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Number Systems and Operations - Mathematics - 6th Grade

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Are you a math teacher struggling to keep your 6th-grade students engaged in learning about number systems and operations? Then, please pay attention, because we have the perfect solution for you! This creative template is designed with interesting illustrations that will capture your students' attention immediately and keep them engaged throughout the lesson. This template includes various exercises, making it the ideal tool to assist in teaching mathematics. Don't waste any more time searching, get this template for Google Slides or PowerPoint now and be 100% ready to teach this important subject matter.

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Number System Presentation

Number system is a way to represent, organize, and manipulate numbers. Different cultures and civilizations have developed their own number systems throughout history. The most commonly used number systems today are decimal, binary, octal, and hexadecimal.

The decimal number system, also known as base-10, uses 10 digits (0-9) to represent numbers. Each digit's value is determined by its position in the number. Decimal numbers can be expanded to include a decimal point and fractional values.

The binary number system, also known as base-2, uses only two digits (0 and 1) to represent numbers. Each digit's value is determined by its position in the number, starting from the rightmost digit. Binary numbers are widely used in computer science and digital electronics.

The octal number system, also known as base-8, uses eight digits (0-7) to represent numbers. Each digit's value is determined by its position in the number. Octal numbers are sometimes used in computer programming and Unix system permissions.

The hexadecimal number system, also known as base-16, uses sixteen digits (0-9 and A-F) to represent numbers. Each digit's value is determined by its position in the number. Hexadecimal numbers are commonly used in computer programming, particularly for representing memory addresses and color codes.

Converting between number systems involves changing the representation of a number from one base to another. Decimal to binary conversion involves repeated division by 2 and noting the remainders. Binary to decimal conversion involves multiplying each digit by the corresponding power of 2 and summing the results.

Decimal to octal conversion involves repeated division by 8 and noting the remainders. Octal to decimal conversion involves multiplying each digit by the corresponding power of 8 and summing the results. Hexadecimal to decimal conversion involves multiplying each digit by the corresponding power of 16 and summing the results.

Binary to octal conversion can be done by grouping binary digits into sets of three and converting each set to its octal equivalent. Octal to binary conversion involves converting each octal digit to its binary equivalent. Hexadecimal to binary conversion involves converting each hexadecimal digit to its binary equivalent.

Number systems are fundamental to mathematics, computer science, and digital electronics. They enable efficient representation and manipulation of numbers and data. Understanding number systems is crucial for programming, data analysis, and other technical fields.

Number systems are ways to represent, organize, and manipulate numbers. Decimal, binary, octal, and hexadecimal are commonly used number systems. Conversion between number systems involves changing the representation of a number from one base to another.

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8th grade Math – Number Systems Unit

Published by Mátyás Mészáros Modified over 5 years ago

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Presentation on theme: "8th grade Math – Number Systems Unit"— Presentation transcript:

Goal When you have finished this presentation you should understand:

Freshman Number Sets Week 3. Review Irrational Numbers Numbers that cannot be written as a fraction Rational Numbers Numbers that can be written as a.

Perfect Square Roots & Approximating Non-Perfect Square Roots

The Mysterious World of Number Identity…

CLASSIFYING NUMBERS 8 th grade Math – Numeration Unit 7 π -√36 ¾ ….

The Real Number System. The natural numbers are the counting numbers, without _______. Whole numbers include the natural numbers and ____________. Integers.

E Birbeck 7/04 Natural Numbers Whole Numbers Integers Rational Numbers.

The Real Number System.  Natural Numbers (AKA Counting Numbers): {1, 2, 3, 4, …}  Whole Numbers (Natural Numbers plus zero): {0, 1, 2, 3, …} NOTE: Both.

 Can be put in fractional form  The decimal form of the number either terminates (ends) or repeats.  Counting numbers, whole numbers, integers and.

The Real Number System -13 O, 1, 2, 3… 1/ π.

What is a Set? A SET is a group of objects that share similar characteristics or properties Example: Clothing.

The Real Number System. Whole numbers Whole numbers Rational numbers Whole numbers Natural numbers Integers / ¾ 18% π √2√2 − ….

-(-7.2) 1-(-3) -9+(-4.5) (-3.4)(-2) -15/3 -2/5 + 3/-5

Exploring Real Numbers. About Real Numbers ● "Real Numbers" are all the numbers that we deal with in math class and in life! ● Real Numbers can be thought.

5-3(D) Real Numbers.

Sets of Real Numbers (0-2)

1-1 REAL NUMBERS Bell-work 1.) 2x + 1 = x + 6.

(as opposed to fake numbers?)

Aim: How do we classify real numbers?

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PPT - Introduction to Number Systems - Computer Science Engineering (CSE) PDF Download

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That's why we've created a set of PowerPoint design templates with a Number System theme. These Number System PowerPoint templates are a great choice for a wide variety of presentation needs. Spend your time wisely - download the Number System PowerPoint Templates today.

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A PowerPoint template is a pattern or blueprint for your slides that you save as a .pptx or .potx file.

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Number Systems

Number systems stone age: knots, some stone marks roman empire: more systematic notation i, ii, iii, iv, v, vi, vii.viii, ix, x, c=100, d=500, m=1000, l=50 – powerpoint ppt presentation.

• Stone Age knots, some stone marks
• Roman Empire more systematic notation I, II, III, IV, V, VI, VII.VIII, IX, X, C100, D500, M1000, L50
• Concept of zero by
• Maya- I century, Hindu-V century
• Positional-value systems decimal, binary, octal, etc..
• The value of a digit (digit from Latin word for finger) depends on its position
• The basis of all digital data is binary representation.
• Binary - means two
• True, False
• We must be able to handle more than just values for real world problems
• True, False, Maybe
• Hot, Cold, LukeWarm, Cool
• On, Off, Leaky
• To talk about binary data, we must first talk about number systems
• The decimal number system (base 10) you should be familiar with!
• A digit in base 10 ranges from 0 to 9.
• A digit in base 2 ranges from 0 to 1 (binary number system). A digit in base 2 is also called a bit.
• A digit in base R can range from 0 to R-1
• A digit in Base 16 can range from 0 to 16-1 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Use letters A-F to represent values 10 to 15. Base 16 is also called Hexadecimal or just Hex.
• The traditional number system is called a positional number system.
• A number is represented as a string of digits.
• Each digit position has a weight assoc. with it.
• Numbers value a weighted sum of the digits
• Octal base 8
• Hexadecimal base 16
• Use A F to represent the values 10 through 16 in each position.
• Useful for representing multi-bit binary numbers because their radices are integer multiples of 2.
• To convert a binary number to a system which is base-2z, group digits together by z and convert each group separately
• 100111.1010 ---gt ( )16
• express number as a power series in I, and add all terms using decimal addition
• Radix-r-to-decimal conversions are easy since we do arithmetic in decimal.
• However, decimal-to-radix-r conversions using decimal arithmetic is harder.
• To do the latter conversion, we convert the integer and fractional parts separately and add the results afterwards.
• Integer part
• Divide the number and all successive quotients by r
• accumulate the remainders
• Fractional part
• Multiply the number and successive fractions by r
• accumulate the integers
• Successively divide number by r, taking remainder as result.
• Example Convert 5710 to binary.
• Successively multiply number by r, taking integer part as result and chopping off integer part before next iteration.
• May be unending!
• Example convert .310 to binary.
• Use same technique as decimal
• Except that the addition and subtraction tables are different
• Already seen addition table
• Truth table for Sum and Cout function
• Not really too different
• But the addition and subtraction tables must be developed.
• More accurately representation of signed numbers
• Signed-magnitude representation
• Radix-complement representation
• 2s-complement representation
• Diminished radix-complement representation
• Ones complement representation
• Excess representations
• Also called, sign-and-magnitude representation
• A number consists of a magnitude and a symbol representing the sign
• Usually 0 means positive, 1 negative
• Usually the entire number is represented with 1 sign bit to the left, followed by a number of magnitude bits
• Takes several steps to add a pair of numbers
• Examine signs of the addends
• If same, add magnitudes and give the result the same sign as the operands
• If different, must
• Compare magnitude of the two operands
• Subtract smaller number from larger
• Give the result the sign of the larger operand
• For this reason the signed-magnitude representation is not as popular as one might think because of its naturalness
• Negates a number by taking its complement instead of negating the sign
• Exact meaning of taking its complement is defined in various ways will see
• Not natural for humans, but better for machine arithmetic
• Will describe 2 complement number systems
• Radix complement very popular in real computers
• Diminished radix-complement not very useful, may skip or not spend much time on it
• Must first decide how many bits to represent the number say n.
• Complement of a number rn number
• Example 4-bit decimal
• Original number 3524
• 10s complement 10000-3524 6476
• 0 and positive numbers 0000-4999
• Negative numbers 5000-9999, where 9999 is minus 1.
• Just radix-complement when radix 2
• Used a lot in computers and other digital arithmetic circuits
• 0 and positive numbers leftmost bit 0
• Negative numbers leftmost bit 1
• To find a numbers complement just flip all the bits and add 1
• See graphical view Fig. 2.3, p. 40.
• Starting from 1000 (-8) on up, each successive 2s comp number all the way to 0111 (7) can be obtained by adding 1 to the previous one, ignoring any carries beyond the 4th bit position
• Since addition is just an extension of ordinary counting, 2s comp numbers can be added by ordinary binary addition!
• No different cases based on operands signs!
• Overflow possible
• Occurs if result is out of range
• To detect happens if operands are the same sign but sum is a different sign of that of the operands
• DECIMAL CODES
• Any encoding needs at least 4 bits/decimal digit
• BCD (8421), a weighted code
• Self-complementing the code for the 9s comp of any digit may be obtained by complementing the individual bits of the digits code word
• Not a weighted code, but is also self-complementing
• Since code follows standard binary counting sequence, standard binary counters can easily be made to count in excess-3
• Uses more than 4 bits
• First 2 bits indicate whether the number is in the range 0-4 or 5-0
• Last 5 bits indicate which of the five numbers in the selected range is represented
• Also one-hot
• Advantage error-detection property. If any 1 bit in a code word is accidentally changed to the opposite value, the resulting code word doesnt represent a decimal digit at all flagged as error.
• GRAY CODES and mechanical encodings
• Error Correcting Codes
• Grammar school method for decimal add a list of shifted multiplicands computed according to the digits of the multiplier
• Same method can be used in binary
• For two unsigned operands, the only possible values of the multiplier digits are 0 and 1
• Thus its trivial to form the shifted multiplicands
• More convenient to add each shifted multiplicand as it is created to a partial product
• Will do an example.
• In general when we multiply an n-bit number by an m-bit number, the result requires at most nm bits to express
• The shift-and-add algorithm requires m partial products and additions to obtain result, but the 1st addition is trivial (adding to 0)
• Convert the following binary numbers to decimal
• 1011011.0110
• 00110.11001
• 1 101 011 110 111
• 11 011.101 1
• Calculate 191141 (Lets first convert these to binary as an exercise.) Verify in decimal
• 210 109, calculate first binary numbers. Verify.
• Discuss how to convert hex, binary integers to Decimal
• Discuss how to convert decimal integers to hex, binary
• Discuss how to convert hex to binary, binary to Hex
• Explain why N binary digits can represent 2N values, unsigned integers 0 to 2N-1.
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Presentation Remote with Air Mouse Function, Wireless Presenter Clicker RF 2.4GHz USB Laser Pointer Presenter Control PowerPoint Presentation Clicker for Mac, Laptop, Computer

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The Real Number System

Jul 31, 2014

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The Real Number System. These are all the numbers that we know and use. . IRRATIONAL NUMBERS. RATIONAL NUMBERS. REAL NUMBERS. INTEGERS. WHOLE NUMBERS. Rational numbers organized into more specific categories…. Real numbers are split into two major categories…. NATURAL NUMBERS.

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The Real Number System These are all the numbers that we know and use.

IRRATIONAL NUMBERS RATIONAL NUMBERS REAL NUMBERS INTEGERS WHOLE NUMBERS Rational numbers organized into more specific categories… Real numbers are split into two major categories… NATURAL NUMBERS Each category can be considered a set of numbers.

Note: There are sets that are nested inside other sets for a reason. If an entire set is contained inside another set, then all members of the set must belong to both sets. REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS INTEGERS WHOLE NUMBERS NATURAL NUMBERS So all whole numbers are integers, rational numbers, and real numbers.

When classifying numbers, you need to consider the definitions of each set of numbers. • Rational numbers are any real numbers that can be written as fractions. (Any rational number can also be written as a repeating or terminating decimal.) • Irrational numbers are any real numbers that are not rational. An example of an irrational number would be pi or the square root of a number that does not give you a repeating or terminating decimal like the square root of 2. • Integers are positive and negative numbers without fractional parts and zero. • Whole numbers are all positive integers and zero. • Natural numbers are all positive integers. These can also be called counting numbers.

Any number that falls in the natural numbers would also be a whole number, an integer, a rational number and a real number… IRRATIONAL NUMBERS RATIONAL NUMBERS REAL NUMBERS REAL NUMBERS INTEGERS WHOLE NUMBERS Rational numbers organized into more specific categories… NATURAL NUMBERS Like the number 4 is… Real numbers are split into two major categories… 4

REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS What about the number -12? REAL NUMBERS INTEGERS WHOLE NUMBERS -12 Rational numbers organized into more specific categories… NATURAL NUMBERS Real numbers are split into two major categories… -12 is an integer, a rational number and a real number, but not a whole or natural number.

Example #1: Place the following numbers in the diagram to determine what kind of numbers they are. REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS REAL NUMBERS √‾ 5 ½ INTEGERS WHOLE NUMBERS Rational numbers organized into more specific categories… NATURAL NUMBERS 5.7 Real numbers are split into two major categories… -45 π 0 √‾ 9

There are standard symbols that can be used to represent the following sets of numbers: • Real Numbers: • Rational Numbers: • Integers: • Natural Numbers: ℝ ℚ ℤ ℕ

We will use: W for whole numbers I for irrational numbers. This will make things easier…

Listing the Sets of Rational Numbers: These are all infinite sets of numbers. Integers: Z = {… , -3, -2, -1, 0, 1, 2, 3, …} Whole Numbers: W = {0, 1, 2, 3, …} Natural Numbers: N = {1, 2, 3, …} Just another way to look at these…

Example 2: Write the symbol of all the sets to which the following numbers belongs. 0: ¾: -8: 6½: -4.3: 9.2: ℚ W ℤ ℝ Don’t be afraid to use the diagram to help with this! ℚ ℝ I ℚ ℤ ℝ ℚ ℝ ℚ ℝ ℝ ℚ ℝ ℚ ℤ ℚ ℝ W ℕ W ℤ ℚ ℝ ℕ I ℝ

Example #3: Which of the following is an irrational number? Why? =12 =9 4.79583152331271 =25 This is irrational because 23 is a prime number and if you take the square root of a prime number you will get a non-repeating decimal.

Example #4: Which of the following are not natural numbers? Why? =8 -3 is not a natural number because no negative numbers are in the natural numbers. 2/3 is not a natural number because no fractions are natural numbers. Remember we sometimes call natural numbers counting numbers we do not count objects using negative or fractional numbers.

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