In algebra, real numbers have a number of properties in algebra which allow us to simplify and solve mathematical problems. Its density property, for example, implies that between any two real numbers provided there is always another real number. These are, for example, 9.1 to 9.2, 9.11, 9.12, 9.13 and so on. Its identity property says that any number added to zero is equal to itself. Real numbers have a variety of other properties including distributive, commutative, reflexive, and symmetric among others. They all serve the same purpose of helping us in problem solving.
For easier computation it is always important to cluster like terms together. The words like must have the same variable lifted to the same exponent in more complex expressions.
During the simplification of the given expressions, properties of real numbers will be utilized and identified. The mathematical workings will be aligned on the left while the right side will discuss the properties used.
2a(a-5)+4(a-5) The given expression
2a2-10a+4a-20 The parentheses are removed by use of the distributive property
2a2-6a-20 Coefficients are added to enable combination of like terms
The above is absolutely simplified, and no more calculation is needed. The words like that were already grouped together, so there was no need to rearrange the order as such.
2w-3+3(w-4)-5(w-6) The given expression
2w-3+3w-12-5w+30 Parentheses removed by distributive property
2w+3w-5w-12+30 Commutative property of real numbers is used to arrange the like terms together. -12 and 30 are like constant terms while 2w, 3w and -5w are like variable terms. These can then all be added or subtracted.
5w-5w+18 Two of both variable terms and constant terms are added
18 The remaining pair of like variable terms is added
The expression above is fully simplified. The variable terms cancel each other out through subtraction and a single constant term remains as the simplified form.
0.05(0.3m+35n)-0.8(-0.09n-22m) The given expression
0.015m+1.75n+0.072n+17.6m Removal of parentheses using the distributive property
0.015m+17.6m+1.75n+0.072n The commutative property enables us arrange like terms together. Only variable terms with the same variable may be combined
17.615m+1.822n Addition of coefficients enable combination of like terms
This given problem looked complicated but it is not substantially different from the others as it simply employs the usage of the decimal numbers.
From the three expressions above it is clear that knowing the properties of real numbers is important. It is also important to arrange mathematical workings in a chronological and organized manner to enable faster understanding and conceptualization of how to tackle different mathematical problems that are encountered.
- Poole, D. (2014). Linear algebra: A modern introduction. Cengage Learning.
- Samuel, P. (2013). Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger. Courier Dover Publications.