Subtraction is the operation which finds the difference between the given two numbers. This is also one of the basic operations. There are two terms used in subtraction they are minuend and subtrahend. Where minuend is the term to be subtracted and subtrahend is the term which subtracts the minuend. When the value of the minuend is greatest then the result is positive. When the value of the subtrahend is greatest then the value is negative.
Subtraction Formula:
We shall learn on subtraction formula. The fixed names for the parts of the formula:
c − b = a
Are minuend(c) –subtrahend (b) = difference (a).
As a substitute that c and −b are terms, and subtract as addition of the opposite. The solution is still called the difference.
As a substitute that c and −b are terms, and subtract as addition of the opposite. The solution is still called the difference.
Let us understand with examples
The following problems are simple subtraction.
1) 9 - 9 = 0
The value 6 minus value 6 equal to 0
The solution is 0(zero).
2) 12 – 24 = - 12
The value 12 minus value 24 equal to minus 12
The solution is minus 12
3) 6 – 4 = 2
The value 6 minus value 4 equal to 2
The solution is 2
4) 10 – 7 = 3
The value 10 minus value 7 equal to 3
The solution is 3
5) 5.3 – 1.1 = 4.2
The decimal value 5.3 minus decimal value 1.1 equal to 4.2
A monomial is an algebraic expression in which the only operations that appear between the variables are the product and the power of the natural exponent.
2x2y3z
Parts of a Monomial
Coefficient
The coefficient of a monomial is the number that multiplies the variable(s).
Literal Part
The literal part is constituted by the letters and its exponents.
Degree
The degree of a monomial is the sum of all exponents of the letters or variables.
The degree of 2x2 y3 z is: 2 + 3 + 1 = 6
Similar Monomials
Two monomials are similar when they have the same literal part.
2x2 y3 z is similar to 5x2 y3 z
Degree of a monomial:
The sum of the degrees/exponents of each of the variables in a monomial is called the degree of monomial.
Note: In a non-variable monomial, that is a constant which is any number other than zero, degree is always zero. Like 6, 3, 4, 2 etc. Have a degree zero!
Degree of a Monomial: The degree of a monomial is the sum of the exponents of all its variables. For example, the degree of xy is 2, the degree a^3 b of is 4, the degree of 12 is 0.
Degree of a Polynomial: The degree of a polynomial is the greatest degree of any term in the polynomial. For example, the degree of x^5 - 5x^ 2 is 5, the degree of x^3+y^4-3x^2 = 2y is 4.
Solved Example on Degree
Find the degree of x^8y^6 + x^9 y^6.
Choices:
A. 8
B. 9
C. 16
D. 15
correct answer is d
Solution
Step 1: The degree of a polynomial is the greatest degree of any term in the polynomial.
Step 2: The degree of the second term in x^8y^6 + x^9 y^6 is ‘15’, which is the highest. So, the degree of the given polynomial is 15.
Related Terms for Degree
Monomial
Exponent
Polynomial
Variable
Sum
The degree of a monomial is the sum of the degrees in each of its variables.
The degree of monomial is the sum of the exponents of the variable in the monomial.
The degree of a nonzero constant term is 0.
The constant 0 does not have a degree.
Example: Find the degree of monomial x7y3z2.
Solution: Sum of the degree of in each variables is = 7+3+2 = 12.
Therefore the degree of the monomial is 12.
In our next we shall learn about degree of a polynomial
Algebraic expressionsare formed from variables and constants. We simply use the combination of operations of addition, subtraction, multiplication and division on the variables and constants to form expressions. Algebraic expressions does not contain equal sign.
Some examples of algebraic expressions:
2x + 2y , 2xy , x2 - y2 , 3m + n - 5.
There are one or more terms in the algebraic expressions. For example, 3x2 + 4x + 2 is algebraic expression. It has three terms.
Let us discuss about some example problems on Algebraic expressions.
An algebraic expression is a number, a variable or a combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots.
For eg.: 4x, 3p, 2a+b, 6z/5
Introduction about Algebraic expressions:
Expressions are a central concept in algebra. A variablecan take various values. Its value is not fixed. On the other hand, a constanthas a fixed value. We combine variables and constants to make algebraic expressions. For this, we use the operations of addition, subtraction, multiplication and division. We have already come across simple algebraic expressions like x + 3, y– 5, 4x + 5, 10y– 5 and so on. The above expressions were obtained by combining variables with constants. We can also obtain expressions by combining variables with themselves or with other variables.In this article we will see some definitions, facts about algebraic expressions
Terms about Algebraic Expressions
Properties about algebraic Expressions:
Like terms: When the terms are same variables then that terms are called as like terms. Otherwise it is called as unlike terms
2xy and 5xy because the factors of 2xy are 2 and x and y. and factors of 5xy are 5 and x and y both factors are same. So these are like terms.
Unlike terms: The terms 2xy and –3x , have different algebraic factors. They are unliketerms. Similarly, the terms, 2xy and 4 are unlike terms. Also, the terms –3x and -4 are unlike term.
Coefficient of the Term: The value in front of the variable is called as the coefficient of the term. The above example the coefficient of x is -3.
Facts about Algebraic Expressions
Monomial: The number of terms in an expression is one then the expression is called as monomial. For example 5x
Binomial:The number of terms in an expression is two then the expression is called as binomial. For example,
x + y, m – 5
Trinomial: The number of terms in an expression is three then the expression is called as trinomial. for example, the expressions x + y + 7, a b + a +b
Polynomial:An expression with one or more terms is called a polynomial. Thus a monomial, a binomial and a trinomial are all polynomials. For Example: 2x3+5x2+15x+22.
Finding the value of an expression:
This is nothing but substituting the values to the variables in the expression. That gives the solution to the expression. For example the area of a square is length2, where l is the length of a side of the square. If l = 8 cm., the area is 82 cm2or 64 cm2; if the side is 12 cm, the area is 122 cm2 or 144 cm2 and so on.
When an expression is written using words or phrases, it is a verbal expression. It is in a statement form. The symbols and statements used have the same meaning as they have in arithmetic.
Signs and symbols:
‘=’ means ‘is equal to’
‘≠’ means ‘is not equal to’
‘<’ means ‘is less than’
‘>’ means ‘is greater than’
‘≤’ means ‘is less than or equal to’
‘≥’ means ‘is greater than or equal to’
The following are the list of words used and their meaning:
1. Addition : sum, plus, add to, more than, increased by, total
2. Subtraction: difference of, minus, subtracted from, less than , decreased by, less
3. Multiplication: product, times, multiply, twice, of
4. Division: quotient , divide, into, ratio
Variable expressions to algebraic expressions:
In Algebra an expression is a term which consists of both symbols and numbers. There are two types in that. They are Algebraic and Verbal expressions. Algebraic expression consists of signs and symbols. The signs involve signs of several operations like addition, subtraction, multiplication and division. The symbol involves all Arabic numerals and literal numbers. Ex: 3ax, 4yz…Verbal expression is the expression which consists of verbal phrases like sum, product, less than, etc…..
Verbal Expressions into Algebraic Expressions Explanation:
Some of the important operations that are used for this verbal expression into algebraic expression are,
In addition with these basic operations some other operations like less than and greater than...
Less than <
Greater than >
Less than or equal <=
Greater than or equal >=
Types of Algebraic Expressions
Monomial
A monomial is an algebraic expression formed by a single term.
Binomial
A binomial is an algebraic expression formed by two terms.
Trinomial
A trinomial is an algebraic expression formed by three terms.
Polynomial
A polynomial is an algebraic expression consists of more than one term.
In math, An algebraic expression is an expression containing symbols, variables and constants together. In math, Verbal expression is a sentence forming from an algebraic expression. Some of the verbal phrases for arithmetic operation is given below. Using these, we can frame the verbal expression for given an algebraic expression.
Let us see brief about verbal expression in math.
Verbal Expression in Math:
The verbal expression for an algebraic expression a + b may write as following,
a plus b , a added to b, a is increased by b, the sum of a and b, b is added to a, b more than a.
The verbal expression for an algebraic expression a - b may write as following,
a minus b , a is decreased by b, b subtracted from a, b less than a, a diminished by b, a reduced by b, the difference between a and b.
The verbal expression for an algebraic expression a x b may write as following,
a times b , the product of a and b, b is multiplied by a.
The verbal expression for an algebraic expression a ÷ b may write as following,
The quotient of a and b, a is divided by b.
Let us learn about how to translate algebraic expression into verbal expression.
Introduction to verbal expression in math:
In math, An algebraic expression is an expression containing symbols, variables and constants together. In math, Verbal expression is a sentence forming from an algebraic expression. Some of the verbal phrases for arithmetic operation is given below. Using these, we can frame the verbal expression for given an algebraic expression.
Let us see brief about verbal expression in math.
Verbal Expression in Math:
The verbal expression for an algebraic expression a + b may write as following,
a plus b , a added to b, a is increased by b, the sum of a and b, b is added to a, b more than a.
The verbal expression for an algebraic expression a - b may write as following,
a minus b , a is decreased by b, b subtracted from a, b less than a, a diminished by b, a reduced by b, the difference between a and b.
The verbal expression for an algebraic expression a x b may write as following,
a times b , the product of a and b, b is multiplied by a.
The verbal expression for an algebraic expression a ÷ b may write as following,
The quotient of a and b, a is divided by b.
Let us learn about how to translate algebraic expression into verbal expression.
Example Problems of Verbal Expression in Math:
Problem 1:
Translate this algebraic expression into verbal expression: 3 + b
Solution:
We can write it as,
b more than 3, 3 plus b, 3 added to b, b is increased by 3.
Problem 2:
Translate this algebraic expression into verbal expression: 3 - b
Solution:
We can write it as,
b less than 3, 3 minus b, b subtracted from 3, b is decreased by 3.
Decimal number is a number. It has two parts. That is whole number part and a fractional part. The “.” is called the decimal point.
Ordering decimals:
Suppose if we have a two or more decimal, we need to arrange the decimals using tenths, hundreds and thousands place.
Comparing decimals:
Suppose if we have a two decimal numbers we can compare them. Normally one number is greater than, less than or equal to another number. Here we are going to see some examples for how to compare and order decimal values.
Mathematical Concepts in Comparing and Ordering Decimals
To compare decimals start at the left and compare digits in the same places, find first the place where the decimals differs,
Equal decimals can be written by adding zeros or deleting zeros from the right of the last non zero digit.
Adding zeros at the end of the decimal does not change it value. On the other hand, adding zeros before the first decimals digit changes its value.
To compare and order decimals having different number of digits, put zeros at the end of the shorter decimals to have the same number of digits, then compare and order the numbers as if the decimal points.
Comparing and Ordering Decimals:
Compare and order decimals through hundred thousandths.
Use the symbols >, <, or = correctly in comparing decimals.
List decimal in the order least to greatest.
Problem in comparing and ordering decimals
When you compare decimals, think about place value you can add or remove zeros at the end of the decimal without changing its meaning.
0.2 = 0.20 =0.200 =0.2000
0.54 = 0.540 = 0.5400 = 0.54000
3.5 = 3.50 = 3.500 = 3.5000
The ordering decimal is 3.5, 0.2,0.54.
Rule : To compare two decimals, first convert the given decimals into like decimals and then compare the whole number parts. The decimal with greater whole number part is greater. If the whole number parts are equal, then compare the tenths digits. The decimal with the bigger digit at tenths place is greater. If the tenths digits are equal, then compare the hundredth digits and so on.
i)In 2.5 and 3.5
Here we can notice that both are like decimals number and the tenths place is same so, comparing the whole number parts of two decimals, we have 3 > 2
Hence, 3.5 > 2.5
ii) In 15.12 and 15.39
Here we can notice that both are like decimals num
ber so, comparing the whole number parts of two decimals, we have 15=15
Now comparing the digits at tenths place, we have 1 <>
Hence, 15.12 <>
iii)In 5.147 and 5.14
Here we need to convert the unlike decimal to like decimal i.e., 5.14 = 5.140, so compare 5.147 and 5.140
Comparing the whole number parts of two decimals, we have 5 = 5
Now, comparing the digits at tenths place, we have 1 = 1
Next, comparing the digits at hundredths place, we have 4 = 4
Finally, comparing the digits at thousandths place, we have 7 > 0
Hence, 5.147 > 5.14.
iv)In 3.159 and 3.15
Here we need to convert the unlike decimal to like decimal i.e., 3.15 = 3.150, so compare 3.159 and 3.150
Comparing the whole number parts of two decimals, we have 3 = 3
Now, comparing the digits at tenths place, we have 1 =1
Next, comparing the digits at hundredths place, we have 5 = 5
Finally, comparing the digits at thousandths place, we have 9 > 0
Hence, 3.159 > 3.150.
5)Arrange 12.4, 135.25, 14.648, 115.35 in ascending order and descending order.
Converting the given decimals into like decimals, we have,
12.400, 135.250, 14.648, 115.350.
Here we notice that in all the numbers the whole number part is different so we need to arrange the whole number from smaller to bigger.
Thus arranging in ascending order, 12.400 <>
Hence, the given decimals in ascending order are :
12.400 <>
Thus arranging in descending order we need to arrange the whole number from bigger to smaller,
135.250 > 115.350> 14.648 > 12.400 since 135 > 115>14>12.
Hence, the given decimals in descending order are :
This step explains the kids that how to compare the decimal valuesWhen 0's are add to the right of the decimal number, the number is not changed.
To compare decimals, we compare the values,
Step 1: Start at the left side; Solve the first place where the numbers are different
Step 2: Compare the digits that are different.
If there are two decimal numbers we can compare them. One number is either greater than, less than or equal to the other number.
A decimal number is just a fractional number. Comparing 0.7 and 0.07 is clearer if we compared 7/10 to 7/100. The fraction 7/10 is equivalent to 70/100 which is clearly larger than 7/100.
place and then hundredths place, etc. If one decimal has a higher number in the tenths place then it is larger than a decimal with fewer tenths. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare. If each decimal place value is the same then the decimals are equal.
Place Value
To understand decimal numbers you must first know about
When we write numbers, the position (or "place") of each number is important.
In the number 327:
the "7" is in the Units position, meaning just 7 (or 7 "1"s),
the "2" is in the Tens position meaning 2 tens (or twenty),
and the "3" is in the Hundreds position, meaning 3 hundreds.
As we move left, each position is 10 times bigger! From Units, to Tens, to Hundreds
... and ...
As we move right, each position is 10 times smaller
From Hundreds, to Tens, to Units
And that is a Decimal Number!
Decimal Point
The decimal point is the most important part of a Decimal Number. It is exactly to the right of the Units position. Without it, we would be lost ... and not know what each position meant.
Now we can continue with smaller and smaller values, from tenths, to hundredths, and so on, like in this
example:
Large and Small
So, our Decimal System lets us write numbers as large or as small as we want, using the decimal point. Numbers can be placed to the left or right of a decimal point, to indicate values greater than one or less than one.
17.591
The number to the left of the decimal point is a whole number
As we move further left, every number place gets 10 times bigger.
The first digit on the right meanstenths (1/10).
As we move further right, every number place gets 10 times smaller (one tenth as big).
