Mathematics Properties: What Are They? An Intensive Overview

Meta: Mathematics properties lend a sturdy basis for more accurate computations. So what are they exactly? Check out this article for our helpful pointers.

Intro

At first glance, mathematics properties seem like an alien concept, which confounds numerous readers. Fortunately, they turn out to be more commonplace than anticipated. You might even encounter them on a startling basis with your scheduled Math assignments. 

So what precisely are they? Our article will shed light on this burning issue.

What Are Mathematics Properties?

It is a trait of a quantifiable set – Source: pixabay

A mathematical property is any trait of a quantifiable set in mathematics. In a nutshell, properties pose several mathematical laws, which require us to observe certain rules [the properties] while tackling a particular math issue. 

For instance, one property p – specified for all members of a set Z – is often defined as a functional [p: Z true, false] that is assessed to be correct whenever the property exists. Similarly, as the subgroup of Z by which p applies (i.e., the set z | p(z) = true); p is its marker function.

Most mathematical subjects – including Geometry and Algebra – adheres to these characteristics. On the other hand, any calculation that does not stick to the commutative property (as in a.b = b.a) is not considered genuine Mathematics.

However, one may argue that such a definition only describes the extent of the property. It fails to imply anything about what drives this property to prevail for those specific values. Mathematicians are still debating over the most appropriate description for the term till now.

What Are Types of Mathematics Properties?

1. Commutative Property

Commutative rules – Source: pixabay

What Is Commutative Property?

It is crucial to assess the label “commutative” for better comprehension. This term is derived from “commute”, which signifies movement or position change. In mathematics, “commutative” means to modify the sequence of two numerals or variables in an equation.

So what is the significance of commutative property? It indicates that the additions or multiplications of two variants will produce intact results – even when their places are switched.

Commutative Properties of Additions

Adjustments in the sequence or orientation of two integers put together do not shift the outcome. Suppose we have two positive numbers, X and Y. The formula will be C + D = D + C.

Example 1: C = 5 and D = 11

Hence:

C + D +  = 5 + 11 = 16

D + C = 11 + 5 = 16

C + D = D + C

Example 2: C = 20 and D = 6

Therefore:

C + D = 20 + 6 = 26

D + C = 6 + 20 = 26

C + D = D + C

These two illustrations depict that the total sum always stays fixed, despite the variant positions.

Commutative Properties of Multiplications

Similarly, the position switch of two multiplied integers bears no impact on the results. With two positive integers, W and Z, the property is expressed as WZ = ZW.

Example 1: W = 5 and Z = 12

WZ = 5. 12 = 60

ZW = 12. 5 = 60

WZ = ZW = 60

Example 2: W = 22 and Z = 7

WZ = 22. 7 = 154

ZW = 7. 22 = 154

WZ = ZW = 154

Is There A Commutative Property for Subtractions and Divisions?

Subtractions:

Let’s assume that the subtraction properties are legitimate. Then they will be represented as K – L = L – K, where both variants are positive integers.

Example 1: K = 4 and L = 3

K — L = 4 – 3 = 1

L — K = 3 – 4 = -1

K — L > L – K

Example 2: K = 10 and L = 2

K – L = 10 — 2 = 8

L – K = 2 – 10 = -8

K — L > L – K

From these examples, we may infer that the integer reversions result in significant outcome differences. Hence, subtraction properties are not feasible. There is one exception, however, presented in Example 3.

Example 3: K = 10 and L = 10

K – L = 10 – 10 = 0

L – K = 10 – 10 = 0

K – L = L – K = 0

The subtraction rules will apply if K and L are identical.

Divisions:

How about divisions (M / N = N / M)? As is with subtractions, we will verify its authenticity through two tangible illustrations below:

Example 1: M = 9 and N = 3

M / N = 9 / 3 = 3

N / M = 3 / 9 = 1/3

M / N > N /M

Example 2: M = 16 and N = 4

M / N = 16 / 4 = 4

N / M = 4 / 16 = 0.25

M/ N > N / M 

Likewise, commutative properties and divisions are not compatible – except for Ns and Ms with equal values:

Example 3: M = N = 8

M / N = 8 : 8 = 1

N / M = 8: 8 = 1

M / N = N / M = 1

2. Associative Property

Associative formula – Source: pixabay

What Is Associative Property?

First of all, we will assess the term “Associate”. To “Associate” is to group or assemble. Here is a straightforward example: once several people get together for the same purpose, they form an association.

Such meanings also persist in Mathematics. The phrase “Associative Property” in Math refers to the notion that: when an assertion has three components, you may arrange them in any structure to solve the equation. 

The order in which integers are grouped does not hamper the outcome of their operation. This associative condition is functional for addition and multiplication processes. Neither the total nor the product shifts in value.

The Core Principle

Associative property poses a mathematical rule of numerical and algebraic concepts in essential terms. It asserts that when three elements (integers or variables) are added or multiplied, the result stays fixed regardless of the component organization.

So how does the grouping process work? The parentheses or square brackets ‘()’ are employed to cluster different items.

Consider these illustrations to grasp the group procedure: 

  • a + b + c is a standard ungrouped expression.
  • (a + b) + c is not different from the ungrouped a and b.
  • a + (b + c), equates to ungrouped b and c.
  • (a + c) + b is the same as the uncombined a and c.

Associative Properties of Additions

As per the Associative Properties of Additions, three or more integers yield persistent sums, remaining unchanged no matter how the digits are organized.

Suppose we have integer values: M, L, and K. In these instances, the following equations will be employed to explain the association properties of additions:

 K + (L + M) = (K + L + M).

Now we have a more tangible example: 10 + 4 + 6 = 20. Applying them to the aforementioned formula, we have (10 + 4) + 6 = 10 + (4 +6 ). 14 + 6 = 20 is obtained by solving the equation from the left-hand side, while 10 + 10 = 20 stems from our right-hand calculation. 

Here is another illustration: 2 + 6 + 5. Whether you group it as (2 + 5) + 6, (2 + 6) + 5, or 2 + (6 + 5), the results undergo no alteration.

Associative Properties of Multiplications

Similarly, in terms of multiplication, the product of three or more digits remains unchanged irrespective of how the integers are arranged.

Again, suppose we have three numbers: M, L, and K. In these instances, we have:

  1. ( L. K)= (M. L). K

One example is 240 = 10. 4. 6. The rule of Associative Property indicates that  (10. 4). 6 = 10. (4. 6). If you choose to solve the equation from left to right, we have got 40. 6 = 240. The reversion (from right to left) will be 10.  24 = 240.  

Is There An Associative Property of Subtractions and Divisions?

Subtractions:

Let’s tackle the subtraction issues first. Since they retain no stated formulation, we shall define the formula as (K – L) – M = K – (L – M).

Now, consider this equation: 18 – 7 – 10 = 1. Here is what happens when we apply our assumed subtraction property:

  • (18 – 7) – 10 = 1 
  • 18 – (7 – 10) = 18 + 3 = 21

Different combinations of the three variables yield different results. Thus, this law is nonbinding.

Divisions:

Our next tactic is to verify whether the property applies to division. Suppose that (K / L ) / M = K / (L / M) is the law. We will insert the expression 32 : 2 : 4 = 4 into this formula:

  • (32 : 2) : 4 = 4
  • 32 : (2 : 4) = 32. 2 = 64

These uneven results are palpable proof of the formula’s invalidity. In short, it is safe to say that the associative property for subtractions and divisions does not prevail.

3. Distributive Property

Divisions – Source: Free SVG

What Is Distributive Property?

When it boils down to definitions, the overall concept is quite basic. Let’s say you need to share a bar of chocolate with your group of friends. Cutting it into different parts is a must, correct? The same notions also apply to Mathematics.

Distributive properties relating to numbers and algebra are a well-known sentiment in the field. As their name implies, these rules emphasize the dispersion and division of a particular integer value under the right conditions.

Distributive Properties of Multiplications for Additions

The rule applies when we multiply an integer (the operand) with the superposition of two numbers (the addend): a (b + d) = ab + ad.

Let’s take 4 x (10 + 4) as an illustration. We might multiply each number by four before proceeding with the remainder of the equation. 

Therefore, it yields: 4 x (10 + 4) = 4 x 10 + 4 x 4 = 40 + 16 + 56.

Distributive Properties of Multiplications for Subtractions

Likewise, the subtraction law emerges in the multiplication of integers (operand) with a gap between two other numbers (addend): a.(b – d) = ab – ad.

For instance, to solve the equation 4 x (10 – 4), we employ the distributive principle and multiply each addend by four: 4 x (10 – 4) = 4 x 10 – 4 x 4 = 40 – 16 = 24.

Distributive Properties of Divisions

Our overall framework remains identical; one major exception is the replacement of multiplication symbols. Larger values are split into smaller segments (addends), while its divisor serves as the operand. This illustration below will help clear some clouds for you:

Example 1: 36 / 12 = ?

Answer:

36 = 24 + 12

=> 36 / 12 = (24 + 12) / 12

=> 36 / 12 = 24 / 12 + 12 / 12 = 2 + 1 = 3

Example 2: 45 / 15 = ?

Answer:

45 = 60 – 15

=> 45/ 15 = (60 – 15) / 15

=> 45 / 15 = 60 / 15 – 15 / 15 = 4 – 1 = 3

4. Identity Property

0 poses no value – Source: pixabay. 

What Is Identity Property?

In several different aspects of mathematics, the phrase “identity” refers to the same concept: a calculation that, given certain conditions, is valid regardless of the number entered into it. 

Identity Properties of Additions

This property indicates that the addition of 0 to any integer is identical to the number itself. Such a sentiment might be portrayed as: e + 0 = 0 + e = e, where e is an integer. No matter the e value, adding 0 to the combination will always result in e.

As per the commutative properties of addition, the position of 0 does not affect the result; after all, 0 poses no value. Hence, any sequence will yield the same amount as the original quantity. This phenomenon is known as “additive identification.”

Example 1: e = 2

e + 0 = 2 + 0 = 2

0 + e = 0 + 2 = 2

e + 0 = 0 + e = 2

Example 2: x = 100.000

e + 0 = 100,000 + 0 = 100,000

0 + e = 0 + 100,000 = 100,000

e + 0 = 0 + e = 100,000

Identity Properties of Multiplications

Multiplying “1” by any integer does not alter the number’s identity; it will retain its original form. Hence, experts regard “1” as the identity of multiplications. This principle, in essence, asserts that when a number is multiplied by 1, the result will always be the initial value.

Such establishments apply to any variable: integers, complex, rational, and real numbers, among other types. Its formula is y x 1 = y.

Example 1: y = 5

y x 1 = 5 x 1 = 5

1 x y = 1 x 5 = 5

y x 1 = 1 x y = 5

Example 2: y = 1000

y x 1 = 1000 x 1 = 1000

1 x y = 1 x 1000 = 1000

y x 1 = 1 x y = 1000

Conclusion

This article has shed light on the most critical mathematics properties. These technical terms might seem quite overwhelming at first glance, but they are pretty basic once you get the hang of their functions. Such properties serve as the groundwork for more intricate formulae of higher levels.

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