If there are both positive and negative numbers, then we count the number of the latter:
   • an even number is the result with “plus”;
   • odd number – result with “minus”.

For example:

• 5 (-4) 3 (-8) = 480
• 15 (-1) (-3) (-10) 12 = -5400

Division

As in the case of multiplication, we perform an action with modules of numbers, then we put the appropriate sign, taking into account the rules described in the paragraph above.

For example:

• 12: 4 = 3
• 48 : (-6) = -8
• 50 : (-2) : (-5) = 5
• 128 : (-4) : (-8) : (-1) = -4

Exponentiation

Raising a rational number a в n is the same as multiplying this number by itself nth number of times. Spelled like a ⁿ.

Wherein:

• Any power of a positive number results in a positive number.
• An even power of a negative number is positive, an odd power is negative.

For example:

• 2⁶ = 2 2 2 2 2 2 = 64
• -3⁴ = (-3) · (-3) · (-3) · (-3) = 81
• -6³ = (-6) · (-6) · (-6) = -216