Contents

# Reduce|Definition & Meaning

## Definition

To reduce an **equation** means to convert it into a **simpler** (usually the simplest) form. For example, we can reduce (3x + 2) / (9x + 3) = 3x + 2 to its simplest form by **cross** multiplying 3x + 2 on both sides, followed by cross multiplying 9x + 3 on both sides, which would give 9x + 3 = 1. Then, we could **subtract** one on both sides to get 9x + 2 = 0, which is the **simplest** form.

Figure 1 – Reduce Transformation

## What Does Reduce Do in Math?

In mathematics, the term “reduce” can have different **meanings** depending on the context. In general, it refers to the process of **simplifying** or bringing an expression or equation to a simpler or more **standard** form.

For example, in algebra, **reducing** an expression means to simplify it by combining **like** terms, such as combining **3x** and **4x** into **7x**.

Another example is reducing a fraction, which means to **divide** both the numerator and denominator by their greatest common factor (**GCF**) to make the fraction in its simplest form.

In trigonometry and calculus, reducing a **function** means to simplify it by applying algebraic and trigonometric **identities**, or by applying the properties of **derivatives** and integrals.

In **linear** algebra, reducing a **matrix** means to transform it into a simpler form by applying elementary **row** operations such as swapping rows, multiplying rows, and adding rows.

In **number** theory, reducing a number means to divide it by a common **divisor** to obtain the greatest common divisor (**GCD**) of two or more numbers.

In general, reducing a mathematical expression means to **simplify** it by applying the appropriate mathematical **rules** and procedures to make it easier to understand, solve, or **manipulate**.

## Advantages and Disadvantages of Reducing in Math

Advantages of reducing a **mathematical** expression or equation:

- Simplifies the
**expression**or equation, making it easier to**understand**, solve, or manipulate. - Allows for the identification of patterns or
**relationships**within the expression or equation. - Allows for the identification of
**equivalent**expressions or equations, which can be useful in solving problems. - Can make it easier to check the
**correctness**of a solution.

**Disadvantages** of reducing a mathematical expression or equation:

- Can be
**time-consuming**, especially for more complex expressions or equations. - Can be
**error-prone**, if the steps are not done correctly, it might lead to wrong results. - Can make the expression or equation more
**difficult**to understand for someone who is not familiar with the reduction method used. - Can make it more difficult to check the
**correctness**of the solution if the expression or equation is highly reduced.

In general, reducing mathematical expressions and **equations** is a powerful tool for solving problems and understanding mathematical concepts. However, it is important to consider the context and the intended **audience** when reducing an expression or equation, as it can sometimes be more beneficial to leave it in a **less-reduced** form.

Figure 2 – Reduce Fractions

## How Do I Reduce a Fraction?

To reduce a fraction means to divide both the numerator and denominator by their greatest common factor (**GCF**) to make the fraction in its simplest form. Here is the **process** of reducing a fraction:

Find the **GCF** of the numerator and denominator: The GCF is the **largest** number that divides both the numerator and denominator without leaving a **remainder**. You can find the GCF using the **Euclidean** algorithm or by listing the prime factors of both the numerator and denominator and seeing which ones are** common**.

Divide both the numerator and denominator by the GCF: Once you have found the GCF, **divide** both the numerator and denominator by it. This will give you the **reduced** form of the fraction.

**Example**: Reduce the fraction 15/45.

First, we find the **GCF** of 15 and 45.

Factors of 15: 1, 3, 5, **15**

Factors of 45: 1, 3, **15**, 45

Divide both the numerator and denominator by the **GCF**, so 15/15 = **1** (numerator) and 45/15 = **3** (denominator). The reduced form of the fraction is **1/3**.

It’s important to note that if the **GCF** is **1**, then the fraction is already in its **simplest** form, and no further reduction is needed.

It’s also possible to use a **calculator** to reduce a fraction. Most of them have a specific button or function to reduce a fraction.

In general, reducing a fraction is important in order to make the fraction **easier** to understand and work with, and it’s a **fundamental** step in solving many mathematical problems.

## What Happens in a Reduction?

In a reduce operation, a function is repeatedly applied to the elements of a **sequence** or iterable, such as a list or an array, in order to combine or “reduce” them into a **single** value. The function that is applied is often called the “reduce function” or “**accumulator** function,” and it should take two arguments and return a single value.Â

The reduce operation starts with the first two **elements** of the sequence, applies the reduce function to them, and then uses the **result** of that as the first argument for the next application of the reduce function, and so on until a single **value** is obtained.

## Is Reduce Positive or Negative?

In general, “reduce” can have both positive and negative **connotations** depending on the context in which it is used.

Positive connotations of “reduce” might include things like reducing **waste** or reducing pollution, which is generally considered good for the **environment** or society as a whole.

Negative connotations of “reduce” might include things like reducing **staff** or reducing benefits, which can have negative impacts on **individuals** or groups.

In summary, “reduce” is a **neutral** term, but its connotation depends on the **context** of the usage.

Figure 3 – Reduce Method of Drawing

## What Is the Reduced Method of Drawing?

The reduced method of **drawing**, also known as “reductive drawing,” is a technique in which an artist begins with a detailed, fully **rendered** image and then gradually removes or simplifies elements to create a more **minimal** and abstract final product.Â

This method is often used in **charcoal** drawing, but it can also be applied to other mediums such as pencil, ink, or even **digital** drawing.

The process starts by drawing a **detailed** image, then the artist will erase or cover parts of the image, gradually **simplifying** and removing details. The goal is to reveal the essential forms and shapes of the **subject** and to simplify the image to its most basic, yet still **recognizable**, form.

Reductive drawing is a great way to **explore** the underlying structure of a subject and to experiment with different ways of simplifying and **abstracting** an image. It can also help artists to develop their ability to see and understand the basic **shapes** and forms that make up an image, which is an important skill for any artist to have.

## Some Examples of Reduce

### Example 1

Reduce the following **expressions**:

- 2x + 3x – 5x.
- 8/12
- (x + y)
^{2} - 2x + 3 = 7

### Solution

Answer to part (**a**): 0x

Answer to part (**b**):2/3

Answer to part (**c**): x^{2} + 2xy + y^2

Answer to part (**d**): x = 2

### Example 2

Try to **reduce** the following expression:

- sin
^{2}(x) + cos^{2}(x) - x
^{2}+ 5x + 6 - log(a
^{3}* b^{2}) - 3(2x + 4) – 5(x – 2)

### Solution

Answer to part (**a**): 1

Answer to part (**b**): (x + 3)(x + 2)

Answer to part (**c**): log(a^{3}) + log(b^{2})

Answer to part (**d**): 6x + 6 – 5x + 10 = x + 16

*All images were created with GeoGebra.*