Many of us wondered about the advantages of Mathematics during our childhood days. Many of us were not able to comprehend the benefits of mathematics beyond the daily usage of calculating simple numbers. Let us see in detail what are some of the benefits of learning mathematics and marveling at this arduous subject at early age.
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The importance of mathematics is two-fold, it is important in the advancement of science and two, it is important in our understanding of the workings of the universe. And in here and now it is important to individuals for personal development, both mentally and in the workplace.
Mathematics equips pupils with a uniquely powerful set of tools to understand and change the world. These tools include logical reasoning, problem-solving skills, and the ability to think in abstract ways. Mathematics is important in everyday life, many forms of employment, science and technology, medicine, the economy, the environment and development, and in public decision-making.
One should also be aware of the wide importance of Mathematics, and the way in which it is advancing at a spectacular rate. Mathematics is about pattern and structure; it is about logical analysis, deduction, calculation within these patterns and structures. When patterns are found, often in widely different areas of science and technology, the mathematics of these patterns can be used to explain and control natural happenings and situations. Mathematics has a pervasive influence on our everyday lives, and contributes to the wealth of the individual.
The study of mathematics can satisfy a wide range of interests and abilities. It develops the imagination. It trains in clear and logical thought. It is a challenge, with varieties of difficult ideas and unsolved problems, because it deals with the questions arising from complicated structures. Yet it also has a continuing drive to simplification, to finding the right concepts and methods to make difficult things easy, to explaining why a situation must be as it is. In so doing, it develops a range of language and insights, which may then be applied to make a crucial contribution to our understanding and appreciation of the world, and our ability to find and make our way in it.
Increasingly, employers are looking for graduates with strong skills in reasoning and problem solving - just the skills that are developed in a mathematics and statistics degree.
Let us look at a few examples. The computing industry employs mathematics graduates; indeed, many university computing courses are taught by mathematicians. Mathematics is used to create the complex programming at the heart of all computing. Also cryptography, a form of pure mathematics, is deployed to encode the millions of transactions made hourly via the Internet and when we use debit or credit cards. Mathematics and Computer Science is a popular degree choice, and four-year degrees with a placement in industry are also available. The latter give graduates plenty of relevant experience to increase their employability.
Mathematics led to the perfect ratios shown in Renaissance painting. The study of astronomy in the early times of its inception demanded the expansion of our understanding of mathematics and made possible such realizations as the size and weight of the earth, our distance from the sun, the fact that we revolve around it, and other discoveries that allowed us to move forward in our body of knowledge without which we would not have any of our modern marvels of technology.
The computer itself is a machine built upon the principles of mathematics, being an invention so important as to bring about an economic revolution of efficiency in data communication and processing.
Exponents comprise a juicy tidbit of basic-math-facts material. Exponents allow us to raise numbers, variables, and even expressions to powers, thus achieving repeated multiplication. The ever present exponent in all kinds of mathematical problems requires that the student be thoroughly conversant with its features and properties. Here we look at the laws, the knowledge of which, will allow any student to master this topic.
In the expression 3^2, which is read "3 squared," or "3 to the second power," 3 is the base and 2 is the power or exponent. The exponent tells us how many times to use the base as a factor. The same applies to variables and variable expressions. In x^3, this mean x*x*x. In (x + 1)^2, this means (x + 1)*(x + 1). Exponents are omnipresent in algebra and indeed all of mathematics, and understanding their properties and how to work with them is extremely important. Mastering exponents requires that the student be familiar with some basic laws and properties.
Product Law
When multiplying expressions involving the same base to different or equal powers, simply write the base to the sum of the powers. For example, (x^3)(x^2) is the same as x^(3 + 2) = x^5. To see why this is so, think of the exponential expression as pearls on a string. In x^3 = x*x*x, you have three x's (pearls) on the string. In x^2, you have two pearls. Thus in the product you have five pearls, or x^5.
Quotient Law
When dividing expressions involving the same base, you simply subtract the powers. Thus in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on the cancellation property of the real numbers. This property says that when the same number or variable appears in both the numerator and denominator of a fraction, then this term can be canceled. Let us look at a numerical example to make this completely clear. Take (5*4)/4. Since 4 appears in both the top and bottom of this expression, we can kill it---well not kill, we don't want to get violent, but you know what I mean---to get 5. Now let's multiply and divide to see if this agrees with our answer: (5*4)/4 = 20/4 = 5. Check. Thus this cancellation property holds. In an expression such as (y^5)/(y^3), this is (y*y*y*y*y)/(y*y*y), if we expand. Since we have 3 y's in the denominator, we can use those to cancel 3 y's in the numerator to get y^2. This agrees with y^(5-3) = y^2.
Power of a Power Law
In an expression such as (x^4)^3, we have what is known as a power to a power. The power of a power law states that we simplify by multiplying the powers together. Thus (x^4)^3 = x^(4*3) = x^12. If you think about why this is so, notice that the base in this expression is x^4. The exponent 3 tells us to use this base 3 times. Thus we would obtain (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same power and can thus use our first property to get x^(4 + 4+ 4) = x^12.
Distributive Property
This property tells us how to simplify an expression such as (x^3*y^2)^3. To simplify this, we distribute the power 3 outside parentheses inside, multiplying each power to get x^(3*3)*y^(2*3) = x^9*y^6. To understand why this is so, notice that the base in the original expression is x^3*y^2. The 3 outside parentheses tells us to multiply this base by itself 3 times. When you do that and then rearrange the expression using both the associative and commutative physics 補習 properties of multiplication, you can then apply the first property to get the answer.
Zero Exponent Property
Any number or variable---except 0---to the 0 power is always 1. Thus 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To see why this is so, let us consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression over itself yields this result. Using our quotient property, we see this is equal to x^(3 - 3) = x^0. Since both expressions must yield the same result, we get that x^0 = 1.
Negative Exponent Property
When we raise a number or variable to a negative integer, we end up with the reciprocal. That is 3^(-2) = 1/(3^2). To see why this is so, let us consider the expression (3^2)/(3^4). If we expand this, we obtain (3*3)/(3*3*3*3). Using the cancellation property, we end up with 1/(3*3) = 1/(3^2). Using the quotient property we that (3^2)/(3^4) = 3^(2 - 4) = 3^(-2). Since both of these expressions must be equal, we have that 3^(-2) = 1/(3^2).
Understanding these six properties of exponents will give students the solid foundation they need to tackle all kinds of pre-algebra, algebra, and even calculus problems. Often times, a student's stumbling blocks can be removed with the bulldozer of foundational concepts. Study these properties and learn them. You will then be on the road to mathematical mastery.