程序代写代做代考 compiler algorithm flex mips Java x86 assembly computer architecture assembler Microsoft PowerPoint – CSE220 Unit03 MIPS Assembly – Basics.pptx
Microsoft PowerPoint – CSE220 Unit03 MIPS Assembly – Basics.pptx
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1Kevin McDonnell Stony Brook University – CSE 220
CSE 220:
Systems Fundamentals I
Unit 3:
MIPS Assembly:
Basic Instructions,
System Calls, Endianness
2Kevin McDonnell Stony Brook University – CSE 220
Computer Instructions
• Recall that a computer’s architecture is the programmer’s
view of a computer
• The architecture is defined, in part, by its instruction set
• These instructions are encoded in binary as the
architecture’s machine language
• Because reading and writing machine language is tedious,
instructions are represented using mnemonics (“neh-
MAHN-icks”) as assembly language
• You are going to be learning the fundamentals of the MIPS
architecture in this course, so that means we will be
covering MIPS assembly language
3Kevin McDonnell Stony Brook University – CSE 220
MIPS Architecture Design Principles
1. Simplicity favors regularity
• Simple, similar instructions are easier to encode and
handle in hardware
2. Make the common case fast
• MIPS architecture includes only simple, commonly used
instructions
3. Smaller is faster
• Smaller, simpler circuits will execute faster than large,
complicated ones that implement a complex instruction
set
4. Good design demands good compromises
• We just try to minimize their number
4Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly
• Why MIPS and not x86?
• Like x86, MIPS CPUs are used in real products, but mostly
embedded systems like network routers
• The MIPS architecture is simpler than x86 architecture,
which makes the assembly language simpler and easier to
learn
• Good textbooks and educational resources exist for MIPS
• Once you learn one architecture and its assembly language,
others are easy to learn
• Fun fact: MIPS was invented by John Hennessy, the former
president of Stanford University and alumnus of Stony
Brook’s MS and PhD programs in computer science
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5Kevin McDonnell Stony Brook University – CSE 220
Instructions: Addition
• Java code: MIPS assembly code:
a = b + c; add a, b, c
• add: mnemonic indicates operation to perform
• b, c: source operands (on which the operation is to be
performed)
• a: destination operand (to which the result is written)
• In MIPS, a, b and c are actually CPU registers. More on this
soon.
6Kevin McDonnell Stony Brook University – CSE 220
Instructions: Subtraction
• Java code: MIPS assembly code:
a = b – c; sub a, b, c
• sub: mnemonic
• b, c: source operands
• a: destination operand
• It should come as no surprise that the code is virtually the
same as for an addition
1. Simplicity favors regularity
• Consistent instruction format
• Same number of operands (two sources and one
destination)
• Easier to encode and handle in hardware
7Kevin McDonnell Stony Brook University – CSE 220
Multiple Instructions
• More complex code is handled by multiple MIPS instructions
• Java code: MIPS assembly code:
a = b + c – d; add t, b, c # t = b + c
sub a, t, d # a = t – d
• In MIPS assembly, the # symbol denotes a comment
2. Make the common case fast
• More complex instructions (that are less common) are
performed using multiple simple instructions
• MIPS is a reduced instruction set computer (RISC), with
a small number of simple instructions
• Other architectures, such as Intel’s x86, are complex
instruction set computers (CISC)
8Kevin McDonnell Stony Brook University – CSE 220
Operands
• Operand location: physical location in computer
• Registers: MIPS has thirty-two 32-bit registers
• Faster than main memory, but much smaller
3. Smaller is faster: reading data from a small set of
registers is faster than from a larger one (simpler
circuitry)
• MIPS is a 32-bit architecture because it operates on
32-bit data
• Memory
• Constants (also called immediates)
• Included as part of an instruction
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9Kevin McDonnell Stony Brook University – CSE 220
MIPS Register Set
Name Register Number Usage
$0 0 the constant value 0
$at 1 assembler temporary
$v0-$v1 2-3 function return values
$a0-$a3 4-7 function arguments
$t0-$t7 8-15 temporaries
$s0-$s7 16-23 saved variables
$t8-$t9 24-25 more temporaries
$k0-$k1 26-27 OS temporaries
$gp 28 global pointer
$sp 29 stack pointer
$fp 30 frame pointer
$ra 31 function return address
10Kevin McDonnell Stony Brook University – CSE 220
Operands: Registers
• Registers:
• $ before name
• Example: $0, “register zero”, “dollar zero”
• Registers are used for specific purposes:
• $0 always holds the constant value 0.
• The saved registers, $s0-$s7, are used to hold
variables
• The temporary registers, $t0-$t9, are used to hold
intermediate values during a larger computation
• We will discuss other registers later
• Programming in MIPS assembly demands that you follow
certain rules (called conventions) when using registers.
11Kevin McDonnell Stony Brook University – CSE 220
Instructions with Registers
• Let’s revisit the add instruction
• Java code: MIPS assembly code:
a = b + c; # $s0 = a, $s1 = b, $s2 = c
add $s0, $s1, $s2
• When programming in a high-level language like Java, you
generally don’t (and shouldn’t) comment every single line
of code
• With MIPS assembly you should!
• Assign meaning to registers and calculations
• Even simple formulas will have to be implemented using
at least several lines of assembly code
• So comment EVERYTHING
12Kevin McDonnell Stony Brook University – CSE 220
Operands: Memory
• A typical program uses too much data to fit in only 32
registers
• Store more data in memory
• Memory is large, but slow
• Commonly used variables kept in registers
• Less frequently used values will need to be copied from
registers to memory for “safe keeping” when we run out
of registers
• Later we will need to copy the values from memory back
to the register file when we need to do a calculation with
them
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13Kevin McDonnell Stony Brook University – CSE 220
Operands: Memory
• Each 32-bit data word has a unique 32-bit address
• A word is the unit of data used natively by a CPU
• A possible logical structure of main memory (word-
addressable), which is not how MIPS actually works:
14Kevin McDonnell Stony Brook University – CSE 220
Operands: Memory
• Most computer architectures cannot read individual bits
from memory
• Rather, the architecture’s instruction set can process only
entire words or individual bytes
• If the smallest unit of data we can read from memory is a
word, we say that the memory is word-addressable
• In this case, memory addresses would be assigned
sequentially, as in the previous figure
• The MIPS architecture, in contrast, is byte-addressable
• Each byte has its own memory address
15Kevin McDonnell Stony Brook University – CSE 220
Operands: Memory
• Byte-addressable memory (what MIPS uses)
16Kevin McDonnell Stony Brook University – CSE 220
Reading Word-Addressable Memory
• A memory read is called a load
• Mnemonic: load word (lw)
• Format: lw $s0, 16($t1)
• Address calculation:
• add base address ($t1) to the offset (16)
• so a register first needs to have the base address that we
want to add to the offset
• effective address = ($t1 + 16)
• Result: $s0 holds the value at address ($t1 + 16)
• Any register may be used to hold the base address
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17Kevin McDonnell Stony Brook University – CSE 220
Reading Word-Addressable Memory
• Example: suppose we want to read a word of data at
memory address 8 into $s3
• address = ($0 + 8) = 8
• So what we want is for
$s3 to hold 0x01EE2842
• Assembly code:
# read memory
# word 2 into $s3
lw $s3, 8($0)
18Kevin McDonnell Stony Brook University – CSE 220
Writing Word-Addressable Memory
• A memory write is called a store
• Mnemonic: store word (sw)
• Example: suppose we wanted to write (store) the value in
register $t4 into memory address 8
• offset for loads and stores can be written in decimal
(default) or hexadecimal
• add the base address ($0) to the offset (0x8)
• address: ($0 + 0x8) = 8
• Assembly code:
sw $t4, 0x8($0) # write the value in
# $t4 to memory addr 8
19Kevin McDonnell Stony Brook University – CSE 220
Big-Endian & Little-Endian Memory
• Each 32-bit word has 4 bytes. How should we number the
bytes within a word?
• Little-endian: byte numbers start at the little (least
significant) end
• Big-endian: byte
numbers start at the big
(most significant) end
• LSB = least significant
byte; MSB = most
significant byte
• Word address is the
same in either case
20Kevin McDonnell Stony Brook University – CSE 220
Big-Endian & Little-Endian Memory
• Suppose $t0 initially contains 0x23456789
• After following code runs, what value is in $s0?
sw $t0, 0($0)
lb $s0, 1($0)
• Big-endian: 0x00000045
• Little-endian: 0x00000067
• The MIPS simulator we will use is little-endian
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21Kevin McDonnell Stony Brook University – CSE 220
Byte-Addressable Memory
• Each data byte has a unique address
• Load/store words or single bytes: load byte (lb) and store
byte (sb)
• 32-bit word = 4 bytes,
so word addresses
increment by 4
• So when performing a
lw or sw, the effective
address must be a
multiple of 4
22Kevin McDonnell Stony Brook University – CSE 220
Byte-Addressable Memory
• When loading a byte, what do we do with the other 24 bits
in the 32-bit register?
• lb sign-extends to fill the upper 24 bits
• Suppose the byte loaded is zxxx xxxx � 8 bits
• The bit z is copied into the upper 24 bits
• Normally with characters do not want to sign-extend the
byte, but rather prepend zeroes
• This is called zero-extension
• MIPS instruction that does zero-extension when loading
bytes:
• load byte unsigned: lbu
23Kevin McDonnell Stony Brook University – CSE 220
Reading Byte-Addressable Memory
• The address of a memory word must be a multiple of 4
• For example,
• the address of memory word #2 is 2 × 4 = 8
• the address of memory word #10 is 10 × 4 = 40 (0x28)
• So do not forget this: MIPS is byte-addressed, not word-
addressed!
• To read/write a word from/to memory, your lw/sw
instruction must provide an effective address that is
word-aligned
24Kevin McDonnell Stony Brook University – CSE 220
Instruction Formats
4. Good design demands good compromises
• Multiple instruction formats allow flexibility
• add, sub: use 3 register operands
• lw, sw: use 2 register operands and a constant
• Number of instruction formats kept small…
• …to adhere to design principles 1 and 3 (simplicity
favors regularity and smaller is faster)
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25Kevin McDonnell Stony Brook University – CSE 220
Instruction Formats
• lw and sw use constants or immediates
• Immediately available from instruction
• The immediate value is stored in the instruction as a 16-bit
two’s complement number
• addi: add immediate
• Is subtract immediate (subi) necessary?
• Java code: MIPS assembly code:
# $s0 = a, $s1 = b
a = a + 4; addi $s0, $s0, 4
b = a – 12; addi $s1, $s0, -12
26Kevin McDonnell Stony Brook University – CSE 220
Machine Language
• Binary representation of instructions
• Computers only “understand” 1’s and 0’s
• 32-bit instructions
• Simplicity favors regularity: 32-bit data & instructions
• 3 instruction formats:
• R-Type: register operands (register-type instruction)
• I-Type: immediate operand (immediate-type
instruction)
• J-Type: for jumping (jump-type instruction) – more on
this later
27Kevin McDonnell Stony Brook University – CSE 220
R-Type Instructions
• 3 register operands:
• rs, rt: source registers
• rd: destination register
• Other fields:
• op: the operation code or opcode (0 for R-type)
• funct: the function; with opcode, tells CPU what
operation to perform
• shamt: the shift amount for shift instructions;
otherwise it’s 0
28Kevin McDonnell Stony Brook University – CSE 220
R-Type Examples
• Note the order of registers in the assembly code:
add rd, rs, rt
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29Kevin McDonnell Stony Brook University – CSE 220
I-Type Instructions
• 3 operands:
• rs, rt: register operands
• imm: 16-bit two’s complement immediate
• Other fields:
• op: the opcode
• Simplicity favors regularity: all instructions have opcode
• Operation is completely determined by opcode
30Kevin McDonnell Stony Brook University – CSE 220
I-Type Examples
Note the differing
order of registers in
assembly and machine
codes:
addi rt, rs, imm
lw rt, imm(rs)
sw rt, imm(rs)
31Kevin McDonnell Stony Brook University – CSE 220
J-Type Instructions
• 26-bit address operand (addr)
• Used for jump instructions (j)
• if-statements, loops, functions
32Kevin McDonnell Stony Brook University – CSE 220
Review: Instruction Formats
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33Kevin McDonnell Stony Brook University – CSE 220
Power of the Stored Program
• 32-bit instructions and data are stored in memory
• To run a new program:
• No rewiring required
• Simply store new program in memory
• Affords general purpose computing
• Program execution:
• Processor fetches (reads) instructions from memory in
sequence
• Processor performs the specified operation and fetches
the next instruction
34Kevin McDonnell Stony Brook University – CSE 220
Interpreting Machine Code
• Start with opcode: tells how to parse the rest
• If opcode all 0’s we have an R-type instruction
• Function bits (funct) indicate operation
• Otherwise, opcode tells operation
35Kevin McDonnell Stony Brook University – CSE 220
Interpreting Machine Code
36Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly Programming
• There’s a lot more to the MIPS instruction set still to cover,
but we (almost) know enough now to write some simple
programs that do computations
• Every statement is divided into fields:
• [Label:] operation [operands] [#comment]
• Parts in square brackets are optional
• A label is a sequence of alphanumeric characters,
underscores and dots. Cannot begin with a number. Ends
with a colon.
• After the assembler has assembled (processed) your
code, the label refers to the address of where the line of
MIPS code is stored in memory
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37Kevin McDonnell Stony Brook University – CSE 220
MIPS Memory Layout
Data Segment (static)
Dynamic Data
Text Segment
(program)
Reserved
(for OS functions)
Grows
this way
Memory addresses (byte addresses)
0x7FFFFFFC
0x1000FFFF
0x10000000
0x00000000
0x00400000
In MARS, static data starts at
0x10010000 and dynamic
data starts at 0x10040000.
Stack Segment
Grows
this way
38Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly Programming
• The main label indicates the start of a program
• Labels are also used to give names to locations in memory
where we want to store data (we will see this shortly)
• Assembly programs also include assembler directives,
which start with a dot and give commands to the
assembler, but are not assembly language instructions
• .text: beginning of the text segment
• .data : beginning of data segment
• .asciiz: declares an ASCII string terminated by NULL
• .ascii: an ASCII string, not terminated by NULL
• .word: allocates space for one or more 32-bit words
• .globl: the name that follows is visible outside the file
39Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly Programming
• Strings (.asciiz and.ascii) are enclosed in quotes
• They recognize escape sequences:
, , ,
, etc.
• Finally, we need some way of doing basic input and output
• The computer’s architecture does not handle those
responsibilities, but relies on the operating system
• A system call is a request made by the program for the OS
to perform some service, such as to read input, print
output, quit the program, etc.
• In our MIPS assembly programs we write syscall to
perform a system call
• We have to include a numerical code (loaded into $v0) to
indicate the service requested
40Kevin McDonnell Stony Brook University – CSE 220
MIPS System Calls
Service
System
Call Code
Arguments Result
print_int 1 $a0=integer
print_float 2 $f12=float
print_double 3 $f12=double
print_string 4 $a0=string
read_int 5 integer (in $v0)
read_float 6 float (in $f0)
read_double 7 double (in $f0)
read_string 8 $a0=buffer,
$a1=length
sbrk 9 $a0=amount
exit 10
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41Kevin McDonnell Stony Brook University – CSE 220
MIPS System Calls
• sbrk allocates memory in the heap (e.g., large chunks of
memory)
• These are the original MIPS system calls
• The SBU MARS simulator has a few custom ones you will
learn about later. These system calls are not available in the
“vanilla” version of MARS publicly available on the web.
42Kevin McDonnell Stony Brook University – CSE 220
Generating Constants
• 16-bit constants using addi:
• Java code: MIPS assembly code:
// int is a 32-bit # $s0 = a
// signed word addi $s0, $0, 0x4f3c
int a = 0x4f3c;
• 32-bit constants use load upper immediate (lui) and ori
(more on ori in a few minutes):
• Java code: MIPS assembly code:
int a = 0xFEDC8765; # $s0 = a
lui $s0, 0xFEDC
ori $s0, $s0, 0x8765
43Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly Pseudoinstructions
• MIPS implements the RISC concept
• Relatively few, simple instructions
• But there are some operations that assembly language
programmers need to do frequently that are not so natural
to write in “native” MIPS assembly instructions
• These instructions can instead be written as a single
pseudoinstruction
• Example: to load a 32-bit integer into a register requires
lui and ori
• Instead we can use the li (load immediate)
pseudoinstruction
• Example: li $v0, 4 # loads 4 into $v0
44Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly Pseudoinstructions
• Another useful pseudoinstruction is la (load address)
• Example: assume that str is a label (i.e., a memory
address)
• la $a0, str # loads addr of str into $a0
• The move pseudoinstruction copies the contents of one
register to another
• move $1, $2 # equiv to add $1, $2, $0
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45Kevin McDonnell Stony Brook University – CSE 220
MIPS Program: Hello World
• No introduction to a new programming language would be
complete without the obligatory “hello world” program
• Let’s see how this is done in MIPS
46Kevin McDonnell Stony Brook University – CSE 220
Multiplication and Division
• Special registers: lo, hi
• 32-bit × 32-bit multiplication produces a 64-bit result
• mult $s0, $s1
• Result in {hi, lo}
• 32-bit division produces a 32-bit quotient and a 32-bit
remainder
• div $s0, $s1
• Quotient in lo
• Remainder in hi
• Instructions to move values from lo/hi special registers
• mflo $s2
• mfhi $s3
47Kevin McDonnell Stony Brook University – CSE 220
MIPS Assembly Pseudoinstructions
• Another useful pseudoinstruction that will help us write up
a program:
• mul d, s, t # d = s * t
• mul d, s, t is equivalent to:
mult s, t
mflo d
• Similar pseudoinstruction for div
48Kevin McDonnell Stony Brook University – CSE 220
MIPS Program: Compute ��� + �� + �
• For the first version of this program we will hard-code the
values for the three coefficients and x
• Major steps of our program:
1. Load values of A, B, C and x from memory into registers
2. Compute ��� + �� + �
• Requires 5 total arithmetical operations
3. Print the result with an appropriate message
• Requires several system calls
• I am going to comment nearly every single line of MIPS
assembly code I write. You should do the same on your
homework!
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49Kevin McDonnell Stony Brook University – CSE 220
MIPS Program: Compute ��� + �� + �
• For version 2 of the program we will add prompts to ask
the user to enter the values for x, A, B and C
• Each prompt requires two system calls:
• One to print the prompt message on the screen (if one is
desired/required)
• Another to read the input
50Kevin McDonnell Stony Brook University – CSE 220
Logical Instructions
• and, or, xor, nor
• and: useful for masking bits
• Masking out (excluding) all but the least significant byte
of a value: 0xF234012E AND 0x000000FF = 0x0000002E
• Why? Let’s see:
0xF234012E AND 0x000000FF
1111 0010 0011 0100 0000 0001 0010 1110
0000 0000 0000 0000 0000 0000 1111 1111
0000 0000 0000 0000 0000 0000 0010 1110
0 0 0 0 0 0 2 E
51Kevin McDonnell Stony Brook University – CSE 220
Logical Instructions
• or: useful for combining bit fields
• Combine 0xF2340000 with 0x000012BC:
• 0xF2340000 OR 0x000012BC = 0xF23412BC
• Written as bits:
0xF2340000 OR 0x000012BC
1111 0010 0011 0100 0000 0000 0000 0000
0000 0000 0000 0000 0001 0010 1011 1100
1111 0010 0011 0100 0001 0010 1011 1100
F 2 3 4 1 2 B C
52Kevin McDonnell Stony Brook University – CSE 220
Logical Instructions
• nor: useful for inverting bits
• A NOR $0 = NOT A
• andi, ori, xori
• 16-bit immediate is zero-extended (not sign-extended)
• nori not needed (can use ori and nor)
• Examples in a moment…
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53Kevin McDonnell Stony Brook University – CSE 220
Logical Instructions Examples
54Kevin McDonnell Stony Brook University – CSE 220
Logical Instructions Examples
55Kevin McDonnell Stony Brook University – CSE 220
Shift Instructions
• Allow you to shift the value in a register left or right by up to
31 bits
• sll: shift left logical
• Example: sll $t0, $t1, 5 # $t0 = $t1 << 5 • Shifts bits to the left, filling least significant bits with zeroes • srl: shift right logical • Example: srl $t0, $t1, 5 # $t0 = $t1 >>> 5
• Shifts zeroes into most significant bits
• sra: shift right arithmetic
• Example: sra $t0, $t1, 5 # $t0 = $t1 >> 5
• Shifts sign bit into most significant bits
56Kevin McDonnell Stony Brook University – CSE 220
Shift Instructions Examples
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57Kevin McDonnell Stony Brook University – CSE 220
Variable-Shift Instructions
• These R-type instructions shift bits by number in a register
• sllv: shift left logical variable
• sllv rd, rt, rs (note: rs and rt reversed)
• rt has value to be shifted
• 5 least significant bits of rs give amount to shift (0-31)
• Example: sllv $t0, $t1, $t2 # $t0 = $t1 << $t2 • srlv: shift right logical variable • Example: srlv $t0, $t1, $t2 # $t0 = $t1 >>> $t2
• srav: shift right arithmetic variable
• Example: srav $t0, $t1, $t2 # $t0 = $t1 >> $t2
• shamt field is ignored
58Kevin McDonnell Stony Brook University – CSE 220
Variable-Shift Instructions Examples
59Kevin McDonnell Stony Brook University – CSE 220
Rotate or Circular Shift
• Bits are not lost when we rotate them (i.e., do a “circular
shift”)
• They wrap around and enter the register from the other
end
• These are pseudo-instructions:
• rol: rotate left
• ror: rotate right
• Example: rol $t2, $t2, 4
• Rotate left bits of $t2 by 4 positions:
• 1101 0010 0011 0100 0101 0110 0111 1000
• 0010 0011 0100 0101 0110 0111 1000 1101
60Kevin McDonnell Stony Brook University – CSE 220
Applications of Bitwise Operators
• The bitwise operations are useful in situations where we
have a set of Yes/No properties and using many Boolean
variables would waste memory
• Example: file access flags in Unix/Linux
• Network protocols: packets have very specific formats,
which may include many bits that need to be extracted to
determine how to process a packet
• Compression algorithms sometime work on a bit-by-bit
basis
• Implementing a mathematical set of values: item � is
present in the set if bit i is 1; not present if the bit is 0
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61Kevin McDonnell Stony Brook University – CSE 220
Bitwise Operator Examples
• Suppose we want to isolate byte 0 (rightmost 8 bits) of a
word in $t0. Simply use this:
andi $t0, $t0, 0xFF
0001 0010 0011 0100 0101 0110 0111 1000
0000 0000 0000 0000 0000 0000 0111 1000
62Kevin McDonnell Stony Brook University – CSE 220
Bitwise Operator Examples
• Suppose we want to isolate byte 1 (bits 8 to 15) of a word
in $t0. (Bits are numbered right-to-left.) We can use:
andi $t0,$t0,0xFF00
but then we still need to do a logical shift to the right by 8
bits. Why? To move the byte we have isolated into byte 0
and also to set bytes 1, 2 and 3 to all zeroes.
• Could use instead:
• sll $t0,$t0,16 *
• srl $t0,$t0,24 **
• 0001 0010 0011 0100 0101 0110 0111 1000
• 0101 0110 0111 1000 0000 0000 0000 0000 *
• 0000 0000 0000 0000 0000 0000 0101 0110 **
63Kevin McDonnell Stony Brook University – CSE 220
Bitwise Operator Examples
• In binary, multiplying by 2 is same as shifting left by 1 bit:
• 11� × 10� = 110�
• Multiplying by 4 is same as shifting left by 2 bits:
• 11� × 100� = 1100�
• Multiplying by 2� is same as shifting left by n bits
• Since shifting may be faster than multiplication, a good
compiler (e.g., C or Java) will recognize a multiplication by
a power of 2 and compile it as a shift:
• a = a∗∗∗∗8; would compile as sll $s0,$s0,3
• Likewise, shift right to do integer division by powers of 2.
• Remember to use sra, not srl. Why?
64Kevin McDonnell Stony Brook University – CSE 220
MIPS Programming Tips
• Initialize all your variables as needed (e.g., use li)
• The MARS simulator fills the memory with zeroes, but
this is merely a convenience and luxury
• When we test your homework programs, we may fill the
registers and main memory with garbage to make sure
you initialize registers with values!
• Use the MARS debugger to fix problems with your code
• The Registers view (on right) is especially useful
• Use $s0-$s7 for local variables, and $t0-$t9 for
temporary values, such as intermediate results
• We will see just how important this distinction is when
we study functions!
