A kosarad üres
Usage of letters in mathematics
We often use letters when generalising a mathematical problem. These can be called variables, indefinites or unknowns depending on the problem.
For example by denoting the length of two adjacent sides of a rectangle by \latex{a} and \latex{b} (Figure 1), then the perimeter of the rectangle is \latex{P=2 \times (a\times b)}, and its area is \latex{A=a\times b}. With the help of these formulas the perimeter and the area of all rectangles can be calculated, if the length of the sides is known.
\latex{a}
\latex{b}
\latex{A=a\times b}
Figure 1
The use of letters is practical e.g. when generally describing the properties of addition and multiplication. For all real numbers \latex{a, b, c}:
commutative:
associative:
\latex{a+b = b+a}
\latex{(a+b)+c = a+(b+c)}
\latex{(a\times b)\times c = a\times(b\times c)}
\latex{a\times b = b\times a}
ADDITION
MULTIPLICATION
The multiplication is distributive over the addition: (Figure 2)
\latex{a\times (b+c)=a\times b+ a\times c}
\latex{a\times b}
\latex{a\times c}
\latex{b}
\latex{c}
\latex{a}
\latex{b+c}
Figure 2
Example 1
Which numbers can be given by the expression \latex{2k + 1?}
Solution
If \latex{k\in \N}, then the expression gives all the odd natural numbers. If \latex{k\in \Z}, then we get all the odd integers. And if \latex{k\in \Q}, then all we can say is that the given expression gives a number one greater than the double of a rational number. In this way all rational numbers can be generated.
When using expressions with letters it is always important to give the number set the elements of which are substituted by the letters used. This number set is the fundamental set.
Example 2
How is it possible to give all the positive integers which leave \latex{ 3 } as a remainder when divided by \latex{ 7? }
Solution
Such numbers are for example 3; 10; 17; 24; 31; etc. Since \latex{3 = 7 \times0 + 3}; \latex{10 = 7 \times 1 + 3}; \latex{17 = 7 \times 2 + 3}; \latex{\dotsc} , the letters help us now too, as these numbers are \latex{ 3 } greater than the multiples of \latex{ 7 }, i.e. they have the form of \latex{7m + 3}. It is important to add that \latex{m \in \N}.
If we write the numbers in the following form: \latex{3 = 7 \times 1 – 4; 10 = 7 \times 2 – 4; 17 = 7 \times 3 - 4}; \latex{\dotsc} , then we get the form \latex{7n -4}, where \latex{n \in \N^{+}} . This expression generates the same numbers.
Example 3
How much money are we left with if we take \latex{ a } \latex{ euros } to the shop, and we buy \latex{b} books for \latex{c} \latex{ euros } each?
Solution
The price of the books is \latex{b×c} \latex{ euros }, with \latex{a – b×c} \latex{ euros } remaining.
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We get an algebraic expression if we apply the four basic operations for numbers or letters finitely many times. We talk about an expression with one variable if it contains only one letter. The expressions containing several different letters are called multivariable expressions.
An example of an expression with one variable: \latex{\; 7k + 3}; \latex{\;2k + 1}.
An example of an expression with two variables: \latex{ \;ab; \; 2\times (a+b)}.
An example of an expression with four variables: \latex{\; 13a^2+4b-3c+d^2}.
If we substitute specific numbers of the fundamental set for the variables appearing in the algebraic expression, then after doing the operations we get the substitution value of the expression.
\latex{a} | \latex{b} | \latex{a\times b} | \latex{2\times(a+ b)} |
12 | 15 | \latex{12\times 15=180} | \latex{2\times(12+15)=54} |
7 | 13 | 91 | 40 |
2.8 | 0.62 | 1.736 | 6.84 |
We talk about an integral algebraic expression if there are no fractions in the algebraic expression or there is a fraction but there is no variable in the denominator of the fraction. We talk about a fractional algebraic expression if there is a variable in the denominator of the fraction appearing in the algebraic expression.
For example \latex{\frac{4.5y}{x^3}}, \latex{\frac{2a-4}{a-2}} and \latex{\frac{3x-y}{2y-1}} are fractional expressions.
Expressions \latex{\frac{2x}{5}}, and \latex{\frac{3x-1}{17}} are not algebraic fractions.
In the case of algebraic fractions we cannot always calculate the substitution value, since the denominator of the fractions cannot be equal to \latex{ 0 }.
The domain of an algebraic fraction is the largest subset of the set of real numbers, substituting the elements of which for the variables, the operations appearing in the expression can be done.
We talk about a monomial if the numbers and letters are connected by the operation multiplication in the expression.
For example \latex{5}\latex{\times a}; \latex{-\frac{2}{3}}\latex{\times b}; 3.8\latex{\times xyz}; \latex{-2\times3.6}\latex{\times klm}.
Numbers 5; \latex{-\frac{2}{3}}; 3.8; \latex{-2\times 3.6=-7.2} are called coefficients.
In the expression \latex{\frac{2x}{5}} the coefficient is \latex{\frac{2}{5}}. In the expression \latex{x} the coefficient is 1. And in expression –\latex{x} the coefficient is –1.
The sum of algebraic integral expressions with one term (monomials) is called integral algebraic expression with several terms, or polynomial. There are special names for polynomials with \latex{ 2 } or \latex{ 3 } terms: binomial and trinomial.
For example: \latex{3x+5ab};\latex{\;} \latex{-\frac{2x}{5}+3xyz};\latex{\;} \latex{7a-\frac{3b}{7}-4.8c}.
Example 4
a) For which natural numbers does the expression
\latex{3b-\frac{4a-2}{b+1}} have a meaning?
b) For which real numbers does the expression have a meaning?
Solution
a) The expression has a meaning for all positive natural numbers a, b, since \latex{b+1\gt 0}.
b) The expression has a meaning on the set of real numbers only if \latex{b\neq -1}.
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We often use letters when writing down physical and chemical results, when we formulate general laws. For example:
- An object moving at a given constant speed v travels a distance of \latex{s=v\times t} in time t.
Example 5
If we mix two sugar solutions, one solution with a mass of m and a mass percentage of p and another solution with a mass of k and a mass percentage of q, then what is the mass percentage of the resulting mixture?
Solution
In the first solution the amount of sugar solute is \latex{m\times\frac{p}{100}},
in the second one it is \latex{k\times\frac{q}{100}}.
in the second one it is \latex{k\times\frac{q}{100}}.
The total amount of solute is \latex{m\times\frac{p}{100}+k\times\frac{q}{100}} in a solution with a mass of \latex{m+k}.
Thus the mixture will have a mass percentage of \latex{\dfrac{m\times\dfrac{p}{100}+k\times\dfrac{q}{100}}{m+k}\times 100 = \frac{mp+kq}{m+k}}.
Exercises
{{exercise_number}}. Which numbers are denoted by the expression \latex{5m + 2} if
- \latex{ m } represents a natural number;
- \latex{ m } represents an integer;
- \latex{ m } represents a rational number?
{{exercise_number}}. What numbers are denoted by the notation \latex{\frac{p}{q}} if p and q are integers and \latex{q\neq 0?}
{{exercise_number}}. Give an expression for all the positive integers which leave 1 as a remainder when divided by 4.
{{exercise_number}}. Give an expression for the coefficients appearing in the following expressions.
\latex{-\frac{3x}{7}};
\latex{-7.83ab};
\latex{2.8\times5xy};
\latex{-2\times 5.3yz};
\latex{2x7y};
\latex{-4.2x5y.}
{{exercise_number}}. Which expression is greater if \latex{a=2, b=\frac{2}{3}, c=-\frac{1}{5}}:
- \latex{3a^2-4a+1} or \latex{\frac{4a-2}{a-1}};
- \latex{-3ab+18ab^2-a^3} or \latex{\frac{1}{2}a-12b};
- \latex{2abc- 4ab^2c+4c^2} or \latex{\frac{3-c}{2-b}-\frac{c}{a+1}?}
{{exercise_number}}. Where do the below expressions have a meaning if the fundamental set is the set of real numbers?
- \latex{x+\frac{1}{x}};
- \latex{\frac{2x}{5}+\frac{11}{x}-\frac{7}{x^2}};
- \latex{\frac{2x-3}{5x+4}-\frac{4x+2}{3x-2}+2x};
- \latex{-\frac{6x+2}{7x}+\frac{1-x}{2x+5}-\frac{x+3}{3+2x}};
- \latex{\frac{2}{x^2}+\frac{3x-5}{1-3x}+\frac{3}{x^2-4}}.
{{exercise_number}}. Calculate the substitution value of the following expressions.
- \latex{\frac{x-2}{x}-\frac{5}{x}}, if \latex{x=1}
- \latex{3-\frac{2}{p}-\frac{4}{p+2}}, if \latex{p=2}
- \latex{\frac{2a-3}{a+1}-\frac{3a+2}{a-1}}, if \latex{a=3}
- \latex{\frac{2b+1}{4b-3}-\frac{b+1}{2b-3}+\frac{4}{b}}, if \latex{b=\frac{1}{2}}
- \latex{\frac{4}{b+2}-\frac{2b}{b-2}+\frac{3b-1}{b^2-4}}, if \latex{b=-5}
f) \latex{\frac{4x^5}{x+3}+\frac{7x+11}{x^2-9}-\frac{8x^3}{x-3}}, if \latex{x=-3}
{{exercise_number}}. A car travelled for \latex{ t } \latex{ hours } at a speed of \latex{ v }\latex{ \frac{km}{h} }, then for \latex{(t+1)} \latex{ hours } at a speed of \latex{(v-3) \frac{km}{h}}. Express the length of the travelled distance.
{{exercise_number}}. Solve the following exercises.
- \latex{t} pupils attend a class. We give everyone k books, and m books are still left. How many books do we want to distribute?
- \latex{t} pupils attend a class. We give \latex{k} books to each pupil, but then j pupils do not get any books. How many books have we distributed?
{{exercise_number}}. In a garden of apple trees, there are \latex{a} number of trees. Every tree yields at least \latex{ l } \latex{ kg } and at most \latex{f} \latex{ kg } of apple. What are the limits of the garden's apple harvest?
